This document introduces hyperpolyadic structures, which are n-ary analogs of binary division algebras like the reals, complexes, quaternions, and octonions. It proposes two constructions:
1) A matrix polyadization procedure that increases the dimension of a binary algebra to obtain a corresponding n-ary algebra by using cyclic shift block matrices.
2) An "imaginary tower" construction on subsets of binary division algebras that gives nonderived ternary division algebras of half the original dimension, called "half-quaternions" and "half-octonions."
We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element we define a new norm which is polyadically multiplicative, and the corresponding map is a n-ary homomorphism. We define a polyadic analog of the Cayley-Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, that is proportional to the intermediate arities. Second, a new polyadic product of vectors in any vector space is defined. Endowed with this product the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process ("imaginary tower"), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call "half-quaternions" and "half-octonions". The latter are not subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced "half-quaternion" norm we obtain the ternary analog of the sum of two squares identity. We prove that the unitless ternary division algebra of imaginary "half-octonions" we have introduced is ternary alternative.
https://arxiv.org/abs/2312.01366
https://www.amazon.com/s?k=duplij
This article continues the study of concrete algebra-like structures in our polyadic approach, when the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions. In this way, the associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, the dimension of the algebra can be not arbitrary, but "quantized"; the polyadic convolution product and bialgebra can be defined, when algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to the quantum group theory, we introduce the polyadic version of the braidings, almost co-commutativity, quasitriangularity and the equations for R-matrix (that can be treated as polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented in the block-diagonal matrix form (Wedderburn decomposition), a general form of polyadic structures is given by block-shift matrices. We combine these forms in a special way to get a general shape of semisimple nonderived polyadic structures. We then introduce the polyadization concept (a "polyadic constructor") according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The "deformation" by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given.
Abstract: Algebraic structures in which the property of commutativity is substituted by the me- diality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or e-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with (n-1) associators of the arity (2n-1) satisfying a (n^2-1)-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.
https://arxiv.org/abs/2006.07865
A general mechanism for "breaking" commutativity in algebras is proposed: if the underlying set is taken to be not a crisp set, but rather an obscure/fuzzy set, the membership function, reflecting the degree of truth that an element belongs to the set, can be incorporated into the commutation relations. The special "deformations" of commutativity and e-commutativity are introduced in such a way that equal degrees of truth result in the "nondeformed" case. We also sketch how to "deform" e-Lie algebras and Weyl algebras. Further, the above constructions are extended to n-ary algebras for which the projective representations and e-commutativity are studied.
https://arxiv.org/abs/2201.08479
We propose a generalization of the external direct product concept to polyadic algebraic structures which introduces novel properties in two ways: the arity of the product can differ from that of the constituents, and the elements from different multipliers can be "entangled" such that the product is no longer componentwise. The main property which we want to preserve is associativity, which is gained by using the associativity quiver technique provided earlier. For polyadic semigroups and groups we introduce two external products: 1) the iterated direct product which is componentwise, but can have arity different from the multipliers; 2) the hetero product (power) which is noncomponentwise and constructed by analogy with the heteromorphism concept introduced earlier. It is shown in which cases the product of polyadic groups can itself be a polyadic group. In the same way the external product of polyadic rings and fields is generalized. The most exotic case is the external product of polyadic fields, which can be a polyadic field (as opposed to the binary fields), when all multipliers are zeroless fields, which can lead to a new category of polyadic fields. Many illustrative concrete examples are presented.
A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.
Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. The relations between operations appearing from the structure definitions lead to restrictions, which determine their arity shape and lead to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. As one application, elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, operator norms, isometries and projections, as well as polyadic C*-algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators are introduced. Another application is connected with number theory, and it is shown that the congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced, see Definition 6.16. Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat’s last theorem are formulated. For the nonderivedpolyadic ring operations (polyadic numbers) neither of these holds, and counterexamples are given. A procedure for obtaining new solutions to the equal sums of like powers equation over polyadic rings by applying Frolov’s theorem for the Tarry-Escott problem is presented.
We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element we define a new norm which is polyadically multiplicative, and the corresponding map is a n-ary homomorphism. We define a polyadic analog of the Cayley-Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, that is proportional to the intermediate arities. Second, a new polyadic product of vectors in any vector space is defined. Endowed with this product the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process ("imaginary tower"), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call "half-quaternions" and "half-octonions". The latter are not subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced "half-quaternion" norm we obtain the ternary analog of the sum of two squares identity. We prove that the unitless ternary division algebra of imaginary "half-octonions" we have introduced is ternary alternative.
https://arxiv.org/abs/2312.01366
https://www.amazon.com/s?k=duplij
This article continues the study of concrete algebra-like structures in our polyadic approach, when the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions. In this way, the associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, the dimension of the algebra can be not arbitrary, but "quantized"; the polyadic convolution product and bialgebra can be defined, when algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to the quantum group theory, we introduce the polyadic version of the braidings, almost co-commutativity, quasitriangularity and the equations for R-matrix (that can be treated as polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented in the block-diagonal matrix form (Wedderburn decomposition), a general form of polyadic structures is given by block-shift matrices. We combine these forms in a special way to get a general shape of semisimple nonderived polyadic structures. We then introduce the polyadization concept (a "polyadic constructor") according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The "deformation" by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given.
Abstract: Algebraic structures in which the property of commutativity is substituted by the me- diality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or e-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with (n-1) associators of the arity (2n-1) satisfying a (n^2-1)-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.
https://arxiv.org/abs/2006.07865
A general mechanism for "breaking" commutativity in algebras is proposed: if the underlying set is taken to be not a crisp set, but rather an obscure/fuzzy set, the membership function, reflecting the degree of truth that an element belongs to the set, can be incorporated into the commutation relations. The special "deformations" of commutativity and e-commutativity are introduced in such a way that equal degrees of truth result in the "nondeformed" case. We also sketch how to "deform" e-Lie algebras and Weyl algebras. Further, the above constructions are extended to n-ary algebras for which the projective representations and e-commutativity are studied.
https://arxiv.org/abs/2201.08479
We propose a generalization of the external direct product concept to polyadic algebraic structures which introduces novel properties in two ways: the arity of the product can differ from that of the constituents, and the elements from different multipliers can be "entangled" such that the product is no longer componentwise. The main property which we want to preserve is associativity, which is gained by using the associativity quiver technique provided earlier. For polyadic semigroups and groups we introduce two external products: 1) the iterated direct product which is componentwise, but can have arity different from the multipliers; 2) the hetero product (power) which is noncomponentwise and constructed by analogy with the heteromorphism concept introduced earlier. It is shown in which cases the product of polyadic groups can itself be a polyadic group. In the same way the external product of polyadic rings and fields is generalized. The most exotic case is the external product of polyadic fields, which can be a polyadic field (as opposed to the binary fields), when all multipliers are zeroless fields, which can lead to a new category of polyadic fields. Many illustrative concrete examples are presented.
A new kind of quantum gates, higher braiding gates, as matrix solutions of the polyadic braid equations (different from the generalized Yang–Baxter equations) is introduced. Such gates lead to another special multiqubit entanglement that can speed up key distribution and accelerate algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates, which can be related with qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, star and circle ones, such that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the introduced classes is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by the higher braid operators are given. Finally, we show, that for each multiqubit state, there exist higher braiding gates that are not entangling, and the concrete conditions to be non-entangling are given for the obtained binary and ternary gates.
Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. The relations between operations appearing from the structure definitions lead to restrictions, which determine their arity shape and lead to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. As one application, elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, operator norms, isometries and projections, as well as polyadic C*-algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators are introduced. Another application is connected with number theory, and it is shown that the congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced, see Definition 6.16. Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat’s last theorem are formulated. For the nonderivedpolyadic ring operations (polyadic numbers) neither of these holds, and counterexamples are given. A procedure for obtaining new solutions to the equal sums of like powers equation over polyadic rings by applying Frolov’s theorem for the Tarry-Escott problem is presented.
We propose a generalization of the external direct product concept to polyadic algebraic structures which introduces novel properties in two ways: the arity of the product can differ from that of the constituents, and the elements from different multipliers can be “entangled” such that the product is no longer componentwise. The main property which we want to preserve is associativity, which is gained by using the associativity quiver technique, which was provided previously. For polyadic semigroups and groups we introduce two external products: (1) the iterated direct product, which is componentwise but can have an arity that is different from the multipliers and (2) the hetero product (power), which is noncomponentwise and constructed by analogy with the heteromorphism concept introduced earlier. We show in which cases the product of polyadic groups can itself be a polyadic group. In the same way, the external product of polyadic rings and fields is generalized. The most exotic case is the external product of polyadic fields, which can be a polyadic field (as opposed to the binary fields), in which all multipliers are zeroless fields. Many illustrative concrete examples are presented. Thу proposed construction can lead to a new category of polyadic fields.
This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but "quantized". The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. The relations between operations appearing from the structure definitions lead to restrictions, which determine their arity shape and leads to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. As one application, elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, oper- ator norms, isometries and projections, as well as polyadic C* -algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators are introduced. Another application is connected with num- ber theory, and it is shown that the congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced, see Definition 6.16. Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat’s last theorem are formulated. For the derived polyadic ring operations neither of these holds, and counterexamples are given. A procedure for obtaining new solutions to the equal sums of like powers equation over polyadic rings by applying Frolov’s theorem for the Tarry-Escott problem is presented.
https://arxiv.org/abs/1703.10132
https://arxiv.org/abs/1905.01927
A quantum mechanical model that realizes the Z2xZ2-graded generalization of the one-dimensional supertranslation algebra is proposed. This model shares some features with the well-known Witten model and is related to parasupersymmetric quantum mechanics, though the model is not directly equivalent to either of these. The purpose of this paper is to show that novel "higher gradings" are possible in the context of non-relativistic quantum mechanics.
A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented as block diagonal matrices (resulting in the Wedderburn decomposition), general forms of polyadic structures are given by block-shift matrices. We combine these forms to get a general shape of semisimple nonderived polyadic structures (“double” decomposition of two kinds). We then introduce the polyadization concept (a “polyadic constructor”), according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The “deformation” by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given.
Quantum gravitational corrections to particle creation by black holesSérgio Sacani
We calculate quantum gravitational corrections to the amplitude for the emission of a Hawking particle
by a black hole. We show explicitly how the amplitudes depend on quantum corrections to the exterior
metric (quantum hair). This reveals the mechanism by which information escapes the black hole. The
quantum state of the black hole is reflected in the quantum state of the exterior metric, which in turn
influences the emission of Hawking quanta.
The document presents research on using a binary reproducing kernel Hilbert space (RKHS) approach to solve a Wick-type stochastic Korteweg-de Vries (KdV) equation with variable coefficients. It introduces the stochastic KdV equation model and discusses previous work analyzing it. The research aims to formulate white noise functional solutions for the stochastic KdV equations by applying Hermite transform, white noise theory, and binary RKHS. It explores representing the exact solution in a reproducing kernel space and investigating uniform convergence of approximate solutions.
Higher braiding gates, a new kind of quantum gate, are introduced. These are matrix solutions of the polyadic braid equations (which differ from the generalized Yang-Baxter equations). Such gates support a special kind of multi-qubit entanglement which can speed up key distribution and accelerate the execution of algorithms. Ternary braiding gates acting on three qubit states are studied in detail. We also consider exotic non-invertible gates which can be related to qubit loss, and define partial identities (which can be orthogonal), partial unitarity, and partially bounded operators (which can be non-invertible). We define two classes of matrices, the star and circle types, and find that the magic matrices (connected with the Cartan decomposition) belong to the star class. The general algebraic structure of the classes introduced here is described in terms of semigroups, ternary and 5-ary groups and modules. The higher braid group and its representation by higher braid operators are given. Finally, we show that for each multi-qubit state there exist higher braiding gates which are not entangling, and the concrete conditions to be non-entangling are given for the binary and ternary gates discussed.
We generalize the Grothendieck construction of the completion group for a monoid (being the starting point of the algebraic $K$-theory) to the polyadic case, when an initial semigroup is $m$-ary and the corresponding final class group $K_{0}$ can be $n$-ary. As opposed to the binary case: 1) there can be different polyadic direct products which can be built from one polyadic semigroup; 2) the final arity $n$ of the class groups can be different from the arity $m$ of initial semigroup; 3) commutative initial $m$-ary semigroups can lead to noncommutative class $n$-ary groups; 4) the identity is not necessary for initial $m$-ary semigroup to obtain the class $n$-ary group, which in its turn can contain no identity at all. The presented numerical examples show that the properties of the polyadic completion groups are considerably nontrivial and have more complicated structure than in the binary case.
In book: S. Duplij, "Polyadic Algebraic Structures", 2022, IOP Publishing (Bristol), Section 1.5. See https://iopscience.iop.org/book/978-0-7503-2648-3
https://arxiv.org/abs/2206.14840
This document summarizes Michael Kreisel's dissertation on the connection between Gabor frames for quasicrystals, the topology of the hull of a quasicrystal, and K-theory of an associated twisted groupoid algebra. The author constructs a finitely generated projective module over this algebra, where any multiwindow Gabor frame for the quasicrystal can be used to construct a projection representing this module in K-theory. As an application, results are obtained on the twisted version of Bellissard's gap labeling conjecture for quasicrystals.
This document summarizes a research paper on spin modular categories. The paper studies algebraic structures on modular categories that allow refinements of quantum 3-manifold invariants involving cohomology classes or generalized spin and complex spin structures. A key role is played by invertible objects under tensor product. The paper defines H-refinable and H-spin modular categories for a subgroup H of invertible objects. It shows such categories provide topological invariants of pairs (M,σ) where M is a 3-manifold and σ is a generalized spin structure. The paper establishes splitting formulas for these refined invariants, generalizing known decompositions of quantum invariants.
In this paper a new form of the Hosszu-Gluskin theorem is presented in terms of polyadic powers and using the language of diagrams. It is shown that the Hosszu-Gluskin chain formula is not unique and can be generalized ("deformed") using a parameter q which takes special integer values. A version of the "q-deformed" analog of the Hosszu-Gluskin theorem in the form of an invariance is formulated, and some examples are considered. The "q-deformed" homomorphism theorem is also given.
This document provides lecture notes on analytic geometry. It begins with an introduction discussing the goals of building an algebraic geometry framework for analytic situations by replacing topological abelian groups with condensed abelian groups. Condensed sets are defined as sheaves on the pro-étale site of the point, and behave like generalized topological spaces. The notes establish that quasiseparated condensed sets correspond to ind-compact Hausdorff spaces. This provides the needed abelian category structure to build an analytic geometry in parallel to algebraic geometry over schemes.
Density theorems for Euclidean point configurationsVjekoslavKovac1
1. The document discusses density theorems for point configurations in Euclidean space. Density theorems study when a measurable set A contained in Euclidean space can be considered "large".
2. One classical result is that for any measurable set A contained in R2 with positive upper Banach density, there exist points in A whose distance is any sufficiently large real number. This has been generalized to higher dimensions and other point configurations.
3. Open questions remain about determining all point configurations P for which one can show that a sufficiently large measurable set A contained in high dimensional Euclidean space must contain a scaled copy of P.
Reciprocity Law For Flat Conformal Metrics With Conical SingularitiesLukasz Obara
The document is a thesis that establishes an analogue of Weil's reciprocity law for flat conformal metrics with conical singularities on Riemann surfaces. It introduces the necessary mathematical background, including definitions of isothermal coordinates and metrics with conical singularities. The main result proves a relationship between three flat conformally equivalent metrics, each with different conical singularities.
This document summarizes the use of the Ritz method to approximate the critical frequencies of a tapered hollow beam. It begins by introducing the governing equations and describing the uniform beam solution. It then outlines the Ritz method, which uses the uniform beam eigenfunctions as a basis to approximate the tapered beam solution. The method is applied numerically to predict the first three critical frequencies of the tapered beam, which are found to match well with finite element analysis results. The Ritz method is concluded to be an effective way to approximate critical frequencies for more complex beam geometries.
Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a ’heteromorphism’ which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed. Ternary algebras and Hopf algebras are defined, and their properties are studied. At the end some ternary generalizations of quantum groups and the Yang-Baxter equation are presented.
Berezin-Toeplitz Quantization On Coadjoint orbitsHassan Jolany
This document discusses Berezin-Toeplitz quantization on coadjoint orbits. It begins by relating the inner product used in quantization to Kirillov's character formula through the metaplectic correction. It then establishes that coadjoint orbits are Kähler manifolds, allowing for the application of Berezin-Toeplitz quantization. Specifically, quantizable coadjoint orbits can be embedded in projective spaces, and Berezin-Toeplitz quantization maps smooth functions on the orbit to families of endomorphisms of the spaces of global holomorphic sections.
This document presents a closed-form solution for a class of discrete-time algebraic Riccati equations (DTAREs) under certain assumptions. It begins with background on Riccati equations and their importance in control theory. It then provides the assumptions considered, including that the A matrix eigenvalues are distinct. The main result is a closed-form solution for the DTARE when R=1. Extensions discussed include the solution's behavior as Q approaches zero and for repeated eigenvalues. Comparisons with numerical solutions verify the closed-form solution's accuracy.
This is a journal concise version (without diagrams and figures) of the preprint arXiv:1308.4060.
Abstract: Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed. Ternary algebras and Hopf algebras are defined, and their properties are studied. At the end some ternary generalizations of quantum groups and the Yang-Baxter equation are presented.
We introduce a polyadic analog of supersymmetry by considering the polyadization procedure (proposed by the author) applied to the toy model of one-dimensional supersymmetric quantum mechanics. The supercharges are generalized to polyadic ones using the n-ary sigma matrices defined in earlier work. In this way, polyadic analogs of supercharges and Hamiltonians take the cyclic shift block matrix form, and they can describe multidegenerated quantum states in a way that is different from the N-extended and multigraded SQM. While constructing the corresponding supersymmetry as an n-ary Lie superalgebra (n is the arity of the initial associative multiplication), we have found new brackets with a reduced arity of 2<=m<n and a related series of m-ary superalgebras (which is impossible for binary superalgebras). In the case of even reduced arity m we obtain a tower of higher order (as differential operators) even Hamiltonians, while for m odd we get a tower of higher order odd supercharges, and the corresponding algebra consists of the odd sector only.
https://arxiv.org/abs/2406.02188
We generalize σ-matrices to higher arities using the polyadization procedure proposed by the author. We build the nonderived n-ary version of SU(2) using cyclic shift block matrices. We define a new function, the polyadic trace, which has an additivity property analogous to the ordinary trace for block diagonal matrices and which can be used to build the corresponding invariants. The elementary Σ-matrices introduced here play a role similar to ordinary matrix units, and their sums are full Σ-matrices which can be treated as a polyadic analog of σ-matrices. The presentation of n-ary SU(2) in terms of full Σ-matrices is done using the Hadamard product. We then generalize the Pauli group in two ways: for the binary case we introduce the extended phase shifted σ-matrices with multipliers in cyclic groups of order 4q (q>4), and for the polyadic case we construct the correspondent finite n-ary semigroup of phase-shifted elementary Σ-matrices of order 4q(n-1)+1, and the finite n-ary group of phase-shifted full Σ-matrices of order 4q. Finally, we introduce the finite n-ary group of heterogeneous full Σ^het-matrices of order (4q(n-1))^4. Some examples of the lowest arities are presented.
https://arxiv.org/abs/2403.19361. *) https://www.researchgate.net/publication/360882654_Polyadic_Algebraic_Structures, https://iopscience.iop.org/book/978-0-7503-2648-3.
CONTENTS 1. INTRODUCTION 2. PRELIMINARIES 3. POLYADIC SU p2q 4. POLYADIC ANALOG OF SIGMA MATRICES 4.1. Elementary Σ-matrices 4.2. Full Σ-matrices 5. TERNARY SUp2q AND Σ-MATRICES 6. n-ARY SEMIGROUPS AND GROUPS OF Σ-MATRICES 6.1. The Pauli group 6.2. Groups of phase-shifted sigma matrices 6.3. The n-ary semigroup of elementary Σ-matrices 6.4. The n-ary group of full Σ-matrices 7. HETEROGENEOUS FULL Σ-MATRICES REFERENCES
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We propose a generalization of the external direct product concept to polyadic algebraic structures which introduces novel properties in two ways: the arity of the product can differ from that of the constituents, and the elements from different multipliers can be “entangled” such that the product is no longer componentwise. The main property which we want to preserve is associativity, which is gained by using the associativity quiver technique, which was provided previously. For polyadic semigroups and groups we introduce two external products: (1) the iterated direct product, which is componentwise but can have an arity that is different from the multipliers and (2) the hetero product (power), which is noncomponentwise and constructed by analogy with the heteromorphism concept introduced earlier. We show in which cases the product of polyadic groups can itself be a polyadic group. In the same way, the external product of polyadic rings and fields is generalized. The most exotic case is the external product of polyadic fields, which can be a polyadic field (as opposed to the binary fields), in which all multipliers are zeroless fields. Many illustrative concrete examples are presented. Thу proposed construction can lead to a new category of polyadic fields.
This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions ("arity freedom principle"). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but "quantized". The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). Finally, we propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.
Concrete two-set (module-like and algebra-like) algebraic structures are investigated from the viewpoint that the initial arities of all operations are arbitrary. The relations between operations appearing from the structure definitions lead to restrictions, which determine their arity shape and leads to the partial arity freedom principle. In this manner, polyadic vector spaces and algebras, dual vector spaces, direct sums, tensor products and inner pairing spaces are reconsidered. As one application, elements of polyadic operator theory are outlined: multistars and polyadic analogs of adjoints, oper- ator norms, isometries and projections, as well as polyadic C* -algebras, Toeplitz algebras and Cuntz algebras represented by polyadic operators are introduced. Another application is connected with num- ber theory, and it is shown that the congruence classes are polyadic rings of a special kind. Polyadic numbers are introduced, see Definition 6.16. Diophantine equations over these polyadic rings are then considered. Polyadic analogs of the Lander-Parkin-Selfridge conjecture and Fermat’s last theorem are formulated. For the derived polyadic ring operations neither of these holds, and counterexamples are given. A procedure for obtaining new solutions to the equal sums of like powers equation over polyadic rings by applying Frolov’s theorem for the Tarry-Escott problem is presented.
https://arxiv.org/abs/1703.10132
https://arxiv.org/abs/1905.01927
A quantum mechanical model that realizes the Z2xZ2-graded generalization of the one-dimensional supertranslation algebra is proposed. This model shares some features with the well-known Witten model and is related to parasupersymmetric quantum mechanics, though the model is not directly equivalent to either of these. The purpose of this paper is to show that novel "higher gradings" are possible in the context of non-relativistic quantum mechanics.
A generalization of the semisimplicity concept for polyadic algebraic structures is proposed. If semisimple structures can be presented as block diagonal matrices (resulting in the Wedderburn decomposition), general forms of polyadic structures are given by block-shift matrices. We combine these forms to get a general shape of semisimple nonderived polyadic structures (“double” decomposition of two kinds). We then introduce the polyadization concept (a “polyadic constructor”), according to which one can construct a nonderived polyadic algebraic structure of any arity from a given binary structure. The polyadization of supersymmetric structures is also discussed. The “deformation” by shifts of operations on the direct power of binary structures is defined and used to obtain a nonderived polyadic multiplication. Illustrative concrete examples for the new constructions are given.
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We generalize the Grothendieck construction of the completion group for a monoid (being the starting point of the algebraic $K$-theory) to the polyadic case, when an initial semigroup is $m$-ary and the corresponding final class group $K_{0}$ can be $n$-ary. As opposed to the binary case: 1) there can be different polyadic direct products which can be built from one polyadic semigroup; 2) the final arity $n$ of the class groups can be different from the arity $m$ of initial semigroup; 3) commutative initial $m$-ary semigroups can lead to noncommutative class $n$-ary groups; 4) the identity is not necessary for initial $m$-ary semigroup to obtain the class $n$-ary group, which in its turn can contain no identity at all. The presented numerical examples show that the properties of the polyadic completion groups are considerably nontrivial and have more complicated structure than in the binary case.
In book: S. Duplij, "Polyadic Algebraic Structures", 2022, IOP Publishing (Bristol), Section 1.5. See https://iopscience.iop.org/book/978-0-7503-2648-3
https://arxiv.org/abs/2206.14840
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This is a journal concise version (without diagrams and figures) of the preprint arXiv:1308.4060.
Abstract: Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed. Ternary algebras and Hopf algebras are defined, and their properties are studied. At the end some ternary generalizations of quantum groups and the Yang-Baxter equation are presented.
Similar to "Hyperpolyadic structures" by S. Duplij, arxiv:2312.01366 (20)
We introduce a polyadic analog of supersymmetry by considering the polyadization procedure (proposed by the author) applied to the toy model of one-dimensional supersymmetric quantum mechanics. The supercharges are generalized to polyadic ones using the n-ary sigma matrices defined in earlier work. In this way, polyadic analogs of supercharges and Hamiltonians take the cyclic shift block matrix form, and they can describe multidegenerated quantum states in a way that is different from the N-extended and multigraded SQM. While constructing the corresponding supersymmetry as an n-ary Lie superalgebra (n is the arity of the initial associative multiplication), we have found new brackets with a reduced arity of 2<=m<n and a related series of m-ary superalgebras (which is impossible for binary superalgebras). In the case of even reduced arity m we obtain a tower of higher order (as differential operators) even Hamiltonians, while for m odd we get a tower of higher order odd supercharges, and the corresponding algebra consists of the odd sector only.
https://arxiv.org/abs/2406.02188
We generalize σ-matrices to higher arities using the polyadization procedure proposed by the author. We build the nonderived n-ary version of SU(2) using cyclic shift block matrices. We define a new function, the polyadic trace, which has an additivity property analogous to the ordinary trace for block diagonal matrices and which can be used to build the corresponding invariants. The elementary Σ-matrices introduced here play a role similar to ordinary matrix units, and their sums are full Σ-matrices which can be treated as a polyadic analog of σ-matrices. The presentation of n-ary SU(2) in terms of full Σ-matrices is done using the Hadamard product. We then generalize the Pauli group in two ways: for the binary case we introduce the extended phase shifted σ-matrices with multipliers in cyclic groups of order 4q (q>4), and for the polyadic case we construct the correspondent finite n-ary semigroup of phase-shifted elementary Σ-matrices of order 4q(n-1)+1, and the finite n-ary group of phase-shifted full Σ-matrices of order 4q. Finally, we introduce the finite n-ary group of heterogeneous full Σ^het-matrices of order (4q(n-1))^4. Some examples of the lowest arities are presented.
https://arxiv.org/abs/2403.19361. *) https://www.researchgate.net/publication/360882654_Polyadic_Algebraic_Structures, https://iopscience.iop.org/book/978-0-7503-2648-3.
CONTENTS 1. INTRODUCTION 2. PRELIMINARIES 3. POLYADIC SU p2q 4. POLYADIC ANALOG OF SIGMA MATRICES 4.1. Elementary Σ-matrices 4.2. Full Σ-matrices 5. TERNARY SUp2q AND Σ-MATRICES 6. n-ARY SEMIGROUPS AND GROUPS OF Σ-MATRICES 6.1. The Pauli group 6.2. Groups of phase-shifted sigma matrices 6.3. The n-ary semigroup of elementary Σ-matrices 6.4. The n-ary group of full Σ-matrices 7. HETEROGENEOUS FULL Σ-MATRICES REFERENCES
178 pages, 6 Chapters. DOI: 10.1088/978-0-7503-5281-9. This book presents new and prospective approaches to quantum computing. It introduces the many possibilities to further develop the mathematical methods of quantum computation and its applications to future functioning and operational quantum computers. In this book, various extensions of the qubit concept, starting from obscure qubits, superqubits and other fundamental generalizations, are considered. New gates, known as higher braiding gates, are introduced. These new gates are implemented as an additional stage of computation for topological quantum computations and unconventional computing when computational complexity is affected by its environment. Other generalizations are considered and explained in a widely accessible and easy-to-understand way. Presented in a book for the first time, these new mathematical methods will increase the efficiency and speed of quantum computing.Part of IOP Series in Coherent Sources, Quantum Fundamentals, and Applications. Key features • Provides new mathematical methods for quantum computing. • Presents material in a widely accessible way. • Contains methods for unconventional computing where there is computational complexity. • Provides methods to increase speed and efficiency. For the light paperback version use MyPrint service here: https://iopscience.iop.org/book/mono/978-0-7503-5281-9, also PDF, ePub and Kindle. For the libraries and direct ordering from IOP: https://store.ioppublishing.org/page/detail/Innovative-Quantum-Computing/?K=9780750352796. Amazon ordering: https://www.amazon.de/gp/product/0750352795
Quantum field theory has achieved some extraordinary successes over the past sixty years; however, it retains a set of challenging problems. It is not yet able to describe gravity in a mathematically consistent manner. CP violation remains unexplained. Grand unified theories have been eliminated by experiment, and a viable unification model has yet to replace them. Even the highly successful quantum chromodynamics, despite significant computational achievements, struggles to provide theoretical insight into the low-energy regime of quark physics, where the nature and structure of hadrons are determined. The only proposal for resolving the fine-tuning problem, low-energy supersymmetry, has been eliminated by results from the LHC. Since mathematics is the true and proper language for quantitative physical models, we expect new mathematical constructions to provide insight into physical phenomena and fresh approaches for building physical theories.
https://www.mdpi.com/books/book/6455
We investigate finite field extensions of the unital 3-field, consisting of the unit element alone, and find considerable differences to classical field theory. Furthermore, the structure of their automorphism groups is clarified and the respective subfields are determined. In an attempt to better understand the structure of 3-fields that show up here we look at ways in which new unital 3-fields can be obtained from known ones in terms of product structures, one of them the Cartesian product which has no analogue for binary fields.
https://arxiv.org/abs/2212.08606
In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become (m, n)-rings. Second, we introduce a possible p-adic analog of the residue class modulo a p-adic integer. Then, we find the relations which determine, when the representatives form a (m, n)-ring. At the very short spacetime scales such rings could lead to new symmetries of modern particle models.
Книга «Поэфизика души» представляет собой полное, на момент издания 2022 г., собрание прозаических произведений автора. Как рассказы, так и миниатюры на полстраницы, пронизаны эмоциями и искренними чувствами на грани срыва, что заставляет возвращаться к ним вновь. Буквально каждое слово рисует уникальные картины нетривиальных внутренних миров автора, которые перетекают друг в друга поэтическими образами, воплощенными в прозе. Также включены юмористические путевые заметки о поездке в Китай.
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The book is devoted to the thorough study of polyadic (higher arity) algebraic structures, which has a long history, starting from 19th century. The main idea was to take a single set, closed under one binary operation, and to 'generalize' it by increasing the arity of the operation, called a polyadic operation. Until now, a general approach to polyadic concrete many-set algebraic structures was absent. We propose to investigate algebraic structures in the 'concrete way' and provide consequent 'polyadization' of each operation, starting from group-like structures and finishing with the Hopf algebra structures. Polyadic analogs of homomorphisms which change arity, heteromorphisms, are introduced and applied for constructing unusual representations, multiactions, matrix representations and polyadic analogs of direct product. We provide the polyadic generalization of the Yang–Baxter equation, find its constant solutions, and introduce polyadic tensor categories.
Suitable for university students of advanced level algebra courses and mathematical physics courses.
Key features
• Provides a general, unified approach
• Widens readers perspective of the possibilities to develop standard algebraic structures
• Provides the new kind of homomorphisms changing the arity, heteromorphisms, are introduced and applied for construction of new representations, multiactions and matrix representations
• Presents applications of 'polyadization' approach to concrete algebraic structures
This document proposes a new mechanism for "deforming" or breaking commutativity in algebras called "membership deformation". It involves taking the underlying set of an algebra to be an "obscure/fuzzy set" with elements having membership functions between 0 and 1 rather than a crisp set. The membership functions are incorporated into the commutation relations such that elements with equal membership functions commute, while others do not. This provides a continuous way to deform commutativity. The approach is then generalized to ε-commutative algebras and n-ary algebras. Projective representations of n-ary algebras are also studied in relation to this new type of deformation.
We generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions. Finally, we introduce the sandwich higher polyadic regularity with generalized idempotents.
This document discusses generalizing the concept of regularity for semigroups in two ways: higher regularity and higher arity (polyadic semigroups).
For binary semigroups, higher n-regularity is defined such that each element has multiple inverse elements rather than a single inverse. However, for binary semigroups this reduces to ordinary regularity. For polyadic semigroups, several definitions of regularity and higher regularity are introduced to account for the higher arity operations. Idempotents and identities are also generalized for polyadic semigroups. It is shown that the definitions of regularity for polyadic semigroups cannot be reduced in the same
In this note we generalize the regularity concept for semigroups in two ways simul- taneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions.
We propose a concept of quantum computing which incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), in a natural way by introducing new entities, obscure qudits (e.g. obscure qubits), which are characterized simultaneously by a quantum probability and by a membership function. To achieve this, a membership amplitude for quantum states is introduced alongside the quantum amplitude. The Born rule is used for the quantum probability only, while the membership function can be computed from the membership amplitudes according to a chosen model. Two different versions of this approach are given here: the \textquotedblleft product\textquotedblright\ obscure qubit, where the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the \textquotedblleft Kronecker\textquotedblright\ obscure qubit, where quantum and vagueness computations are to be performed independently (i.e. quantum computation alongside truth evaluation). The latter is called a double obscure-quantum computation. In this case, the measurement becomes mixed in the quantum and obscure amplitudes, while the density matrix is not idempotent. The obscure-quantum gates act not in the tensor product of spaces, but in the direct product of quantum Hilbert space and so called membership space which are of different natures and properties. The concept of double (obscure-quantum) entanglement is introduced, and vector and scalar concurrences are proposed, with some examples being given.
Online first: https://www.intechopen.com/online-first/obscure-qubits-and-membership-amplitudes
We first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.
Abstract: In this note we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.
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"Hyperpolyadic structures" by S. Duplij, arxiv:2312.01366
1. arXiv:2312.01366v2
[math.RA]
5
Dec
2023
HYPERPOLYADIC STRUCTURES
STEVEN DUPLIJ
Center for Information Technology (CIT), Universität Münster, Röntgenstrasse 7-13
D-48149 Münster, Deutschland
ABSTRACT. We introduce a new class of division algebras, hyperpolyadic algebras, which correspond to the
binary division algebras R, C, H, O without considering new elements. First, we use the proposed earlier
matrix polyadization procedure which increases the algebra dimension. The obtained algebras obey the
binary addition and nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For
each invertible element we define a new norm which is polyadically multiplicative and the corresponding map
is n-ary homomorphism. We define a polyadic analog of the Cayley-Dickson construction which corresponds
to the consequent embedding of monomial matrices from the polyadization procedure. Then we obtain
another series of n-ary algebras corresponding to the binary division algebras which have more dimension,
that is proportional to intermediate arities, and they are not isomorphic to those obtained by the previous
constructions. Second, we propose a new iterative process (we call it “imaginary tower”), which leads to
nonunital nonderived ternary division algebras of half dimension, we call them “half-quaternions” and “half-
octonions”. The latter are not subalgebras of the binary division algebras, but subsets only, since they have
different arity. Nevertheless, they are actually ternary division algebras, because allow division, and their
nonzero elements are invertible. From the multiplicativity of the introduced “half-quaternion” norm we
obtain the ternary analog of the sum of two squares identity. We prove that the introduced unitless ternary
division algebra of imaginary “half-octonions” is ternary alternative.
CONTENTS
1. INTRODUCTION 2
2. PRELIMINARIES 2
3. MATRIX POLYADIZATION 3
4. POLYADIZATION OF DIVISION ALGEBRAS 6
5. POLYADIC NORMS 13
6. POLYADIC ANALOG OF CAYLEY-DICKSON CONSTRUCTION 15
6.1. Abstract (tuple) approach 15
6.2. Concrete (hyperembedding) approach 16
6.3. Polyadic Cayley-Dickson process 17
7. POLYADIC IMAGINARY DIVISION ALGEBRAS 19
7.1. Complex number ternary division algebra 19
7.2. Half-quaternion ternary division algebra 20
7.3. Half-octonion ternary division algebra 21
REFERENCES 23
E-mail address: douplii@uni-muenster.de; sduplij@gmail.com; https://ivv5hpp.uni-muenster.de/u/douplii.
Date: of start September 23, 2023. Date: of completion December 3, 2023.
Total: 37 references.
2010 Mathematics Subject Classification. 11R52, 17A35, 17A40, 17A42, 20N10, 20N15 .
Key words and phrases. n-ary algebra, querelement, quaternion, octonion, Cayley-Dickson construction, division algebra.
2. 1. INTRODUCTION
The field extension is a fundamental concept of algebra MCCOY [1972], ROTMAN [2010], LOVETT
[2016] and number theory BOREVICH AND SHAFAREVICH [1966], NEURKICH [1999], SAMUEL [1972].
Informally, the main idea is to enlarge a given structure by a special way (using elements not from the
underlying set) and try to obtain the resulting algebraic structure with “good” properties. One of the
first well known examples is the field of complex numbers C which is a simple field extension of real
numbers R. The direct generalization of this construction leads to the hypercomplex numbers (see, e.g.
WEDDERBURN [1908], KANTOR AND SOLODOVNIKOV [1989]) defined as finite D-dimensional alge-
bras A over reals with the special basis (which squares restrict to 0, ˘1). Among numerous versions of
hypercomplex number systems HAWKES [1902], TABER [1904] (for modern review, see, e.g. YAGLOM
[1968], BURLAKOV AND BURLAKOV [2020]), only complex numbers A “ C (D “ 2), quaternions
A “ H (D “ 4) and octonions A “ O (D “ 8) are classical division algebras (with no zero divisors and
nilpotents) SCHAFER [1966], SALTMAN [1992], GUBARENI [2021], and the two latter can be obtain by
the Cayley-Dickson doubling procedure DICKSON [1919], ALBERT [1942], SCHAFER [1954].
In this paper we construct nonderived hyperpolyadic structures corresponding to the above di-
vision algebras without introducing new elements (as in, e.g., DUBROVSKI AND VOLKOV [2007],
LIPATOV ET AL. [2008]). First, we use the matrix polyadization procedure proposed by the author DUPLIJ
[2022]. We show that the polyadic analog of Cayley-Dickson construction can lead only to non-division
of more higher dimensions than initial division algebras. For the obtained n-ary algebras we introduce a
new norm which is polyadically multiplicative and is well-defined for invertible elements.
Second, on the subsets of the binary division algebras we propose another new construction (we call it
“imaginary tower”) and the iterative process which gives naturally the corresponding nonderived ternary
division algebras of half dimension. The latter are not subalgebras, because they have another multi-
plication and different arity than initial algebras. We call the obtained nonunital ternary algebras “half-
quaternions” and “half-octonions”. They are actually division algebras, because nonzero elements are
invertible and allow division. From the multiplicity of “half-quaternions” norm we obtain the ternary
analog of the sum of two squares identity. Finally, we show that the introduced unitless ternary division
algebra of imaginary “half-octonions” satisfies the ternary alternativity.
2. PRELIMINARIES
Here we briefly remind notation from DUPLIJ [2022]. A (one-set) polyadic algebraic structure A is a
set A closed with respect to polyadic operations (or n-ary multiplication) µrns
: An
Ñ A (n-ary magma).
We denote polyads POST [1940] as ~
a “ ~
apkq
“ pa1, . . . , akq, ai P A, and ak
“
k
´hkkikkj
a, . . . , a
¯
, a P A (usually,
the value of k follows for the context). A (positive) polyadic power is
axℓµy
“
`
µrns
˘˝ℓµ “
aℓµpn´1q`1
‰
, a P A, ℓµ P N. (2.1)
A polyadic ℓµ-idempotent (or idempotent for ℓµ “ 1) is defined by axℓµy
“ a. A polyadic zero is defined
by µrns
r~
a, zs “ z, z P A, ~
a P An´1
, where z can be on any place. An element of a polyadic algebraic
– 2 –
3. PRELIMINARIES
structure a is called ℓµ-nilpotent (or nilpotent for ℓµ “ 1), if there exist ℓµ such that axℓµy
“ z. A polyadic
(or n-ary) identity (or neutral element) is defined by
µrns
“
a, en´1
‰
“ a, @a P A, (2.2)
where a can be on any place in the l.h.s. of (2.2). In addition, there exist neutral polyads (usually not
unique) satisfying
µrns
ra,~
ns “ a, @a P A. (2.3)
A one-set polyadic algebraic structure
@
A | µrns
D
is totally associative, if
`
µrns
˘˝2
”
~
a,~
b,~
c
ı
“ µrns
”
~
a, µrns
”
~
b
ı
,~
c
ı
“ invariant, (2.4)
with respect to placement of the internal multiplication on any of the n places, and ~
a,~
b,~
c are polyads of
the necessary sizes.
A polyadic semigroup Srns
is a one-set and one-operation structure in which µrns
is totally associative.
A polyadic structure is (totally) commutative, if µrns
“ µrns
˝ σ, for all σ P Sn. A polyadic structure is
solvable, if for all polyads b, c and an element x, one can (uniquely) resolve the equation (with respect
to x) for µrns
”
~
b, x,~
c
ı
“ a, where x can be on any place, and ~
b,~
c are polyads of the needed lengths.
A solvable polyadic structure is called n-ary quasigroup BELOUSOV [1972]. An associative polyadic
quasigroup is called a n-ary (or polyadic) group Grns
(for review, see, e.g. GAL’MAK [2003]). In an n-ary
group the only solution of
µrns
“
an´1
, ã
‰
“ a, a, ã P A, (2.5)
is called a querelement (polyadic analog of inverse) of a and denoted by ã DÖRNTE [1929], where ã
can be on any place. The relation (2.5) can be considered as a definition of the unary queroperation
µ̄p1q
ras “ ã GLEICHGEWICHT AND GŁAZEK [1967].
For further details and references, see DUPLIJ [2022].
3. MATRIX POLYADIZATION
Let us briefly (just to establish notation and terminology) remind that the 2, 4, 8-dimensional algebras
are the only hypercomplex extensions of reals A “ R (Hurwitz’s theorem for composition algebras)
A “ D “ C, H, O which are normed division algebras. The first two ones are associative (and can
be represented by matrices), and only C is commutative (being a field). We use the unified notation
z P D, and if we need to distinguish and concretize, the standard parametrization will be exploited
C Qz “ zp2q “ a ` bi and H Qz “ zp4q “ a ` bi ` cj ` dk, etc., a, d, c, d P R. The standard Euclidean
norm (2-norm) }z} “
?
z˚z (where the conjugate is z˚
p2q “ a´bi, etc.), for z P D (which for C coincides
with the modulus
›
›zp2q
›
› “
ˇ
ˇzp2q
ˇ
ˇ “
?
a2 ` b2, and
›
›zp4q
›
› “
?
a2 ` b2 ` c2 ` d2, etc.) has the properties
}1} “ 1, (3.1)
}λz} “ |λ| }z} , (3.2)
}z1
` z2
} ď }z1
} ` }z2
} , 1, λ P R, 1, z, z1
, z2
P D, (3.3)
and is multiplicative
}z1z2} “ }z1} }z2} P Rě0, z P D, (3.4)
such that the corresponding mapping D Ñ Rě0 is a homomorphism. Each nonzero element of the above
normed unital algebra has the multiplicative inverse z´1
z “ 1, because the norm vanishes only for z “ 0
– 3 –
4. MATRIX POLYADIZATION
and there are no zero divisors, therefore from }z}2
“ z˚
z P Rě0, it follows
z´1
“
z˚
}z}2 , z P Dz t0u . (3.5)
To construct polyadic analogs of the binary hypercomplex algebras A and, in particular, of the binary
division algebras D (over R), we use the polyadization procedure proposed in DUPLIJ [2022] (called
there block-matrix polyadization). That is based on the general structure theorem for polyadic rings (a
generalization of the Wedderburn theorem): any simple p2, nq-ring is isomorphic to the ring of special
cyclic shift block-matrices (of the shape (3.6)) over a division ring NIKITIN [1984].
Let us introduce the pn ´ 1q ˆ pn ´ 1q cyclic shift (weighted) block-matrix with elements from the
algebra A
Z “ Zrns
“
¨
˚
˚
˚
˚
˚
˝
0 z1 . . . 0 0
0 0 z2 . . . 0
0 0
...
...
.
.
.
.
.
.
.
.
.
... 0 zn´2
zn´1 0 . . . 0 0
˛
‹
‹
‹
‹
‹
‚
, zi P A. (3.6)
The set of monomial matrices of the form (3.6) is closed with respect to the ordinary product p¨q of
exactly n matrices, but not less. Therefore, we can define the n-ary multiplication DUPLIJ [2022]
µ
rns
Z
»
–
n
hkkkkkkikkkkkkj
Z1
, Z2
. . . Z3
fi
fl “ Z1
¨ Z2
¨ . . . ¨ Z3
“ Z “ Zrns
, (3.7)
which is nonderived in the sense that the binary (and all ď n ´ 1) product of Zrns
’s is out of the set (3.6).
Remark 3.1. The binary addition in the algebra A transfers in the standard way to the matrix addition (as
componentwise addition), and so we will pay attention on the multiplicative part mostly, implying that
the addition of Z-matrices (3.6) is always binary.
Definition 3.2. We call the following new algebraic structure
Arns
“
A!
Zrns
)
| p`q, µ
rns
Z
E
(3.8)
a n-ary hypercomplex algebra Arns
which corresponds to the binary hypercomplex algebra A by the
matrix polyadization procedure.
Assertion 3.3. The dimension Drns
of n-ary hypercomplex algebra Arns
is
Drns
“ dim Arns
“ D pn ´ 1q . (3.9)
Proof. It obviously follows from (3.6).
– 4 –
5. MATRIX POLYADIZATION
Proposition 3.4. If the binary hypercomplex algebra A is associative (for the dimensions D “ 1, D “ 2
and D “ 4), the n-ary multiplication (3.7) in components has the cyclic product form DUPLIJ [2022]
n
hkkkkkkkkikkkkkkkkj
z1
1z2
2 . . . z3
n´1z4
1 “ z1,
n
hkkkkkkkikkkkkkkj
z1
2z2
3 . . . z3
1 z4
2 “ z2,
.
.
.
n
hkkkkkkkkkkkikkkkkkkkkkkj
z1
n´1z2
1 . . . z3
n´2z4
n´1 “ zn´1, zi, z1
i, . . . , z3
i , z4
i P A. (3.10)
Proof. It follows from (3.6) and (3.7).
Remark 3.5. The cycled product (3.10) can be treated as some n-ary extension of the Jordan pair LOOS
[1975, 1974], which is different from FAULKNER [1995].
Assertion 3.6. If A is unital, then Arns
contains n-ary unit (polyadic identity (2.2)) being the permutation
(cyclic shift) matrix of the form
Erns
“
¨
˚
˚
˚
˚
˚
˝
0 1 . . . 0 0
0 0 1 . . . 0
0 0
...
...
.
.
.
.
.
.
.
.
.
... 0 1
1 0 . . . 0 0
˛
‹
‹
‹
‹
‹
‚
P Arns
, 1 P A. (3.11)
Proof. It follows from (3.6), (3.7) and (3.10).
Consider the polyadization of the hypercomplex two-dimensional algebra of dual numbers.
Example 3.7 (4-ary dual numbers). The commutative and associative two-dimensional algebra Adual “
xtzu | p`q , p¨qy is defined by the element z “ a ` bε with ε2
“ 0, a, b P R. The binary multiplication
µ
r2s
v “ µ
r2s
dual of pairs v2 “
ˆ
a
b
˙
P Atuple
dual “
@
tv2u | p`q , µr2s
D
(addition is componentwise, see Remark
3.1) is
µr2s
v
„ˆ
a1
b1
˙
,
ˆ
a2
b2
˙
“
ˆ
a1
a2
a1
b2
` b1
a2
˙
P Atuple
dual , a1
, a2
, b1
, b2
P R. (3.12)
It follows from (3.12) that Atuple
dual (and so is Adual) is a non-division algebra, because it contains idempo-
tents and zero divisors (e.g.
ˆ
0
b
˙2
“
ˆ
0
0
˙
).
Using the matrix polyadization procedure, we construct 4-ary algebra Ar4s
dual “
A!
Zr4s
)
| p`q , µr4s
E
of the dimension Dr4s
“ 6 (see (3.9)) by introducing the following 3ˆ3 cyclic shift weighted matrix (3.6)
Zr4s
“
¨
˝
0 z1 0
0 0 z2
z3 0 0
˛
‚“
¨
˝
0 a1 ` b1ε 0
0 0 a2 ` b2ε
a3 ` b3ε 0 0
˛
‚P Ar4s
dual, z1, z2, z3 P Adual. (3.13)
– 5 –
6. MATRIX POLYADIZATION
The cyclic product of the components (3.10) becomes
z1
1z2
2z3
3 z4
1 “ z1,
z1
2z2
3z3
1 z4
2 “ z2,
z1
3z2
1z3
2 z4
3 “ z3, zi, z1
i, z2
i , z3
i , z4
i P Adual. (3.14)
In terms of 6-tuples v6 over R (cf. (3.12)) the 4-ary multiplication µr4s
in Ar4s,tuple
dual “
A
tv6u | p`q , µ
r4s
v
E
has the form
µr4s
v
»
—
—
—
—
—
–
¨
˚
˚
˚
˚
˚
˝
a1
1
b1
1
a1
2
b1
2
a1
3
b1
3
˛
‹
‹
‹
‹
‹
‚
,
¨
˚
˚
˚
˚
˚
˝
a2
1
b2
1
a2
2
b2
2
a2
3
b2
3
˛
‹
‹
‹
‹
‹
‚
,
¨
˚
˚
˚
˚
˚
˝
a3
1
b3
1
a3
2
b3
2
a3
3
b3
3
˛
‹
‹
‹
‹
‹
‚
,
¨
˚
˚
˚
˚
˚
˝
a4
1
b4
1
a4
2
b4
2
a4
3
b4
3
˛
‹
‹
‹
‹
‹
‚
fi
ffi
ffi
ffi
ffi
ffi
fl
“
¨
˚
˚
˚
˚
˚
˝
a1
1a2
2a3
3 a4
1
a1
1a2
2a3
3 b4
1 ` a1
1a2
2b3
3 a4
1 ` a1
1b2
2a3
3 a4
1 ` b1
1a2
2a3
3 a4
1
a1
2a2
3a3
1 a4
2
a1
2a3
1 a2
3b4
2 ` a1
2a2
3b3
1 a4
2 ` a1
2b2
3a3
1 a4
2 ` b1
2a2
3a3
1 a4
2
a1
3a2
1a3
2 a4
3
a1
3a2
1a3
2 b4
3 ` a1
3a2
1b3
2 a4
3 ` a1
3b2
1a3
2 a4
3 ` b1
3a2
1a3
2 a4
3
˛
‹
‹
‹
‹
‹
‚
,
(3.15)
which is nonderived and noncommutative due to braidings in the cyclic product (3.14). The polyadic unit
(4-ary unit) in the 4-ary algebra of 6-tuples is (see (2.2))
e6 “
¨
˚
˚
˚
˚
˚
˝
1
0
1
0
1
0
˛
‹
‹
‹
‹
‹
‚
P Ar4s,tuple
dual . (3.16)
It follows from (3.15) that Ar4s,tuple
dual (and so is Ar4s
dual) is a non-division 4-ary algebra, not a field (similar
to Adual), because it contains 4-ary idempotents and zero divisors, for instance,
µr4s
v
»
—
—
—
—
—
—
–
¨
˚
˚
˚
˚
˚
˝
0
b1
0
b2
a3
b3
˛
‹
‹
‹
‹
‹
‚
4fi
ffi
ffi
ffi
ffi
ffi
ffi
fl
“ µr4s
v
»
—
—
—
—
—
—
–
¨
˚
˚
˚
˚
˚
˝
a1
b1
0
b2
0
b3
˛
‹
‹
‹
‹
‹
‚
4fi
ffi
ffi
ffi
ffi
ffi
ffi
fl
“ µr4s
v
»
—
—
—
—
—
—
–
¨
˚
˚
˚
˚
˚
˝
0
b1
a2
b2
0
b3
˛
‹
‹
‹
‹
‹
‚
4fi
ffi
ffi
ffi
ffi
ffi
ffi
fl
“
¨
˚
˚
˚
˚
˚
˝
0
0
0
0
0
0
˛
‹
‹
‹
‹
‹
‚
“ 0v. (3.17)
Thus, the application of the matrix polyadization procedure to the commutative, associative, unital, two-
dimensional, non-division algebra of dual numbers Adual gives noncommutative, 4-ary nonderived totally
associative, unital, 6-dimensional, non-division algebra Ar4s
dual over R, we call that 4-ary dual numbers.
4. POLYADIZATION OF DIVISION ALGEBRAS
Let us consider the matrix polyadization procedure for the division algebras D “ R, C, H, O in more
details paying attention on invertibility and norms.
Recall that the binary division algebra D (without zero element) forms a group (with respect to binary
multiplication) having the inverse (3.5). The polyadic counterpart of the binary inverse is the querelement
(2.5).
– 6 –
7. POLYADIZATION OF DIVISION ALGEBRAS
Theorem 4.1. The nonderived n-ary algebra (3.8) constructed from the binary division algebra D is the
n-ary algebra
Drns
“
A!
Zrns
)
| p`q, µ
rns
Z , r
Z
rns
E
, (4.1)
where r
Z
rns
is the querelement
r
Z “ r
Z
rns
“
¨
˚
˚
˚
˚
˚
˝
0 r
z1 . . . 0 0
0 0 r
z2 . . . 0
0 0
...
...
.
.
.
.
.
.
.
.
.
... 0 r
zn´2
r
zn´1 0 . . . 0 0
˛
‹
‹
‹
‹
‹
‚
, (4.2)
which satisfies (if D is associative)
µ
rns
Z
»
–
n´1
hkkkkikkkkj
Z, Z . . . Z r
Z
fi
fl “
n´1
hkkkkkkkikkkkkkkj
Z ¨ Z ¨ . . . ¨ Z ¨ r
Z “ Z, @Z P Drns
, (4.3)
and r
Z can be on any place, such that
r
zi “ z´1
i´1z´1
i´2 . . . z´1
2 z´1
1 z´1
n´1z´1
n´1 . . . z´1
i`2z´1
i`1, zi P Dz t0u . (4.4)
Proof. The main relation (4.4) follows from (4.3) in components (3.10) as the following cycle products
n
hkkkkkkkkikkkkkkkkj
z1z2 . . . zn´1r
z1 “ z1,
n
hkkkkkkkkkikkkkkkkkkj
z2z3 . . . zn´1z1r
z2 “ z2,
.
.
.
n
hkkkkkkkkkkkikkkkkkkkkkkj
zn´1z1 . . . zn´2r
zn´1 “ zn´1, zi, r
zi P Dz t0u , (4.5)
by applying z´1
i (which exists in D for nonzero z (3.5)) from the left pn ´ 1q times (with suitable indices)
to both sides of each equation in (4.5) to get r
zi.
Corollary 4.2. Each D-dimensional division algebra over reals D “ C, H, O (including R itself as one-
dimensional case) has its n polyadic counterparts (where n is arbitrary), the nonderived n-ary non-division
algebras Drns
(4.1) of the dimension Dpn ´ 1q (having the polyadic unit (3.11) and the querelement (4.2)
for invertible zi P Dz t0u) constructed by the matrix polyadization procedure.
Theorem 4.3. The matrix polyadization changes invertibility property of initial algebra, that is polyadiza-
tion of a binary division algebra D leads to n-ary non-division algebra Drns
with arbitrary n.
Proof. The polyadization procedure is provided by monomial matrices which have determinant which is
proportional to the product of nonzero entries. The nonzero elements of Drns
having some of zi “ 0 are
noninvertible, and therefore Drns
is not a field.
Nevertheless, a special subalgebra of Drns
can be a division n-ary algebra.
Theorem 4.4. The elements of Drns
which have all invertible zi P Dz t0u (or set of invertible Z-matrices
det Z ‰ 0) form a subalgebra Drns
div Ă Drns
which is the division n-ary algebra corresponding to the
division algebra D “ R, C, H, O.
– 7 –
8. POLYADIZATION OF DIVISION ALGEBRAS
The simplest case is polyadization of reals.
Example 4.5 (5-ary real numbers). The 5-ary algebra of real numbers is Rr5s
“
A!
Rr5s
)
| p`q , µ
r5s
R
E
,
where Rr5s
is the cyclic 4 ˆ 4 block-shift matrix (3.6) with real entries
Rr5s
“
¨
˚
˚
˝
0 a1 0 0
0 0 a2 0
0 0 0 a3
a4 0 0 0
˛
‹
‹
‚, (4.6)
det Rr5s
“ a1a2a3a4, ai P R, (4.7)
and the multiplication µ
r5s
R is an ordinary product of 5 matrices. Only with respect to product of 5 ele-
ments Rr5s
the algebra is closed, and therefore Rr5s
is nonderived. In components we have the braiding
cyclic products (3.10) for ai. If ai P Rz t0u, the component equations for the querelement (4.5) (after
cancellation of nonzero ai) become
a2a3a4r
a1 “ 1, (4.8)
a3a4a1r
a2 “ 1, (4.9)
a4a1a2r
a3 “ 1, (4.10)
a1a2a3r
a4 “ 1. (4.11)
The querelement for invertible (4.6) is
r
R
r5s
“
¨
˚
˚
˚
˚
˚
˚
˚
˚
˝
0
1
a2a3a4
0 0
0 0
1
a3a4a1
0
0 0 0
1
a4a1a2
1
a1a2a3
0 0 0
˛
‹
‹
‹
‹
‹
‹
‹
‹
‚
, ai P Rz t0u , (4.12)
and therefore, the algebra Rr5s
of 5-ary real numbers is the non-division algebra, because the elements
with some ai “ 0 are in Rr5s
, but they are noninvertible (due to (4.7)), and so Rr5s
is not a field. But the
subalgebra Rr5s
div Ă Rr5s
of invertible matrices Rr5s
(det Rr5s
‰ 0, with all ai ‰ 0) is a division n-ary
algebra of reals (see Theorem 4.4).
Now we provide the example of 4-ary algebra of complex numbers to compare it with the dual numbers
of the same arity in Example 3.7.
Example 4.6 (4-ary complex numbers). First, we establish notations, as before. The commutative and
associative two-dimensional algebra of complex numbers is C “ xtzu | p`q , p¨qy, where z “ a ` bi with
i2
“ ´1, a, b P R. The binary multiplication µ
r2s
v “ µ
r2s
compl of pairs
v2 “
ˆ
a
b
˙
P Ctuple
“
@
tv2u | p`q , µr2s
v
D
(4.13)
(addition is componentwise, see Remark 3.1) is
µr2s
v
„ˆ
a1
b1
˙
,
ˆ
a2
b2
˙
“
ˆ
a1
a2
´ b2
b1
b2
a1
` b1
a2
˙
P Ctuple
, a1
, a2
, b1
, b2
P R. (4.14)
– 8 –
9. POLYADIZATION OF DIVISION ALGEBRAS
The algebra of pairs Ctuple
does not contain idempotents and zero divisors, its multiplication agrees with
one of C (z1
¨ z2
“ z), and therefore it is a division algebra being isomorphic to C
Ctuple
– C. (4.15)
Using the matrix polyadization procedure, we construct the nonderived 4-ary algebra of complex num-
bers Cr4s
“
A!
Zr4s
)
| p`q , µ
r4s
Z
E
of the dimension Dr4s
“ 6 (see (3.9)) by introducing the following
3 ˆ 3 matrix (3.6)
Z “ Zr4s
“
¨
˝
0 z1 0
0 0 z2
z3 0 0
˛
‚“
¨
˝
0 a1 ` b1i 0
0 0 a2 ` b2i
a3 ` b3i 0 0
˛
‚P Cr4s
, z1, z2, z3 P C. (4.16)
The 4-ary product of Z-matrices (4.16) is
µ
r4s
Z rZ1
, Z2
, Z3
, Z4
s “ Z1
Z2
Z3
Z4
. (4.17)
The corresponding to (4.17) cyclic product in the components (3.10) becomes
z1
1z2
2z3
3 z4
1 “ z1,
z1
2z2
3z3
1 z4
2 “ z2,
z1
3z2
1z3
2 z4
3 “ z3, zi, z1
i, z2
i , z3
i , z4
i P C. (4.18)
In terms of 6-tuples v6 over R (cf. (3.12)) the 4-ary multiplication µ
r4s
v in Cr4s,tuple
“
A
tv6u | p`q , µ
r4s
v
E
has the form (cf. (4.14))
µr4s
v
»
—
—
—
—
—
–
¨
˚
˚
˚
˚
˚
˝
a1
1
b1
1
a1
2
b1
2
a1
3
b1
3
˛
‹
‹
‹
‹
‹
‚
,
¨
˚
˚
˚
˚
˚
˝
a2
1
b2
1
a2
2
b2
2
a2
3
b2
3
˛
‹
‹
‹
‹
‹
‚
,
¨
˚
˚
˚
˚
˚
˝
a3
1
b3
1
a3
2
b3
2
a3
3
b3
3
˛
‹
‹
‹
‹
‹
‚
,
¨
˚
˚
˚
˚
˚
˝
a4
1
b4
1
a4
2
b4
2
a4
3
b4
3
˛
‹
‹
‹
‹
‹
‚
fi
ffi
ffi
ffi
ffi
ffi
fl
(4.19)
“
¨
˚
˚
˚
˚
˚
˝
a1
1a2
2a3
3 a4
1 ´ a1
1b2
2b3
3 a4
1 ´ b1
1a2
2b3
3 a4
1 ´ b1
1b2
2a3
3 a4
1 ´ a1
1a2
2b3
3 b4
1 ´ a1
1b2
2a3
3 b4
1 ´ b1
1a2
2a3
3 b4
1 ` b1
1b2
2b3
3 b4
1
a1
1a2
2b3
3 a4
1 ` a1
1b2
2a3
3 a4
1 ` b1
1a2
2a3
3 a4
1 ` a1
1a2
2a3
3 b4
1 ´ b1
1b2
2b3
3 a4
1 ´ a1
1b2
2b3
3 b4
1 ´ b1
1a2
2b3
3 b4
1 ´ b1
1b2
2a3
3 b4
1
a1
2a2
3a3
1 a4
2 ´ a1
2b2
3b3
1 a4
2 ´ b1
2b2
3a3
1 a4
2 ´ b1
2a2
3b3
1 a4
2 ´ a1
2b2
3a3
1 b4
2 ´ a1
2a2
3b3
1 b4
2 ´ b1
2a2
3a3
1 b4
2 ` b1
2b2
3b3
1 b4
2
a1
2b2
3a3
1 a4
2 ` a1
2a2
3b3
1 a4
2 ` b1
2a2
3a3
1 a4
2 ` a1
2a2
3a3
1 b4
2 ´ b1
2b2
3b3
1 a4
2 ´ a1
2b2
3b3
1 b4
2 ´ b1
2b2
3a3
1 b4
2 ´ b1
2a2
3b3
1 b4
2
a1
3a2
1a3
2 a4
3 ´ b1
3a2
1b3
2 a4
3 ´ a1
3b2
1b3
2 a4
3 ´ b1
3b2
1a3
2 a4
3 ´ a1
3a2
1b3
2 b4
3 ´ b1
3a2
1a3
2 b4
3 ´ a1
3b2
1a3
2 b4
3 ` b1
3b2
1b3
2 b4
3
a1
3a2
1b3
2 a4
3 ` b1
3a2
1a3
2 a4
3 ` a1
3b2
1a3
2 a4
3 ` a1
3a2
1a3
2 b4
3 ´ b1
3b2
1b3
2 a4
3 ´ b1
3a2
1b3
2 b4
3 ´ a1
3b2
1b3
2 b4
3 ´ b1
3b2
1a3
2 b4
3
˛
‹
‹
‹
‹
‹
‚
,
(4.20)
which is nonderived and noncommutative due to braidings in the cyclic product (3.14). The polyadic unit
(4-ary unit) in the 4-ary algebra of 6-tuples is (see (2.2))
e6 “
¨
˚
˚
˚
˚
˚
˝
1
0
1
0
1
0
˛
‹
‹
‹
‹
‹
‚
P Cr4s,tuple
, (4.21)
µr4s
v re6, e6, e6, v6s “ µr4s
v re6, e6, v6, e6s “ µr4s
v re6, v6, e6, e6s “ µr4s
v rv6, e6, e6, e6s “ v6. (4.22)
– 9 –
10. POLYADIZATION OF DIVISION ALGEBRAS
The querelement (2.5), (4.2) for invertible elements of 4-ary algebra of complex numbers Cr4s
has the
matrix form which follows from the equations (4.5)
r
Z
r4s
“
¨
˝
0 1
z2z3
0
0 0 1
z1z3
1
z1z2
0 0
˛
‚P Cr4s
, z1, z2, z3 P Cz t0u . (4.23)
Thus, Cr4s
“
A!
Zr4s
)
| p`q , µr4s
, Ă
p q
E
is the nonderived 4-ary non-division algebra over R obtained
by the matrix polyadization procedure from the algebra C of complex numbers. The 4-ary algebra of
complex numbers Cr4s
is not a field, because it contains noninvertible nonzero elements (with some zi “
0).
In Cr4s,tuple
the querelement is given by the following 6-tuple
r
v6 “
Č
¨
˚
˚
˚
˚
˚
˝
a1
b1
a2
b2
a3
b3
˛
‹
‹
‹
‹
‹
‚
“
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
a2a3 ´ b2b3
pa2b3 ` a3b2q2
` pa2a3 ´ b2b3q2
´
a2b3 ` a3b2
pa2b3 ` a3b2q2
` pa2a3 ´ b2b3q2
a1a3 ´ b1b3
pa1b3 ` a3b1q2
` pa1a3 ´ b1b3q2
´
a1b3 ` a3b1
pa1b3 ` a3b1q2
` pa1a3 ´ b1b3q2
a1a2 ´ b1b2
pa1b2 ` a2b1q2
` pa1a2 ´ b1b2q2
´
a1b2 ` a2b1
pa1b2 ` a2b1q2
` pa1a2 ´ b1b2q2
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
, ai, bi P R. (4.24)
Therefore, Cr4s,tuple
is a division nonderived 4-ary algebra isomorphic to Cr4s
.
Thus, the application of the matrix polyadization procedure to the commutative, associative, unital,
two-dimensional, division algebra of complex numbers C gives noncommutative, 4-ary nonderived totally
associative, unital, 6-dimensional, non-division algebra Cr4s
over R (with the corresponding isomorphic
4-ary non-division algebra of 6-tuples Cr4s,tuple
), we call that 4-ary complex numbers. The subalgebra
Cr4s
div Ă Cr4s
of invertible matrices Zr4s
(det Zr4s
‰ 0, with all zi ‰ 0) is the division 4-ary algebra of
complex numbers (by Theorem 4.4).
Consider now polyadization of the noncommutative quaternion algebra H.
Example 4.7 (Ternary quaternions). The associative four-dimensional algebra of quaternions is
H “ xtqu | p`q , p¨qy , q “ a ` bi ` cj ` dk,
ij “ k, ij “ ´ji, p+cycledq , i2
“ j2
“ k2
“ ijk “ ´1, a, b, c, d P R. (4.25)
The binary multiplication µ
r4s
v of the quadruples
v4 “
¨
˚
˚
˝
a
b
c
d
˛
‹
‹
‚P Htuple
“
@
tv4u | p`q , µr2s
v
D
(4.26)
– 10 –
12. POLYADIZATION OF DIVISION ALGEBRAS
The polyadic unit (ternary unit) e8 in the ternary algebra of 8-tuples is (see (2.2))
e8 “
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
1
0
0
0
1
0
0
0
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
P Hr3s,tuple
, (4.32)
µr3s
v re8, e8, v8s “ µr3s
v re8, v8, e8s “ µr3s
v rv8, e8, e8s “ v8. (4.33)
The querelement (2.5), (4.2) of nonderived noncommutative 8-dimensional ternary algebra of quaternions
Hr3s
has the matrix form which follows from the general equations (4.5)
r
Q
r3s
“
ˆ
0 q´1
2
q´1
1 0
˙
P Hr3s
, q1, q2 P Hz t0u . (4.34)
Therefore, Hr3s
“
A!
Qr3s
)
| p`q , µ
r3s
Q , Ă
p q
E
is the nonderived noncommutative ternary non-division
algebra obtained by the matrix polyadization procedure from the algebra H of quaternions, because the
elements Qr3s
with q1 “ 0 or q2 “ 0 are nonzero, but noninvertible, such that Hr3s
is not a field.
In Hr3s,tuple
the querelement is given by the following 8-tuple
r
v8 “
Č
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
a1
b1
c1
d1
a2
b2
c2
d2
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
“
¨
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
a1
a2
1 ` b2
1 ` c2
1 ` d2
1
´
b1
a2
1 ` b2
1 ` c2
1 ` d2
1
´
c1
a2
1 ` b2
1 ` c2
1 ` d2
1
´
d1
a2
1 ` b2
1 ` c2
1 ` d2
1
a2
a2
2 ` b2
2 ` c2
2 ` d2
2
´
b2
a2
2 ` b2
2 ` c2
2 ` d2
2
´
c2
a2
2 ` b2
2 ` c2
2 ` d2
2
´
d2
a2
2 ` b2
2 ` c2
2 ` d2
2
˛
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‹
‚
, a2
1,2 ` b2
1,2 ` c2
1,2 ` d2
1,2 ‰ 0, ai, bi, ci, di P R. (4.35)
Therefore, Hr3s,tuple
is a non-division ternary algebra isomorphic to Hr3s
.
To conclude, the application of the matrix polyadization procedure to the noncommutative, associative,
unital, four-dimensional, division algebra of quaternions H gives noncommutative, nonderived ternary to-
tally associative, unital, 8-dimensional, non-division algebra Hr3s
over R (with the corresponding isomor-
phic ternary non-division algebra of 8-tuples Hr3s,tuple
), we call that ternary quaternions. The subalgebra
Hr3s
div Ă Hr3s
of invertible matrices Qr3s
(det Qr3s
‰ 0, with all qi ‰ 0) is the division ternary algebra of
quaternions (see Theorem 4.4).
– 12 –
13. POLYADIC NORMS
5. POLYADIC NORMS
The division algebras D “ R, C, H, O are normed as vector spaces, and the corresponding Euclidean
2-norm is multiplicative (3.4) such that the corresponding mapping is the binary homomorphism. It would
be worthwhile to define a polyadic analog of the binary norm } } which would have similar properties.
Definition 5.1. We define the polyadic (n-ary) norm } }rns
in the n-ary algebra Drns
, that is obtained from
D by the matrix polyadization procedure (3.6), as the product (in R) of the component norms
}Z}rns
“ }z1} }z2} . . . }zn´1} P R, Z P Drns
, zi P D. (5.1)
Corollary 5.2. The polyadic norm (5.1) is zero for noninvertible elements of Drns
, having some zi “ 0.
Therefore, it is worthwhile to consider polyadic norm for invertible elements of Drns
only.
Proposition 5.3. The division n-ary subalgebras Drns
div Ă Drns
are normed n-ary algebras with respect to
the polyadic norm } }rns
(5.1).
Let us consider properties of the polyadic norm (5.1).
Assertion 5.4. The introduced polyadic norm } }rns
(5.1) has the properties (for invertible Z P Drns
)
›
›
›Erns
›
›
›
rns
“ 1 P R, Erns
P Drns
, (5.2)
}λZ}rns
“ |λ|n´1
}Z}rns
, λ P R, (5.3)
}Z1
` Z2
}
rns
ď }Z}rns
` }Z}rns
, Z P Drns
, (5.4)
where Erns
is the polyadic unit in n-ary algebra Drns
(3.11).
Proof. The first property is obvious, the second one follows from the definition (5.1) and linearity of the
ordinary Euclidean norm in D (3.2).The polyadic triangle inequality (5.4) follows from the binary triangle
inequality (3.3), because of the binary addition of Z-matrices of the cyclic block-shift form (3.6).
The norms satisfying (5.3) are called norms of higher degree, and they were investigated for the binary
case in PUMPLÜN [2011].
The most important property of any (binary) norm is its multiplicativity (3.4).
Theorem 5.5. The polyadic norm } }rns
defined in (5.1) is n-ary multiplicative (such that the correspond-
ing map Drns
Ñ R is an n-ary homomorphism)
›
›
›
›
›
›
µ
rns
Z
»
–
n
hkkkkkkikkkkkkj
Z1
, Z2
. . . Z3
fi
fl
›
›
›
›
›
›
rns
“
n
hkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkj
}Z1
}
rns
¨ }Z2
}
rns
¨ . . . ¨ }Z3
}
rns
, Z1
, Z2
, Z3
P Drns
div. (5.5)
Proof. Consider the component form (3.10) of each multiplier Z in (5.5), then use the definition (5.1) and
commutativity (as they are in R) and multiplicativity (3.4) of the ordinary binary norms } } to rearrange
the products of norms from l.h.s. to r.h.s. in (5.5). That is
n´1
hkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkj
›
›z1
1z2
2 . . . z3
n´1z4
1
›
› }z1
2z2
3 . . . z3
1 z4
2 } . . .
›
›z1
n´1z2
1 . . . z3
n´2z4
n´1
›
›
“
n
hkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkj
`
}z1
1} }z1
2} . . .
›
›z1
n´1
›
›
˘ `
}z2
1} }z2
2} . . .
›
›z2
n´1
›
›
˘
. . .
`
}z3
1 } }z3
2 } . . .
›
›z3
n´1
›
›
˘ `
}z4
1 } }z4
2 } . . .
›
›z4
n´1
›
›
˘
.
(5.6)
– 13 –
14. POLYADIC NORMS
Remark 5.6. The n-ary multiplicativity (5.5) of the introduced polyadic norm (5.1) is independent of the
concrete form of binary norm } }, only the multiplicativity of the latter is needed.
Proposition 5.7. The polyadic norm of the querelement in n-ary division subalgebra Drns
div is
›
›
›r
Z
›
›
›
rns
“
1
´
}Z}rns
¯n´2 P Rą0, @r
Z P Drns
div. (5.7)
Proof. It follows from the component relations for the querelements }r
zi} (4.5) and multiplicativity of the
binary norm, that
n
hkkkkkkkkkkkkkkikkkkkkkkkkkkkkj
}z1} }z2} . . . }zn´1} }r
z1} “ }z1} ,
n
hkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkj
}z2} }z3} . . . }zn´1} }z1} }r
z2} “ }z2} ,
.
.
.
n
hkkkkkkkkkkkkkkkkkikkkkkkkkkkkkkkkkkj
}zn´1} }z1} . . . }zn´2} }r
zn´1} “ }zn´1} , zi, r
zi P D, }zi} , }r
zi} P Rą0. (5.8)
Multiplying all equations in (5.8) and using the definition of the polyadic norm (5.1), together with the
component form (3.6) and the querelement (4.2), we get
´
}Z}rns
¯n´1 ›
›
›r
Z
›
›
›
rns
“ }Z}rns
, (5.9)
from which it follows (5.7).
Example 5.8. In the division 4-ary algebra of complex numbers Cr4s
div from Example 4.6 the polyadic norm
becomes
}Z}r4s
“
b
pa2
1 ` b2
1q pa2
2 ` b2
2q pa2
3 ` b2
3q, ai, bi P R. (5.10)
The polyadic norm of the querelement r
Z in Cr4s
(5.7) is
›
›
›r
Z
›
›
›
r4s
“
1
pa2
1 ` b2
1q pa2
2 ` b2
2q pa2
3 ` b2
3q
, a2
i ` b2
i ‰ 0, ai, bi P R. (5.11)
Example 5.9. In the division ternary algebra of quaternions Hr3s
div from Example 4.7 the polyadic norm
becomes
}Q}r3s
“
b
pa2
1 ` b2
1 ` c2
1 ` d2
1q pa2
2 ` b2
2 ` c2
2 ` d2
2q, ai, bi P R. (5.12)
The polyadic norm of the querelement r
Q in Hr3s
div (5.7) is
›
›
› r
Q
›
›
›
r3s
“
1
a
pa2
1 ` b2
1 ` c2
1 ` d2
1q pa2
2 ` b2
2 ` c2
2 ` d2
2q
, a2
i ` b2
i ` c2
i ` d2
i ‰ 0, ai, bi P R. (5.13)
Further properties of the polyadic norm } }rns
can be investigated for invertible elements of concrete
n-ary algebras.
– 14 –
15. Abstract (tuple) approach POLYADIC ANALOG OF CAYLEY-DICKSON CONSTRUCTION
6. POLYADIC ANALOG OF CAYLEY-DICKSON CONSTRUCTION
The standard method of obtaining the higher hypercomplex algebras is the Cayley-Dickson construction
SCHAFER [1954], KORNILOWICZ [2012], FLAUT [2019]. It is well known, that all four binary division
algebras D “ R, C, H, O can be built in this way KANTOR AND SOLODOVNIKOV [1989]. Here we
generalize the Cayley-Dickson construction to the polyadic (n-ary) division algebras introduced in the
previous section. As the result, the number of polyadic division algebras becomes infinity (as opposite to
above four in the binary case), because of arbitrary initial and final arities of algebras under consideration.
For illustration, we present several low arity examples, since higher arity cases become too cumbersome
and unobservable. First, we recall in brief (just to install our notation) the ordinary (binary) Cayley-
Dickson doubling process (in our notation convenient for the polyadization procedure).
6.1. Abstract (tuple) approach. Consider the sequence of algebras Aℓ, ℓ ě 0, over reals, starting from
A0 “ R. The main idea is to repeat the doubling process of complex number construction using pairs
(doubles) (4.14) and taking into account the isomorphism (4.15) on each stage Aℓ Ñ Aℓ`1. Let us denote
the binary algebra over R on ℓ-th stage with the underlying set tAℓu as
Aℓ “
A
tAℓu | p`q , µ
r2s
ℓ , p˚ℓq
E
, (6.1)
where p˚ℓq is involution in Aℓ and µ
r2s
ℓ : Aℓ b Aℓ Ñ Aℓ is its binary multiplication, we also will write
µ
r2s
ℓ ” p¨ℓq. The corresponding algebra of doubles (2-tuples)
v2pℓq “
ˆ
apℓq
bpℓq
˙
, apℓq, bpℓq P Aℓ, (6.2)
is denoted by
Atuple
ℓ “
A
v2pℓq
(
| p`q , µ
r2s
vpℓq,
´
˚tuple
ℓ
¯E
, (6.3)
where
´
˚tuple
ℓ
¯
is involution in Atuple
ℓ and µ
r2s
vpℓq : Atuple
ℓ bAtuple
ℓ Ñ Atuple
ℓ is the binary product of doubles.
If ℓ “ 0, then the conjugation is the identical map, as it should be for reals R. The addition and scalar
multiplication are made componentwise in the standard way.
In this notation, the Cayley-Dickson doubling process is defined by the recurrent multiplication formula
µ
r2s
vpℓ`1q
„ˆ
a1
pℓq
b1
pℓq
˙
,
ˆ
a2
pℓq
b2
pℓq
˙
“
¨
˝
µ
r2s
ℓ
”
a1
pℓq, a2
pℓq
ı
´ µ
r2s
ℓ
“`
b2
pℓq
˘˚ℓ
, b1
pℓq
‰
µ
r2s
ℓ
”
b2
pℓq, a1
pℓq
ı
` µ
r2s
ℓ
”
b1
pℓq,
´
a2
pℓq
¯˚ℓ
ı
˛
‚ (6.4)
“
˜
a1
pℓq ¨ℓ a2
pℓq ´
`
b2
pℓq
˘˚ℓ
¨ℓ b1
pℓq
b2
pℓq ¨ℓ a1
pℓq ` b1
pℓq ¨ℓ
´
a2
pℓq
¯˚ℓ
¸
”
ˆ
apℓ`1q
bpℓ`1q
˙
P Atuple
ℓ`1 , (6.5)
and the recurrent conjugation
ˆ
apℓq
bpℓq
˙˚tuple
ℓ`1
“
ˆ
a˚ℓ
pℓq
´bpℓq
˙
P Atuple
ℓ`1 , apℓq, bpℓq P Aℓ. (6.6)
Then, we use the isomorphism (4.15) which is now becomes
Atuple
ℓ`1 – Aℓ`1. (6.7)
To go to the next level of recursion with the obtained Aℓ`1 we use (6.4)–(6.7) with changing ℓ Ñ ℓ ` 1.
The dimension of the algebra Aℓ is
Dpℓq “ 2ℓ
, (6.8)
– 15 –
16. POLYADIC ANALOG OF CAYLEY-DICKSON CONSTRUCTION Concrete (hyperembedding) approach
such that each element can be presented in the form of 2ℓ
-tuple of reals
¨
˚
˚
˝
apℓq,1
apℓq,2
.
.
.
apℓq,2ℓ
˛
‹
‹
‚P Aℓ, (6.9)
and the conjugated 2ℓ
-tuple becomes
¨
˚
˚
˝
apℓq,1
´apℓq2
.
.
.
´apℓq,2ℓ
˛
‹
‹
‚, apℓq,k P R. (6.10)
For clarity, we intensionally mark elements and operations of ℓ-th level manifestly, because they are
really different for different ℓ. The example with ℓ “ 0 is obtaining the algebra of complex numbers A1 ”
C from the algebra of reals A0 ” R (4.13)–(4.15). All the binary division algebras D “ R, C, H, O can
be obtained by the Cayley-Dickson construction KANTOR AND SOLODOVNIKOV [1989].
6.2. Concrete (hyperembedding) approach. Alternatively, one can reparametrize the pairs (6.2) satis-
fying the complex-like multiplication (6.4), as the field extension p
Aℓ “ Aℓ piℓq by one complex-like unit
iℓ on each ℓ-th stage of iteration. The Cayley-Dickson doubling process is given by the iterations
p
Aℓ`1 “
A
tAℓ piℓqu | p`q , p
µ
r2s
ℓ`1, p˚ℓ`1q
E
piℓ`1q , i2
ℓ “ ´1, i2
ℓ`1 “ ´1, iℓ`1iℓ “ ´iℓiℓ`1, ℓ ě 0.
(6.11)
In the component form it is
zpℓ`1q “ zpℓq,1 ` zpℓq,2 ¨ℓ`1 iℓ`1 “ zpℓq,1 ` p
µ
r2s
ℓ`1
“
zpℓq,2, iℓ`1
‰
P p
Aℓ`1, zpℓq,1, zpℓq,2 P p
Aℓ. (6.12)
The product in p
Aℓ`1 can be expressed through the product of previous stage from p
Aℓ and the conjuga-
tion by using anticommutation of the imagery units from different stages iℓ`1iℓ “ ´iℓiℓ`1 (see (6.11)).
Thus, we obtain the standard complex-like multiplication (see (4.14) and (6.5)) on each ℓ-th stage of the
Cayley-Dickson doubling process
z1
pℓ`1q ¨ℓ`1 z2
pℓ`1q “
`
z1
pℓq,1 ¨ℓ z2
pℓq,1 ´
`
z2
pℓq,2
˘˚ℓ
¨ℓ z1
pℓq,2
˘
`
`
z2
pℓq,2 ¨ℓ z1
pℓq,1 ` z1
pℓq,2 ¨ℓ
`
z2
pℓq,1
˘˚ℓ
˘
¨ℓ`1 iℓ`1,
(6.13)
or with the manifest form of different stage multiplications p
µ
r2s
ℓ and p
µ
r2s
ℓ`1 (needed to go on higher arities)
p
µ
r2s
ℓ`1
“
z1
pℓ`1q, z2
pℓ`1q
‰
“
´
p
µ
r2s
ℓ
“
z1
pℓq,1, z2
pℓq,1
‰
´ p
µ
r2s
ℓ
“`
z2
pℓq,2
˘˚ℓ
, z1
pℓq,2
‰¯
` p
µ
r2s
ℓ`1
”´
p
µ
r2s
ℓ
“
z2
pℓq,2, z1
pℓq,1
‰
` p
µ
r2s
ℓ
“
z1
pℓq,2,
`
z2
pℓq,1
˘˚ℓ
‰¯
, iℓ`1
ı
. (6.14)
Let us consider the example of quaternion construction in our notation.
Example 6.1. In the case of quaternions ℓ “ 1 and p
Aℓ`1 “ p
A2 “ H, while p
Aℓ “ p
A1 “ C. The recurrent
relations (6.12) for ℓ “ 0, 1 become
zp2q “ zp1q,1 ` zp1q,2 ¨2 i2, zp1q,1, zp1q,2 P p
A1 “ C, zp2q, i2 P H, (6.15)
zp1q,1 “ zp0q,11 ` zp0q,12 ¨1 i1, zp0q,11, zp0q,12 P p
A0 “ R, zp1q,1, i1 P C, (6.16)
zp1q,2 “ zp0q,21 ` zp0q,22 ¨1 i1, zp0q,21, zp0q,22 P p
A0 “ R, zp1q,2, i1 P C. (6.17)
– 16 –
17. Polyadic Cayley-Dickson process POLYADIC ANALOG OF CAYLEY-DICKSON CONSTRUCTION
After substitution (6.16)-(6.17) into (6.15) we obtain the expression of quaternion with real coefficients
zp0q,αβ P R, α, β “ 1, 2, and two imagery units (from different parts of H) i1 P CzR Ă H and i2 P HzC
zp2q “ zp1q,1 ` zp1q,2 ¨2 i2 “
`
zp0q,11 ` zp0q,12 ¨1 i1
˘
`
`
zp0q,21 ` zp0q,22 ¨1 i1
˘
¨2 i2
“ zp0q,11 ` zp0q,12 ¨1 i1 ` zp0q,21 ¨2 i2 `
`
zp0q,22 ¨1 i1
˘
¨2 i2. (6.18)
To return to the standard notation (4.25), we put zp2q “ q, zp0q,11 “ a, zp0q,12 “ b, zp0q,21 “ c, zp0q,22 “ d,
i1 “ i P C, i2 “ j P HzC, i1 ¨2 i2 “ k P H and get q “ a ` bi ` cj ` dk. Similarly, the complex-like
multiplication (6.13) can be also applied twice with ℓ “ 0, 1 to obtain the quaternion multiplication in
terms of real coefficients (4.27).
6.3. Polyadic Cayley-Dickson process. Now we provide generalization of the Cayley-Dickson con-
struction to the polyadic case in the framework of the second (embedding) approach using the field ex-
tension formalism (see Subsection 6.2). The main iteration relation (6.11) will now contain, instead of
binary hypercomplex algebras p
Aℓ, n-ary algebras p
Arnℓs
ℓ on each stage.
Definition 6.2. The polyadic Cayley-Dickson process is defined as the iteration of n-ary algebras
p
Arnℓ`1s
ℓ`1 “
A!
Arnℓs
ℓ piℓq
)
| p`q , p
µ
rnℓ`1s
ℓ`1
E
piℓ`1q ,
i2
ℓ “ ´1, i2
ℓ`1 “ ´1, iℓ`1iℓ “ ´iℓiℓ`1, iℓ P Arnℓ`1s
ℓ`1 , nℓ ě 2, ℓ ě 0. (6.19)
The concrete representation of the ℓ-th stage n-ary algebras p
Arnℓs
ℓ is not important for the general
recurrence formula (6.19). Nevertheless, here we will use the matrix polyadization to obtain higher n-ary
non-division algebras (Section 3). First, we need the obvious
Lemma 6.3. The embedding of a block-monomial matrix into a block-monomial matrix gives a block-
monomial matrix.
Corollary 6.4. If the binary Cayley-Dickson construction gives a division algebra, then the corresponding
polyadic Cayley-Dickson process gives a nonderived n-ary non-division algebra, because of noninvertible
elements.
Thus, the structure of general algebra obtained by the polyadic Cayley-Dickson process, is more rich,
than one block-shift monomial matrix (3.6), it is the “tower” of such monomial matrices of size pnℓ ´ 1qˆ
pnℓ ´ 1q on ℓ-th stage embedded one into another. The connection of near arities (and matrix sizes) is
nℓ`1 “ κ pnℓ ´ 1q ` 1, (6.20)
where κ is the polyadic power [ ].
Proposition 6.5. If number of stages of the polyadic Cayley-Dickson process is ℓ, then the dimension of
the final algebra is (cf. (6.8))
DCDpℓq “ DpArn0,n1,...,nℓs
CD q “ 2ℓ
pn0 ´ 1q pn1 ´ 1q . . . pnℓ ´ 1q (6.21)
where ni are arities of the intermediate algebras. The size of the final matrix becomes
pn0 ´ 1q pn1 ´ 1q . . . pnℓ ´ 1q ˆ pn0 ´ 1q pn1 ´ 1q . . . pnℓ ´ 1q . (6.22)
Proof. On the ℓ-th stage of the polyadic Cayley-Dickson process, each block-monomial pnℓ ´ 1q ˆ
pnℓ ´ 1q matrix has pnℓ ´ 1q nonzero blocks of the cycle-shift shape (3.6). The 0-th stage corresponds to
reals, which gives 20
pn0 ´ 1q parameters. Then, the simple field extension ℓ “ 1 (6.12) with blocks of
reals gives pn0 ´ 1q ¨ 2 pn1 ´ 1q parameters and so on. Thus, the ℓ-th stage is given by (6.21).
– 17 –
19. Complex number ternary division algebra POLYADIC ANALOG OF CAYLEY-DICKSON CONSTRUCTION
22
¨ 24 “ 96, because of two simple field extensions (6.15)–(6.18). This is 13-ary quaternion non-division
algebra. If there would be no higher stages of the Cayley-Dickson process, and the 24 ˆ 24 monomial
matrix would not be composed, having the form (3.6), then it will give 25-ary algebra.
Thus, we present the polyadic Cayley-Dickson process in the component form (6.12), but instead of
elements zpℓq of ℓ-th stage, we place Z-matrices (3.6) of suitable sizes. The resulting matrix is still
monomial, and so its determinant is proportional to the product of elements. The subalgebra of invertible
matrices corresponds to the division n-ary algebra.
Theorem 6.9. The invertible elements (which are described by Z-matrices with nonzero determinant) of
the polyadic Cayley-Dickson construction Drns
CD (for first stages ℓ “ 0, 1, 2, 3) (6.23) form the subalgebra
Drns
CD,div Ă Drns
CD which is the polyadic division n-ary algebra Drns
CD,div corresponding to the binary division
algebras D “ R, C, H, O.
To conclude, the matrix polyadization procedure applied for division algebras leads, in general, to non-
division algebras, because it is presented by the monomial matrices (representing nonzero noninvertible
elements) which become noninvertible when at least one entry vanishes. However, the subalgebras of
invertible elements can be considered as new n-ary division algebras.
7. POLYADIC IMAGINARY DIVISION ALGEBRAS
Here we introduce non-matrix polyadization procedure, which allows to obtain ternary division al-
gebras from the ordinary binary normed division algebras. Let us exploit the notations of the concrete
hyperembedding approach from Subsection 6.2. First, we show that some version of ternary division
algebra structure can be obtained by a new iterative process without introducing additional variables.
Definition 7.1. We define the ternary “imaginary tower” of algebras by
A
r3s
CD,ℓ`1 “ ACD piℓq ¨ iℓ`1 “
A
zpℓq piℓq ¨ iℓ`1
(
| p`q , µ
r3s
ℓ`1
E
, i2
ℓ “ ´1, iℓ`1iℓ “ ´iℓiℓ`1, ℓ ě 0.
(7.1)
The multiplication µ
r3s
ℓ`1 in (7.1) is nonderived ternary, because i3
ℓ “ ´iℓ for ℓ ě 1.
Theorem 7.2. If the initial algebra ACD piℓq is a normed division algebra, then its iterated imaginary
version A
r3s
CD,ℓ`1 (7.1) is the ternary division algebra of the same (initial) dimension Dpℓq and norm.
Assertion 7.3. The ternary algebras A
r3s
CD,ℓ`1 are not subalgebras of the initial algebras ACD piℓq.
Proof. This is because multiplications in the above algebras have different arities, despite the underlying
sets of imaginary algebras are subsets of the corresponding initial algebras.
Let us present the concrete expressions for the initial division algebras.
7.1. Complex number ternary division algebra. The first algebra (with ℓ “ 0) is the ternary division
algebra of pure imaginary complex numbers having the dimension D p0q “ D pRq “ 1
A
r3s
CD,1 ” C
r3s
“ R ¨ i1, zp1q “ bi1 P C, i2
1 “ ´1, b P R, (7.2)
which is unitless. The multiplication in C
r3s
is ternary nonderived and commutative
µ
r3s
1
“
z1
p1q, z2
p1q, z3
p1q
‰
“ ´b1
b2
b3
i1 P C
r3s
. (7.3)
– 19 –
20. POLYADIC IMAGINARY DIVISION ALGEBRAS Half-quaternion ternary division algebra
The norm is the absolute value in C (module)
›
›zp1q
›
›
p1q
“ |b|, and it is ternary multiplicative (from (7.3))
›
›
›µ
r3s
1
“
z1
p1q, z2
p1q, z3
p1q
‰›
›
›
p1q
“
›
›z1
p1q
›
›
p1q
›
›z2
p1q
›
›
p1q
›
›z3
p1q
›
›
p1q
P R, (7.4)
such that the corresponding map C
r3s
Ñ R is a ternary homomorphism. The querelement of zp1q (2.5) is
now defined by
µ
r3s
1
“
zp1q, zp1q, r
zp1q
‰
“ zp1q, (7.5)
which gives
r
zp1q “
1
zp1q
“ ´
i1
b
, b P Rz t0u . (7.6)
Thus, C
r3s
is a ternary commutative algebra, which is indeed a division algebra, because each nonzero
element has a querelement, i.e. it is invertible, so all equation of type (7.5) with different z1
s have solution.
7.2. Half-quaternion ternary division algebra. The next iteration case (ℓ “ 1) of the “imaginary
tower” (7.1) is more complicated and leads to pure imaginary ternary quaternions of the dimension
D p1q “ D pCq “ 2
A
r3s
CD,2 ” H
r3s
“ C pi1q ¨ i2, zp2q “ pc ` di1q i2 “ ci2 ` di1i2 P H
r3s
,
i2
1 “ i2
2 “ ´1, i1i2 “ ´i2i1, i1 P C, i2 P HzC, c, d P R. (7.7)
We informally can call H
r3s
the ternary algebra of imaginary “half-quaternions”, because in the standard
notation (see Example 6.1) zp2q from (7.7) is q “ a`bi`cj`dk
a“0,b“0
ÝÑ qhalf “ cj`dk. The nonderived
ternary algebra H
r3s
is obviously unitless. The multiplication of the half-quaternions H
r3s
µ
r3s
2
“
z1
p2q, z2
p2q, z3
p2q
‰
“ z1
p2q ¨ z2
p2q ¨ z3
p2q “ pd1
c2
d3
´ c1
d2
d3
´ d1
d2
c3
´ c1
c1
c3
q i2
` pc1
d2
c3
´ d1
c2
c3
´ c1
c2
d3
´ d1
d2
d3
q i1i2 P H
r3s
, (7.8)
is ternary nonderived (the algebra is closed with respect of product of 3 elements, but not less), noncom-
mutative and totally ternary associative (2.4)
µ
r3s
2
”
µ
r3s
2
“
z1
p2q, z2
p2q, z3
p2q
‰
z4
p2q, z41
p2q
ı
“ µ
r3s
2
”
z1
p2q, µ
r3s
2
“
z2
p2q, z3
p2qz4
p2q
‰
, z41
p2q
ı
“ µ
r3s
2
”
z1
p2q, z2
p2q, µ
r3s
2
“
z3
p2qz4
p2q, z41
p2q
‰ı
. (7.9)
The norm is defined by the absolute value in the quaternion algebra H in the standard way
›
›zp2q
›
›
p2q
“
ˇ
ˇzp2q
ˇ
ˇ “
?
c2 ` d2. (7.10)
The norm (7.10) is ternary multiplicative
›
›
›µ
r3s
2
“
z1
p2q, z2
p2q, z3
p2q
‰›
›
›
p2q
“
›
›z1
p2q
›
›
p2q
›
›z2
p2q
›
›
p2q
›
›z3
p2q
›
›
p2q
P R, (7.11)
such that the corresponding map H
r3s
Ñ R is a ternary homomorphism.
Proposition 7.4. Ternary analog of the sum of two squares identity is
pd1
c2
d3
´ c1
d2
d3
´ c1
c1
c3
´ d1
d2
c3
q
2
` pc1
d2
c3
´ d1
c2
c3
´ c1
c2
d3
´ d1
d2
d3
q
2
“
`
pc1
q2
` pd1
q2
˘ `
pc2
q2
` pd2
q2
˘ `
pc3
q2
` pd3
q2
˘
. (7.12)
– 20 –
21. Half-octonion ternary division algebra POLYADIC IMAGINARY DIVISION ALGEBRAS
Proof. It follows from the ternary multiplication (7.8) and the multiplicativity (7.11) of the norm (7.10).
Remark 7.5. The ternary sum of two squares identity (7.12) cannot be derived from the binary sum of
two squares (Diophantus, Fibonacci) identity or from Euler’s sum of four squares identity, while it can be
treated as the intermediate (triple) identity.
The querelement of zp2q (2.5) is now defined by
µ
r3s
2
“
zp2q, zp2q, r
zp2q
‰
“ zp2q, (7.13)
which gives
r
zp2q “ ´
ci2 ` di1i2
c2 ` d2
P H
r3s
, c2
` d2
‰ 0, c, d P R. (7.14)
It follows from (7.14) that H
r3s
(7.7) is a noncommutative nonderived ternary algebra, which is indeed
a division algebra, because each nonzero element zp2q has a querelement r
zp2q, i.e. it is invertible, and
equation of type (7.5) have solutions.
7.3. Half-octonion ternary division algebra. The next case (ℓ “ 2) gives pure imaginary ternary octo-
nions of the dimension D p2q “ D pHq “ 4
A
r3s
CD,3 ” O
r3s
“ H pi1, i2q ¨ i3, zp3q “ pa ` bi1 ` ci2 ` di1i2q i3 “ ai3 ` bi1i3 ` ci2i3 ` di1i2i3,
i2
1 “ i2
2 “ i2
3 “ ´1, i1i2 “ ´i2i1, i1i3 “ ´i3i1, i2i3 “ ´i3i2,
i1 P C, i2 P HzC, i3 P OzH, a, b, c, d P R. (7.15)
We informally can call O
r3s
the nonderived ternary algebra of imaginary “half-octonions” which is ob-
viously unitless. In the standard notation (with seven imaginary units e1 . . . e7) the “half-octonion” is
ohalf “ zp3q “ ae4 ` be5 ` ce6 ` de7.
It is well-known (see, e.g. BAEZ [2002], DRAY AND MANOGUE [2015]), that the algebra of octonions
O is not a field, but a special nonassociative loop (quasigroup with an identity), the Moufang loop (see,
e.g. BRUCK AND KLEINFELD [1951], PUMPLÜN [2014]). Because the ternary algebra O
r3s
is unitless, it
cannot be a ternary loop.
In this way, we introduce the ternary multiplication for “half-octonions” not by a triple product in O
(as in (7.8)) which is not unique due to nonassociativity, but in the following (unique) way
µ
r3s
3
“
z1
p3q, z2
p3q, z3
p3q
‰
“
´
z1
p3q ‚ z2
p3q
¯
‚ z3
p3q ` z1
p3q ‚
´
z2
p3q ‚ z3
p3q
¯
2
, (7.16)
where p‚q is the binary product in O.
Proposition 7.6. The nonderived nonunital ternary algebra of imaginary “half-octonions” (7.15)
O
r3s
“
A
zp3q
(
| p`q , µ
r3s
3
E
(7.17)
is closed under the multiplication (7.16) and ternary nonassociative.
– 21 –
22. POLYADIC IMAGINARY DIVISION ALGEBRAS Half-octonion ternary division algebra
In components the multiplication (7.16) becomes
µ
r3s
3
“
z1
p3q, z2
p3q, z3
p3q
‰
“ pa2
pb1
b3
` c1
c3
` d1
d3
q ´ a3
pb1
b2
` c1
c2
` d1
d2
q ` a1
pb2
b3
` c2
c3
` d2
d3
q ´ a1
a2
a3
q i3
` pb2
pa1
a3
` c1
c3
` d1
d3
q ` b3
pa1
a2
` c1
c2
` d1
d2
q ´ b1
pa2
a3
` c2
c3
` d2
d3
q ´ b1
b2
b3
q i1i3
` pc2
pa1
a3
` b1
b3
` d1
d3
q ´ c3
pa1
a2
` b1
b2
` d1
d2
q ´ c1
pa2
a3
` b2
b3
` d2
d3
q ´ c1
c2
c3
q i2i3
` pd2
pa1
a3
` b1
b3
` c1
c3
q ´ d3
pa1
a2
` b1
b2
` c1
c2
q ´ d1
pa2
a3
` b2
b3
` c2
c3
q ´ d1
d2
d3
q i1i2i3.
(7.18)
The querelement r
zp3q (2.5) of zp3q from O
r3s
is defined by
µ
r3s
3
“
zp3q, zp3q, r
zp3q
‰
“ zp3q, (7.19)
where r
zp3q can be on any place. In components we obtain (cf. half-quaternions (7.14))
r
zp3q “ ´
ai3 ` bi1i3 ` ci2i3 ` di1i2i3
a2 ` b2 ` c2 ` d2
P O
r3s
, a2
` b2
` c2
` d2
‰ 0, a, b, c, d P R. (7.20)
It follows from (7.20) that O
r3s
is a nonderived noncommutative nonassociative ternary algebra, which is
indeed a division algebra, because each nonzero element zp3q has a querelement r
zp3q, i.e. it is invertible.
The algebra of (binary) octonions O is nonassociative, but has a very close property, it is alternative.
Recall that a (binary) algebra A is left and right alternative if two relations hold valid
pxxq y “ x pxyq , (7.21)
x pyyq “ pxyq y, x, y P A. (7.22)
The ternary analog of alternativity requires three set of two relations, in general for n-ary algebra we
will need n sets by pn ´ 1q relations, i.e. in total n pn ´ 1q relations.
Definition 7.7. For a nonassociative ternary algebra Ar3s
`
p`q , µr3s
˘
we define the ternary alternativity
by 3 sets (left, middle and right alternativity) of 2 relations
µr3s
“
µr3s
rx, x, xs , y, t
‰
“ µr3s
“
x, µr3s
rx, x, ys , t
‰
“ µr3s
“
x, x, µr3s
rx, y, ts
‰
, (7.23)
µr3s
“
x, µr3s
ry, y, ys , t
‰
“ µr3s
“
x, y, µr3s
ry, y, ts
‰
“ µr3s
“
µr3s
rx, y, ys , y, t
‰
, (7.24)
µr3s
“
x, y, µr3s
rt, t, ts
‰
“ µr3s
“
x, µr3s
ry, t, ts , t
‰
“ µr3s
“
µr3s
rx, y, ts , t, t
‰
, x, y, t P Ar3s
. (7.25)
Obviously, that a ternary associative algebra (satisfying (7.9)) is ternary alternative, but not vise versa.
Proposition 7.8. The ternary algebra of imaginary “half-octonions” O
r3s
with the product (7.18) is
ternary alternative, i.e. it satisfies (7.23)–(7.25).
Thus, we have constructed the noncommutative nonassociative nonderived ternary division algebra of
half-octonions O
r3s
which is nonunital and ternary alternative.
– 22 –
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– 23 –