This document summarizes a talk on localization and deformation quantization. The talk is organized into sections on localization, deformation quantization, and results. Localization is discussed for both the commutative and noncommutative cases, including Ore conditions. Deformation quantization covers star products and concrete localization examples. The overall goal is to discuss noncommutative localization in the context of deformation quantization.
Contact geometry is the study of certain geometric structures on odd dimensional smooth manifolds. A contact structure is a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non-integrability. The associated one form is called a contact form and uniquely determines a Hamiltonian-like vector field called the Reeb vector field on the manifold. I will give some background on this subject, including motivation from classical mechanics. I will also explain how to make use of J-holomorphic curves to obtain a Floer theoretic contact invariant, contact homology, whose chain complex is generated by closed Reeb orbits. This talk will feature numerous graphics to acclimate people to the realm of contact geometry.
Gabriel Gaudreault presents a functional calculus called λ↔-calculus for doing semantics with pregroup grammars. The calculus allows functional composition as the main reduction operation and respects the incoming direction of inputs. There is a 1-1 correspondence between the calculus and a subset of pregroup grammars relevant for linguistic analysis. The calculus can be used for syntactic and semantic analysis of natural language examples. Future work includes further analysis of the semantic power of the calculus and proving properties like Church-Rosser.
1. The document extends Ehrenfest's theorem to weak values by showing that the semiclassical evaluation of weak values agrees with classical trajectories within the semiclassical accuracy.
2. When the contribution from a unique classical trajectory dominates without quantum interference, the weak value is equal to the observable evaluated along that classical trajectory plus higher order corrections.
3. This suggests that some "anomalous" weak values outside the observable's eigenvalues can be understood classically without invoking quantum effects like interference or entanglement.
Some fundamental theorems in Banach spaces and Hilbert spacesSanjay Sharma
This document provides an overview of functional analysis and some fundamental theorems in Banach and Hilbert spaces. It discusses how functional analysis studies topological-algebraic structures and their applications in mathematics and sciences. It also summarizes key definitions like normed linear spaces and Hilbert spaces. Some fundamental theorems covered include the Hahn-Banach theorem, open mapping theorem, closed graph theorem, Banach-Steinhaus theorem, and Riesz representation theorem.
This document outlines the thesis presentation of Frank Romascavage III on deriving an explicit formula for the mean square of a Dirichlet L-function of prime power modulus. The introduction reviews previous work on power moments of the Riemann zeta function and Dirichlet L-functions. It also motivates studying power moments and notes the main theorem will not include an error term. The outline lists the introduction, main theorem, and sections on the diagonal sum, near-diagonal sums, and off-diagonal sums.
This document discusses properties of injective modules over various non-commutative algebras. It is noted that injective hulls of simple modules over Down-Up algebras may or may not be locally Artinian depending on the algebra, with examples given of algebras where the injective hulls are and are not locally Artinian. The document also examines related properties for Weyl algebras, quantum planes, Heisenberg algebras, and other algebras.
The document discusses abstract model checking techniques for verifying temporal logic properties on abstract models derived from concrete models using abstraction. It presents two dual methods - the classic method and over-approximation method - for defining satisfiability relations on abstract models and analyzes their properties and relationships. The document also proposes techniques for dealing with imprecision and incompleteness that can arise from working with abstract models.
This document summarizes research on Higher Dimensional Automata (HDA) and modal logics for reasoning about concurrency in HDA models. It includes:
- An introduction to HDA and geometrical concurrency models introduced by Pratt and van Glabbeek.
- A discussion of Higher Dimensional Modal Logic (HDML) for reasoning about concurrency in HDA models. HDML is extended to history-aware HDML (hHDML) to capture hereditary history preserving bisimulation.
- The use of ST-configuration structures to represent HDA models, along with examples showing the expressiveness of hHDML compared to other bisimulations and logics.
Contact geometry is the study of certain geometric structures on odd dimensional smooth manifolds. A contact structure is a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non-integrability. The associated one form is called a contact form and uniquely determines a Hamiltonian-like vector field called the Reeb vector field on the manifold. I will give some background on this subject, including motivation from classical mechanics. I will also explain how to make use of J-holomorphic curves to obtain a Floer theoretic contact invariant, contact homology, whose chain complex is generated by closed Reeb orbits. This talk will feature numerous graphics to acclimate people to the realm of contact geometry.
Gabriel Gaudreault presents a functional calculus called λ↔-calculus for doing semantics with pregroup grammars. The calculus allows functional composition as the main reduction operation and respects the incoming direction of inputs. There is a 1-1 correspondence between the calculus and a subset of pregroup grammars relevant for linguistic analysis. The calculus can be used for syntactic and semantic analysis of natural language examples. Future work includes further analysis of the semantic power of the calculus and proving properties like Church-Rosser.
1. The document extends Ehrenfest's theorem to weak values by showing that the semiclassical evaluation of weak values agrees with classical trajectories within the semiclassical accuracy.
2. When the contribution from a unique classical trajectory dominates without quantum interference, the weak value is equal to the observable evaluated along that classical trajectory plus higher order corrections.
3. This suggests that some "anomalous" weak values outside the observable's eigenvalues can be understood classically without invoking quantum effects like interference or entanglement.
Some fundamental theorems in Banach spaces and Hilbert spacesSanjay Sharma
This document provides an overview of functional analysis and some fundamental theorems in Banach and Hilbert spaces. It discusses how functional analysis studies topological-algebraic structures and their applications in mathematics and sciences. It also summarizes key definitions like normed linear spaces and Hilbert spaces. Some fundamental theorems covered include the Hahn-Banach theorem, open mapping theorem, closed graph theorem, Banach-Steinhaus theorem, and Riesz representation theorem.
This document outlines the thesis presentation of Frank Romascavage III on deriving an explicit formula for the mean square of a Dirichlet L-function of prime power modulus. The introduction reviews previous work on power moments of the Riemann zeta function and Dirichlet L-functions. It also motivates studying power moments and notes the main theorem will not include an error term. The outline lists the introduction, main theorem, and sections on the diagonal sum, near-diagonal sums, and off-diagonal sums.
This document discusses properties of injective modules over various non-commutative algebras. It is noted that injective hulls of simple modules over Down-Up algebras may or may not be locally Artinian depending on the algebra, with examples given of algebras where the injective hulls are and are not locally Artinian. The document also examines related properties for Weyl algebras, quantum planes, Heisenberg algebras, and other algebras.
The document discusses abstract model checking techniques for verifying temporal logic properties on abstract models derived from concrete models using abstraction. It presents two dual methods - the classic method and over-approximation method - for defining satisfiability relations on abstract models and analyzes their properties and relationships. The document also proposes techniques for dealing with imprecision and incompleteness that can arise from working with abstract models.
This document summarizes research on Higher Dimensional Automata (HDA) and modal logics for reasoning about concurrency in HDA models. It includes:
- An introduction to HDA and geometrical concurrency models introduced by Pratt and van Glabbeek.
- A discussion of Higher Dimensional Modal Logic (HDML) for reasoning about concurrency in HDA models. HDML is extended to history-aware HDML (hHDML) to capture hereditary history preserving bisimulation.
- The use of ST-configuration structures to represent HDA models, along with examples showing the expressiveness of hHDML compared to other bisimulations and logics.
FACTORIZATION OF OPERATORS AND VECTOR MEASURESesasancpe
This document discusses factorization of operators and vector measures. It begins by stating that an operator on a Banach function space always defines a vector measure, and a vector measure is related to a Banach function space. This provides a unified viewpoint for studying vector measures, operators, and spaces of integrable functions. Some key results mentioned include factorization theorems for operators, analysis of the integration map, and geometric properties related to the structure of L1 spaces of vector measures. Specific topics covered include Riesz representation theorems, Radon-Nikodym theorems, L1 spaces of vector measures, and extensions of operators through semivariation inequalities.
Presentation of calculus on application of derivativeUrwaArshad1
The document discusses the history and applications of derivatives. It begins by covering pioneers in derivative mathematics such as Aryabhata, Bhaskara, Gottfried Leibniz, Isaac Newton, and Sharaf al-Din al-Tusi. It then defines derivatives formally and informally, discusses rules like the product rule and chain rule, and gives examples of derivatives in sciences and daily life such as physics, biology, and analyzing graphs. The document concludes that the use of derivatives is increasing across many fields and professions.
This document describes work on simply typed lambda calculus modulo type isomorphisms. It introduces an equivalence relation between types based on isomorphisms for conjunction, implication, and their interactions. This allows identifying isomorphic types and terms with the same type up to isomorphism. The document outlines the challenges of defining a type-isomorphic proof theory and presents solutions, such as Church-style projections and alpha equivalence rules, to develop a sound and complete operational semantics for the simply typed lambda calculus modulo type isomorphisms.
Vector measures and classical disjointification methodsesasancpe
1. The document discusses applying classical disjointification methods (Bessaga-Pelczynski and Kadecs-Pelczynski) to spaces of p-integrable functions with respect to vector measures.
2. These methods allow working with orthogonality notions in the range space and analyzing disjoint functions.
3. Combining the results provides tools to analyze the structure of subspaces in these spaces of p-integrable functions.
This document discusses different notions of convergence for sequences of graphs as studied in graph theory, statistical physics, and probability. It addresses three main notions of convergence for both dense and sparse graphs:
1) Left convergence, which requires subgraph counts to converge.
2) Convergence of quotients, which requires properties like MaxCut to converge as graphs are colored and collapsed.
3) Right convergence, which requires free energies of graphical models on the graphs to converge.
For sparse graphs with bounded degrees, the document shows these three notions are not equivalent, and introduces a new notion of large deviation convergence, which implies the other three notions. The large deviation principle characterizes the probability distribution of random color
This document discusses various topics related to Fourier series and partial differential equations, including:
- Periodic functions and their properties.
- Fourier series representations of functions over intervals, including the calculation of Fourier coefficients.
- Using Fourier series to solve partial differential equations, including first and second order equations.
- Applications of Fourier series such as image compression using the discrete cosine transform.
This document summarizes a research paper that proposes using symbolic determinants of adjacency matrices to solve the Hamiltonian Cycle Problem (HCP). The HCP involves finding a cycle in a graph that visits each vertex exactly once. The paper shows that the HCP can be reduced to solving a system of polynomial equations related to the graph's adjacency matrix. Specifically, it represents the matrix symbolically and derives equations constraining the symbols to represent a Hamiltonian cycle. Solving this system of equations determines if the original graph has a Hamiltonian cycle.
A Unifying theory for blockchain and AILonghow Lam
This document proposes a unifying theory connecting blockchain and artificial intelligence technologies. It introduces the Lam-Visser theory and how it fits within the Damhof Quadrants framework. The document provides definitions related to the main result, which states that there exists a minimal, ultra-connected, almost everywhere linear and generic solvable, semi-countable polytope if a certain condition is met. It then discusses applications of this theory to questions of associativity and the computation of analytically independent subalgebras.
Discussion of Fearnhead and Prangle, RSS< Dec. 14, 2011Christian Robert
The document discusses approximate Bayesian computation (ABC), a technique used when the likelihood function is intractable. ABC works by simulating data under different parameter values and accepting simulations that are close to the observed data according to a distance measure. The key challenges are choosing a sufficient summary statistic of the data and setting the tolerance level. Later sections discuss using a noisy ABC approach, where the summary statistic is perturbed, and calibrating the method so that the ABC posterior converges to the true parameter as the number of simulations increases. The document examines issues around choosing optimal summary statistics and tolerance levels to minimize errors in the ABC approximation.
Talk at Workshop "Supergeometry and Applications", Luxemburg, Dec. 2017.
Plan:
1. Regular mappings and semisupermanifolds
2. Superconformal-like transformations twisting parity of tangent space
3. Ternary supersymmetry
4. Polyadic analogs of integer number ring Z and field Z/pZ
This document discusses using the sequence of iterates generated by inertial methods to minimize convex functions. It introduces inertial methods and how they can be used to generate sequences that converge to the minimum. While the last iterate is often used, sometimes averaging over iterates or using extrapolations like Aitken acceleration can provide better estimates of the minimum. Inertial methods allow for more exploration of the function space than gradient descent alone. The geometry of the function may provide opportunities to analyze the iterate sequence and obtain improved convergence estimates.
Random knots can be modeled by taking random Fourier series or connecting random points on the unit sphere. For the Fourier model, the limiting curve is continuously differentiable if the Fourier coefficients decay quickly enough. The expected number of self-intersections is also finite in this case. For both models, the Alexander polynomial coefficients appear to concentrate on the unit circle, unlike polynomials with random coefficients. Different random knot models may produce different topological and geometric properties worth further study. Computation of invariants like the Alexander polynomial remains challenging for models with many segments.
This document discusses rational ergodicity properties of rank-one transformations. It proves that all rank-one transformations are subsequence boundedly rationally ergodic, and that there exist rank-one transformations that are not weakly rationally ergodic. It also proves that rank-one transformations with a bounded sequence of cuts satisfy the stronger property of being boundedly rationally ergodic. The document provides background on rational ergodicity, rank-one transformations, and cutting and stacking constructions.
Workshop on Bayesian Inference for Latent Gaussian Models with ApplicationsChristian Robert
ABC methods provide a way to perform Bayesian inference when the likelihood function is intractable or impossible to compute directly. The basic ABC algorithm works by simulating parameters from the prior and simulating data from those parameters, accepting the parameters if the simulated data is "close" to the actual observed data according to some distance measure and tolerance level.
Later advances include ABC-MCMC which uses an MCMC approach to sample from the posterior, and ABC-NP which adjusts the parameters to better match the observed data rather than rejecting simulations. Other variants such as ABC-SMC and ABC-μ extend the framework to include sequential Monte Carlo methods or jointly model the intractable parameters. Overall, ABC methods provide a
This document discusses tests for determining if infinite series converge or diverge, including:
1) The Alternating Series Test, which can be used to determine if alternating series (with terms that are alternately positive and negative) converge.
2) Absolute and conditional convergence, where a series is absolutely convergent if the sum of the absolute values converges, and conditionally convergent if the series converges but the sum of absolute values does not.
3) The Ratio and Root Tests, which can be used to determine if series converge absolutely (and thus converge) based on the ratios or roots of successive terms.
An Analysis and Study of Iteration Proceduresijtsrd
In computational mathematics, an iterative method is a scientific technique that utilizes an underlying speculation to produce a grouping of improving rough answers for a class of issues, where the n th estimate is gotten from the past ones. A particular execution of an iterative method, including the end criteria, is a calculation of the iterative method. An iterative method is called joined if the relating grouping meets for given starting approximations. A scientifically thorough combination investigation of an iterative method is typically performed notwithstanding, heuristic based iterative methods are additionally normal. This Research provides a survey of iteration procedures that have been used to obtain fixed points for maps satisfying a variety of contractive conditions. Dr. R. B. Singh | Shivani Tomar ""An Analysis and Study of Iteration Procedures"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019, URL: https://www.ijtsrd.com/papers/ijtsrd23715.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/computational-science/23715/an-analysis-and-study-of-iteration-procedures/dr-r-b-singh
The document discusses concepts from abstract algebra and topology, including:
- Extending previous results on constructing matrices and graphs to other mathematical objects.
- Defining terms like affine planes, isometries, and homomorphisms.
- Stating theorems about relationships between these concepts, like one stating conditions under which a modulus is analytically separable.
- Citing many previous works in the field and discussing how the present work relates to and builds upon past results.
This document discusses the concept of "generic" elements in infinite groups like mapping class groups and SL(n,Z). It summarizes Kapovich's question about whether a generic element of a mapping class group is pseudo-Anosov, and explains how this relates to Thurston's classification of surface automorphisms. It then discusses different interpretations of what "generic" could mean, and presents the theorem that under one interpretation, a generic matrix in SL(n,Z) or Sp(2g,Z) satisfies certain properties like having an irreducible characteristic polynomial. Experimental results are also presented supporting the theorem.
This document summarizes and reinterprets the proof of the Riemannian Penrose inequality by H. Bray. It introduces a normalized conformal flow that can be interpreted as a model of time evolution in a virtual spacetime. This gives a physical interpretation to Bray's proof by viewing the conformal flow as modeling a virtual gravitational collapse process, with the area of the apparent horizon increasing over time. As an example, the normalized conformal flow is applied to a Schwarzschild slice, showing features consistent with gravitational collapse such as a decreasing expansion rate on the apparent horizon over time.
Optimal order a posteriori error bounds in L∞(L2) norm are derived for semidiscrete semilinear parabolic problems. Standard continuous Galerkin (conforming) finite element method is employed. Our main tools in deriving these error estimates are the elliptic reconstruction technique which is first introduced by Makridakis and Nochetto [5], with the aid of Gronwall’s lemma and continuation argument.
FACTORIZATION OF OPERATORS AND VECTOR MEASURESesasancpe
This document discusses factorization of operators and vector measures. It begins by stating that an operator on a Banach function space always defines a vector measure, and a vector measure is related to a Banach function space. This provides a unified viewpoint for studying vector measures, operators, and spaces of integrable functions. Some key results mentioned include factorization theorems for operators, analysis of the integration map, and geometric properties related to the structure of L1 spaces of vector measures. Specific topics covered include Riesz representation theorems, Radon-Nikodym theorems, L1 spaces of vector measures, and extensions of operators through semivariation inequalities.
Presentation of calculus on application of derivativeUrwaArshad1
The document discusses the history and applications of derivatives. It begins by covering pioneers in derivative mathematics such as Aryabhata, Bhaskara, Gottfried Leibniz, Isaac Newton, and Sharaf al-Din al-Tusi. It then defines derivatives formally and informally, discusses rules like the product rule and chain rule, and gives examples of derivatives in sciences and daily life such as physics, biology, and analyzing graphs. The document concludes that the use of derivatives is increasing across many fields and professions.
This document describes work on simply typed lambda calculus modulo type isomorphisms. It introduces an equivalence relation between types based on isomorphisms for conjunction, implication, and their interactions. This allows identifying isomorphic types and terms with the same type up to isomorphism. The document outlines the challenges of defining a type-isomorphic proof theory and presents solutions, such as Church-style projections and alpha equivalence rules, to develop a sound and complete operational semantics for the simply typed lambda calculus modulo type isomorphisms.
Vector measures and classical disjointification methodsesasancpe
1. The document discusses applying classical disjointification methods (Bessaga-Pelczynski and Kadecs-Pelczynski) to spaces of p-integrable functions with respect to vector measures.
2. These methods allow working with orthogonality notions in the range space and analyzing disjoint functions.
3. Combining the results provides tools to analyze the structure of subspaces in these spaces of p-integrable functions.
This document discusses different notions of convergence for sequences of graphs as studied in graph theory, statistical physics, and probability. It addresses three main notions of convergence for both dense and sparse graphs:
1) Left convergence, which requires subgraph counts to converge.
2) Convergence of quotients, which requires properties like MaxCut to converge as graphs are colored and collapsed.
3) Right convergence, which requires free energies of graphical models on the graphs to converge.
For sparse graphs with bounded degrees, the document shows these three notions are not equivalent, and introduces a new notion of large deviation convergence, which implies the other three notions. The large deviation principle characterizes the probability distribution of random color
This document discusses various topics related to Fourier series and partial differential equations, including:
- Periodic functions and their properties.
- Fourier series representations of functions over intervals, including the calculation of Fourier coefficients.
- Using Fourier series to solve partial differential equations, including first and second order equations.
- Applications of Fourier series such as image compression using the discrete cosine transform.
This document summarizes a research paper that proposes using symbolic determinants of adjacency matrices to solve the Hamiltonian Cycle Problem (HCP). The HCP involves finding a cycle in a graph that visits each vertex exactly once. The paper shows that the HCP can be reduced to solving a system of polynomial equations related to the graph's adjacency matrix. Specifically, it represents the matrix symbolically and derives equations constraining the symbols to represent a Hamiltonian cycle. Solving this system of equations determines if the original graph has a Hamiltonian cycle.
A Unifying theory for blockchain and AILonghow Lam
This document proposes a unifying theory connecting blockchain and artificial intelligence technologies. It introduces the Lam-Visser theory and how it fits within the Damhof Quadrants framework. The document provides definitions related to the main result, which states that there exists a minimal, ultra-connected, almost everywhere linear and generic solvable, semi-countable polytope if a certain condition is met. It then discusses applications of this theory to questions of associativity and the computation of analytically independent subalgebras.
Discussion of Fearnhead and Prangle, RSS< Dec. 14, 2011Christian Robert
The document discusses approximate Bayesian computation (ABC), a technique used when the likelihood function is intractable. ABC works by simulating data under different parameter values and accepting simulations that are close to the observed data according to a distance measure. The key challenges are choosing a sufficient summary statistic of the data and setting the tolerance level. Later sections discuss using a noisy ABC approach, where the summary statistic is perturbed, and calibrating the method so that the ABC posterior converges to the true parameter as the number of simulations increases. The document examines issues around choosing optimal summary statistics and tolerance levels to minimize errors in the ABC approximation.
Talk at Workshop "Supergeometry and Applications", Luxemburg, Dec. 2017.
Plan:
1. Regular mappings and semisupermanifolds
2. Superconformal-like transformations twisting parity of tangent space
3. Ternary supersymmetry
4. Polyadic analogs of integer number ring Z and field Z/pZ
This document discusses using the sequence of iterates generated by inertial methods to minimize convex functions. It introduces inertial methods and how they can be used to generate sequences that converge to the minimum. While the last iterate is often used, sometimes averaging over iterates or using extrapolations like Aitken acceleration can provide better estimates of the minimum. Inertial methods allow for more exploration of the function space than gradient descent alone. The geometry of the function may provide opportunities to analyze the iterate sequence and obtain improved convergence estimates.
Random knots can be modeled by taking random Fourier series or connecting random points on the unit sphere. For the Fourier model, the limiting curve is continuously differentiable if the Fourier coefficients decay quickly enough. The expected number of self-intersections is also finite in this case. For both models, the Alexander polynomial coefficients appear to concentrate on the unit circle, unlike polynomials with random coefficients. Different random knot models may produce different topological and geometric properties worth further study. Computation of invariants like the Alexander polynomial remains challenging for models with many segments.
This document discusses rational ergodicity properties of rank-one transformations. It proves that all rank-one transformations are subsequence boundedly rationally ergodic, and that there exist rank-one transformations that are not weakly rationally ergodic. It also proves that rank-one transformations with a bounded sequence of cuts satisfy the stronger property of being boundedly rationally ergodic. The document provides background on rational ergodicity, rank-one transformations, and cutting and stacking constructions.
Workshop on Bayesian Inference for Latent Gaussian Models with ApplicationsChristian Robert
ABC methods provide a way to perform Bayesian inference when the likelihood function is intractable or impossible to compute directly. The basic ABC algorithm works by simulating parameters from the prior and simulating data from those parameters, accepting the parameters if the simulated data is "close" to the actual observed data according to some distance measure and tolerance level.
Later advances include ABC-MCMC which uses an MCMC approach to sample from the posterior, and ABC-NP which adjusts the parameters to better match the observed data rather than rejecting simulations. Other variants such as ABC-SMC and ABC-μ extend the framework to include sequential Monte Carlo methods or jointly model the intractable parameters. Overall, ABC methods provide a
This document discusses tests for determining if infinite series converge or diverge, including:
1) The Alternating Series Test, which can be used to determine if alternating series (with terms that are alternately positive and negative) converge.
2) Absolute and conditional convergence, where a series is absolutely convergent if the sum of the absolute values converges, and conditionally convergent if the series converges but the sum of absolute values does not.
3) The Ratio and Root Tests, which can be used to determine if series converge absolutely (and thus converge) based on the ratios or roots of successive terms.
An Analysis and Study of Iteration Proceduresijtsrd
In computational mathematics, an iterative method is a scientific technique that utilizes an underlying speculation to produce a grouping of improving rough answers for a class of issues, where the n th estimate is gotten from the past ones. A particular execution of an iterative method, including the end criteria, is a calculation of the iterative method. An iterative method is called joined if the relating grouping meets for given starting approximations. A scientifically thorough combination investigation of an iterative method is typically performed notwithstanding, heuristic based iterative methods are additionally normal. This Research provides a survey of iteration procedures that have been used to obtain fixed points for maps satisfying a variety of contractive conditions. Dr. R. B. Singh | Shivani Tomar ""An Analysis and Study of Iteration Procedures"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019, URL: https://www.ijtsrd.com/papers/ijtsrd23715.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/computational-science/23715/an-analysis-and-study-of-iteration-procedures/dr-r-b-singh
The document discusses concepts from abstract algebra and topology, including:
- Extending previous results on constructing matrices and graphs to other mathematical objects.
- Defining terms like affine planes, isometries, and homomorphisms.
- Stating theorems about relationships between these concepts, like one stating conditions under which a modulus is analytically separable.
- Citing many previous works in the field and discussing how the present work relates to and builds upon past results.
This document discusses the concept of "generic" elements in infinite groups like mapping class groups and SL(n,Z). It summarizes Kapovich's question about whether a generic element of a mapping class group is pseudo-Anosov, and explains how this relates to Thurston's classification of surface automorphisms. It then discusses different interpretations of what "generic" could mean, and presents the theorem that under one interpretation, a generic matrix in SL(n,Z) or Sp(2g,Z) satisfies certain properties like having an irreducible characteristic polynomial. Experimental results are also presented supporting the theorem.
This document summarizes and reinterprets the proof of the Riemannian Penrose inequality by H. Bray. It introduces a normalized conformal flow that can be interpreted as a model of time evolution in a virtual spacetime. This gives a physical interpretation to Bray's proof by viewing the conformal flow as modeling a virtual gravitational collapse process, with the area of the apparent horizon increasing over time. As an example, the normalized conformal flow is applied to a Schwarzschild slice, showing features consistent with gravitational collapse such as a decreasing expansion rate on the apparent horizon over time.
Optimal order a posteriori error bounds in L∞(L2) norm are derived for semidiscrete semilinear parabolic problems. Standard continuous Galerkin (conforming) finite element method is employed. Our main tools in deriving these error estimates are the elliptic reconstruction technique which is first introduced by Makridakis and Nochetto [5], with the aid of Gronwall’s lemma and continuation argument.
The document describes five main families of functions - linear, power, root, reciprocal, and absolute value functions. It provides the name, equation, domain and range for each type of function. It also discusses concepts like piecewise functions, average rate of change, transformations, combinations of functions, and variations.
The document is a presentation about using model theory to prove Hilbert's Weak Nullstellensatz. It begins with introductions to model theory, including definitions of structures, embeddings, elementary extensions, theories, and model-completeness. It then states that the theory of algebraically closed fields has model-completeness. The presentation concludes with a proof of the Weak Nullstellensatz using these model-theoretic concepts, showing there is a tuple in an algebraically closed field that satisfies a given ideal of polynomials.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
This document discusses outgoing and ingoing Klein-Gordon waves near the event horizons of black holes. It first presents the Klein-Gordon equation of motion in the background of the Schwarzschild spacetime metric. Near the event horizon, the radial equation is approximated in the Regge-Wheeler coordinate, leading to oscillatory solutions. The time and radial solutions are then expressed in outgoing and ingoing coordinates, resulting in outgoing and ingoing waves with different analytic properties in the future and past event horizons.
The document discusses various topics related to differential equations including:
1. Linear differential equations of nth order with constant coefficients and the characteristic equation. The general solution depends on whether the roots of the characteristic equation are real/complex, distinct/multiple.
2. Finding the complementary function and particular integral to get the general solution for a second order differential equation with constant coefficients.
3. Solving simultaneous linear differential equations using the D-operator method and Laplace transform method.
4. Changing dependent and independent variables to solve second order differential equations numerically using an approximation technique similar to the trapezoid rule.
The document discusses key concepts in Laplace transforms including:
1) The Laplace transform is defined as an integral transform that transforms a function of time into a function of a complex variable, simplifying analysis of differential equations.
2) Important properties include the Laplace transforms of derivatives and integrals, which allow transforming differential equations into algebraic equations.
3) The existence theorem guarantees a unique solution to initial value problems under certain conditions on the function.
This document summarizes key concepts related to solving the Schrodinger equation for the hydrogen atom. It begins by introducing the hydrogen atom as an important system to study and then presents the time-independent Schrodinger equation in spherical coordinates. It identifies the complete set of commuting observables for the hydrogen atom as the Hamiltonian, angular momentum, and z-component of angular momentum operators. It then separates the radial and angular dependencies of the wavefunction and shows that this leads to separate differential equations that can be solved.
FDM is an older method than FEM that requires less computational power but is also less accurate in some cases where higher-order accuracy is required. FEM permit to get a higher order of accuracy, but requires more computational power and is also more exigent on the quality of the mesh.29-Jun-2017
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What's the difference between FEM and FDM? - FEA for All
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This document summarizes research on encoding Reiter's solution to the frame problem in modal logic. Specifically, it presents a modal logic counterpart to Reiter's regression technique. The paper introduces a version of deterministic PDL with quantification over actions and equality. It then describes how Reiter's approach can be encoded in this logic by representing action preconditions, possible causes of state changes, and successor state axioms that enable regression. The paper claims this provides a way to perform reasoning about actions using a modal logic framework with computational advantages over the Situation Calculus.
This document provides an outline for a lecture on discrete mathematics. It introduces topics like logic, set theory, mathematical reasoning/proof techniques, propositional/predicate calculus, Boolean algebra, induction, algorithms, recursion, counting/probability, and graph theory. The lecture begins with an introduction to logic, including statements, propositions, truth values, and logical operators like negation, conjunction, disjunction, implication, and biconditional. It provides examples of combining propositions using logical operators and truth tables. It also discusses propositional functions, quantification, and counterexamples.
The document discusses linear transformations and their applications in mathematics for artificial intelligence. It begins by introducing linear transformations and how matrices can be used to define functions. It describes how a matrix A can define a linear transformation fA that maps vectors in Rn to vectors in Rm. It also defines key concepts for linear transformations like the kernel, range, row space, and column space. The document will continue exploring topics like the derivative of transformations, linear regression, principal component analysis, and singular value decomposition.
Gauge field theory describes fundamental interactions through the principle of local gauge invariance. Quantum mechanics respects the gauge invariance of electromagnetic fields by requiring a simultaneous change in phase of the wavefunction under gauge transformations of potentials. Insisting on local gauge freedom in quantum mechanics forces the introduction of gauge fields that interact with particles. Yang-Mills theory extends this concept to field theories by demanding local gauge invariance of the Lagrangian density. This dictates that gauge fields belong to the Lie algebra of the symmetry group and interact with matter fields through covariant derivatives. The Lagrangian includes terms for gauge fields constructed from an invariant field strength tensor.
This document summarizes a journal article that proposes an alternative approach to variable selection called the KL adaptive lasso. The KL adaptive lasso replaces the squared error loss used in traditional adaptive lasso with Kullback-Leibler divergence loss. The paper shows that the KL adaptive lasso enjoys oracle properties, meaning it performs as well as if the true underlying model was given. Specifically, it consistently selects the true variables and estimates their coefficients at optimal rates. The KL adaptive lasso can also be solved using efficient algorithms like LARS. The approach is extended to generalized linear models, and theoretical properties are discussed.
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SOFIE - A Unified Approach To Ontology-Based Information Extraction Using Rea...Tobias Wunner
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Noncommutaive localization in smooth deformation quantization
1. Preamble
Localization
Deformation Quantization
Results
Localization and Lie Rinehart algebras in deformation
quantization
Hamilton ARAUJO
Preprint on ArXiv: 2010.15701
Join work with Martin BORDEMANN (Phd supervisor) and
Benedikt HURLE
Université de Haute-Alsace
IRIMAS - Departément de Mathematiques
Arbeitsgruppenseminar Analysis
Universität Potsdam, Deutschland, 04 dez 2020
1/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
5. Preamble
Localization
Deformation Quantization
Results
Preamble
Noncommutative localization
for Star products
Deformation quantization Localization
In this work we talk about:
Algebraic localization
Analytic localization
Compare this two types of localization.
2/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
6. Preamble
Localization
Deformation Quantization
Results
Preamble
Noncommutative localization
for Star products
Deformation quantization Localization
In this work we talk about:
Algebraic localization
Analytic localization
Compare this two types of localization.
⇒ Noncommutative localization is not
very well know.
2/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
9. Preamble
Localization
Deformation Quantization
Results
This talk is organized as follows:
Localization
- Commutative and noncommutative cases
3/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
10. Preamble
Localization
Deformation Quantization
Results
This talk is organized as follows:
Localization
- Commutative and noncommutative cases
- Ore conditions
3/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
11. Preamble
Localization
Deformation Quantization
Results
This talk is organized as follows:
Localization
- Commutative and noncommutative cases
- Ore conditions
Deformation quantization
3/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
12. Preamble
Localization
Deformation Quantization
Results
This talk is organized as follows:
Localization
- Commutative and noncommutative cases
- Ore conditions
Deformation quantization
- Star products
- Concrete localization
3/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
13. Preamble
Localization
Deformation Quantization
Results
This talk is organized as follows:
Localization
- Commutative and noncommutative cases
- Ore conditions
Deformation quantization
- Star products
- Concrete localization
Results
3/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
14. Preamble
Localization
Deformation Quantization
Results
This talk is organized as follows:
Localization
- Commutative and noncommutative cases
- Ore conditions
Deformation quantization
- Star products
- Concrete localization
Results
- and comments
3/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
17. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Domains
Let us start recalling a basic example
Let be R a integral domain (e.g. R = Z or R = R[x]).
Consider the following equivalence relation on R × (R {0R}):
(r, s) ∼ (r′
, s′
) ⇐⇒ rs′
= r′
s.
4/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
18. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Domains
Let us start recalling a basic example
Let be R a integral domain (e.g. R = Z or R = R[x]).
Consider the following equivalence relation on R × (R {0R}):
(r, s) ∼ (r′
, s′
) ⇐⇒ rs′
= r′
s.
The quotient R =
R × (R {0R})
∼
= {(r, s) = r
s , r ∈ R and s ∈ S} -with the
usual operations of sum and product of classes- is called field of fractions of R
(e.g. R = Q or R = R(x)).
4/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
19. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Domains
Let us start recalling a basic example
Let be R a integral domain (e.g. R = Z or R = R[x]).
Consider the following equivalence relation on R × (R {0R}):
(r, s) ∼ (r′
, s′
) ⇐⇒ rs′
= r′
s.
The quotient R =
R × (R {0R})
∼
= {(r, s) = r
s , r ∈ R and s ∈ S} -with the
usual operations of sum and product of classes- is called field of fractions of R
(e.g. R = Q or R = R(x)).
Note: R is naturally included in R
x 7→
x
1
and the elements of R {0R} have become
invertible in R,
x
1
−1
=
1
x
.
4/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
20. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Domains
Let us start recalling a basic example
Let be R a integral domain (e.g. R = Z or R = R[x]).
Consider the following equivalence relation on R × (R {0R}):
(r, s) ∼ (r′
, s′
) ⇐⇒ rs′
= r′
s.
The quotient R =
R × (R {0R})
∼
= {(r, s) = r
s , r ∈ R and s ∈ S} -with the
usual operations of sum and product of classes- is called field of fractions of R
(e.g. R = Q or R = R(x)).
Note: R is naturally included in R
x 7→
x
1
and the elements of R {0R} have become
invertible in R,
x
1
−1
=
1
x
. The morphism x 7→
x
1
is in general not injective.
4/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
22. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Conventions and definitions
From now on -if we have not specified it before- let us consider:
K is a commutative associative unital ring such that 1 = 1K ̸= 0 = 0K.
5/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
23. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Conventions and definitions
From now on -if we have not specified it before- let us consider:
K is a commutative associative unital ring such that 1 = 1K ̸= 0 = 0K.
We will consider associative and unital K-algebras. We shall include unital
K-algebras isomorphic to {0} (for which 1 = 0).
5/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
24. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Conventions and definitions
From now on -if we have not specified it before- let us consider:
K is a commutative associative unital ring such that 1 = 1K ̸= 0 = 0K.
We will consider associative and unital K-algebras. We shall include unital
K-algebras isomorphic to {0} (for which 1 = 0).
Def.: If R is a K-algebra, S ⊂ R is called multiplicative subset if 1 ∈ S and for all
s, s′ ∈ S we have ss′ ∈ S.
5/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
25. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Conventions and definitions
From now on -if we have not specified it before- let us consider:
K is a commutative associative unital ring such that 1 = 1K ̸= 0 = 0K.
We will consider associative and unital K-algebras. We shall include unital
K-algebras isomorphic to {0} (for which 1 = 0).
Def.: If R is a K-algebra, S ⊂ R is called multiplicative subset if 1 ∈ S and for all
s, s′ ∈ S we have ss′ ∈ S.
Def.: A K-algebra morphism ϕ : R → R′ is called S-inverting if ϕ(S) ⊂ U(R′), where
U(R′) denote the group of invertible elements of R′.
5/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
27. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Localization of a commutative K-algebra
Let R be a commutative K-algebra and S ⊂ R a multiplicative subset, then the
following binary relation ∼ on R × S defined by
(r1, s1) ∼ (r2, s2) if and only if ∃ s ∈ S : r1s2s = r2s1s
is an equivalence relation.
6/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
28. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Localization of a commutative K-algebra
Let R be a commutative K-algebra and S ⊂ R a multiplicative subset, then the
following binary relation ∼ on R × S defined by
(r1, s1) ∼ (r2, s2) if and only if ∃ s ∈ S : r1s2s = r2s1s
is an equivalence relation.
RS :=
R × S
∼
is a commutative K-algebra, also called the quotient algebra or
algebra of fractions of R with respect to S.
6/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
29. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Localization of a commutative K-algebra
Let R be a commutative K-algebra and S ⊂ R a multiplicative subset, then the
following binary relation ∼ on R × S defined by
(r1, s1) ∼ (r2, s2) if and only if ∃ s ∈ S : r1s2s = r2s1s
is an equivalence relation.
RS :=
R × S
∼
is a commutative K-algebra, also called the quotient algebra or
algebra of fractions of R with respect to S.
The equivalence classes (r, s) = r
s are also called fractions.
6/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
30. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Localization of a commutative K-algebra
Let R be a commutative K-algebra and S ⊂ R a multiplicative subset, then the
following binary relation ∼ on R × S defined by
(r1, s1) ∼ (r2, s2) if and only if ∃ s ∈ S : r1s2s = r2s1s
is an equivalence relation.
RS :=
R × S
∼
is a commutative K-algebra, also called the quotient algebra or
algebra of fractions of R with respect to S.
The equivalence classes (r, s) = r
s are also called fractions.
There is a ring homomorphism (the numerator morphism) η(R,S) = η : R → RS
given by r 7→ r
1. This map defines a K-algebra stucture of RS.
6/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
32. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Localization of a commutative K-algebra
The construction of the K-algebra RS gives us important properties.
Proposition:
If R is a commutative K-algebra and
S ⊂ R is a multiplicative subset we have:
a. η(R,S)(S) ⊂ U(RS).
b. Every element of RS is written as a
fraction η(r)η(s)−1, for some r ∈ R
and s ∈ S.
c. ker(η(R,S)) = {r ∈ R | rs = 0
for some s ∈ S}.
7/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
33. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Localization of a commutative K-algebra
The construction of the K-algebra RS gives us important properties.
Proposition:
If R is a commutative K-algebra and
S ⊂ R is a multiplicative subset we have:
a. η(R,S)(S) ⊂ U(RS).
b. Every element of RS is written as a
fraction η(r)η(s)−1, for some r ∈ R
and s ∈ S.
c. ker(η(R,S)) = {r ∈ R | rs = 0
for some s ∈ S}.
Moreover, for S ⊂ R as before:
7/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
34. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Localization of a commutative K-algebra
The construction of the K-algebra RS gives us important properties.
Proposition:
If R is a commutative K-algebra and
S ⊂ R is a multiplicative subset we have:
a. η(R,S)(S) ⊂ U(RS).
b. Every element of RS is written as a
fraction η(r)η(s)−1, for some r ∈ R
and s ∈ S.
c. ker(η(R,S)) = {r ∈ R | rs = 0
for some s ∈ S}.
Moreover, for S ⊂ R as before:
The pair (RS, η(R,S)) is universal
in the sense that for any S-inverting
morphism of commutative unital K-algebras
α : R → R′ uniquely factorizes, i.e.
R
η
//
α
RS
f
R′
where f is a morphism of unital K-algebras
determined by α (Universal property).
7/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
36. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
Let R be K-algebra (commut. or not).
KAlgMS KAlg
(R, S)
S ⊂ R mult. subset R
ϕ : (R, S) → (R′, S′)
ϕ(S) ⊂ S′ ϕ : R → R′
8/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
37. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
Let R be K-algebra (commut. or not).
KAlgMS KAlg
(R, S)
S ⊂ R mult. subset R
ϕ : (R, S) → (R′, S′)
ϕ(S) ⊂ S′ ϕ : R → R′
There is an obvious functor
U : KAlg → KAlgMS given by
U(R) = (R, U(R)) and, for the
commutative case, we already get a
localization functor L(R, S) = RS.
8/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
38. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
Let R be K-algebra (commut. or not).
KAlgMS KAlg
(R, S)
S ⊂ R mult. subset R
ϕ : (R, S) → (R′, S′)
ϕ(S) ⊂ S′ ϕ : R → R′
There is an obvious functor
U : KAlg → KAlgMS given by
U(R) = (R, U(R)) and, for the
commutative case, we already get a
localization functor L(R, S) = RS.
(R, S)
| {z }
L
−
−
−
−
−
−
−
→ RS
|{z}
KAlgMS KAlg
z }| {
(R, U(R))
←
−
−
−
−
−
−
U
z}|{
R
8/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
39. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
Let R be K-algebra (commut. or not).
KAlgMS KAlg
(R, S)
S ⊂ R mult. subset R
ϕ : (R, S) → (R′, S′)
ϕ(S) ⊂ S′ ϕ : R → R′
There is an obvious functor
U : KAlg → KAlgMS given by
U(R) = (R, U(R)) and, for the
commutative case, we already get a
localization functor L(R, S) = RS.
(R, S)
| {z }
L
−
−
−
−
−
−
−
→ RS
|{z}
KAlgMS KAlg
z }| {
(R, U(R))
←
−
−
−
−
−
−
U
z}|{
R
Proposition:
The functor L also exists in the
noncommutative case and L is left adjoint to U.
8/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
41. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
From this proposition we have two problems:
=⇒ The general element of RS is a sum of ‘multifractions’
η(r1) η(s1)
−1
· · · η(rN ) η(sN )
−1
where η = η(R,S), ri ∈ R and si ∈ S.
9/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
42. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
From this proposition we have two problems:
=⇒ The general element of RS is a sum of ‘multifractions’
η(r1) η(s1)
−1
· · · η(rN ) η(sN )
−1
where η = η(R,S), ri ∈ R and si ∈ S.
=⇒ The kernel ker(η) can vary: η as well as RS may be trivial although 0 ̸∈ S.
9/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
43. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
From this proposition we have two problems:
=⇒ The general element of RS is a sum of ‘multifractions’
η(r1) η(s1)
−1
· · · η(rN ) η(sN )
−1
where η = η(R,S), ri ∈ R and si ∈ S.
=⇒ The kernel ker(η) can vary: η as well as RS may be trivial although 0 ̸∈ S.
To get rid of these problems: assume that each left fraction η(s)
−1
η(r)
becomes a right fraction η(r′) η(s′)
−1
implying the condition:
9/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
44. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
From this proposition we have two problems:
=⇒ The general element of RS is a sum of ‘multifractions’
η(r1) η(s1)
−1
· · · η(rN ) η(sN )
−1
where η = η(R,S), ri ∈ R and si ∈ S.
=⇒ The kernel ker(η) can vary: η as well as RS may be trivial although 0 ̸∈ S.
To get rid of these problems: assume that each left fraction η(s)
−1
η(r)
becomes a right fraction η(r′) η(s′)
−1
implying the condition:
∀ (r, s) ∈ R × S ∃ (r′
, s′
) ∈ R × S : η(rs′
) = η(sr′
).
9/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
45. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Noncommutative case (According to the book by [T.Y.Lam] (1999) with a ‘categorical tuning’)
From this proposition we have two problems:
=⇒ The general element of RS is a sum of ‘multifractions’
η(r1) η(s1)
−1
· · · η(rN ) η(sN )
−1
where η = η(R,S), ri ∈ R and si ∈ S.
=⇒ The kernel ker(η) can vary: η as well as RS may be trivial although 0 ̸∈ S.
To get rid of these problems: assume that each left fraction η(s)
−1
η(r)
becomes a right fraction η(r′) η(s′)
−1
implying the condition:
∀ (r, s) ∈ R × S ∃ (r′
, s′
) ∈ R × S : η(rs′
) = η(sr′
).
This will motivate the following:
9/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
47. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Ore conditions [Øystein Ore] (1931)
Let R be a unital K-algebra and S ⊂ R be a multiplicative subset.
Def.: S is called a right denominator set if
a. For all r ∈ R and s ∈ S there are
r′ ∈ R and s′ ∈ S such that
rs′ = sr′
(S right permutable or right Ore
set),
b. For all r ∈ R and for all s′ ∈ S: if
s′r = 0 then there is s ∈ S such that
rs = 0 (S right reversible).
10/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
48. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Ore conditions [Øystein Ore] (1931)
Let R be a unital K-algebra and S ⊂ R be a multiplicative subset.
Def.: S is called a right denominator set if
a. For all r ∈ R and s ∈ S there are
r′ ∈ R and s′ ∈ S such that
rs′ = sr′
(S right permutable or right Ore
set),
b. For all r ∈ R and for all s′ ∈ S: if
s′r = 0 then there is s ∈ S such that
rs = 0 (S right reversible).
Def.: ŘS with η̌(R,S) = η̌ : R → ŘS
is said to be a right K-algebra of fractions of
(R, S) if:
a. η̌(R,S) is S-inverting,
b. Every element of ŘS is of the form
η̌(r) η̌(s)
−1
for r ∈ R and s ∈ S;
c. ker(η̌) = {r ∈ R | rs = 0, for some s ∈ S}
=: I(R,S) =: I.
10/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
50. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Theorem: [Ø.Ore] (1931) For (R, S) as before the following is true:
1 ŘS is a right K-algebra of fractions ⇐⇒ S is a right denominator set.
11/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
51. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Theorem: [Ø.Ore] (1931) For (R, S) as before the following is true:
1 ŘS is a right K-algebra of fractions ⇐⇒ S is a right denominator set.
2 If this is the case each such pair (ŘS, η̌) is universal and then each ŘS is
isomorphic to the canonical localized algebra RS.
11/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
52. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Theorem: [Ø.Ore] (1931) For (R, S) as before the following is true:
1 ŘS is a right K-algebra of fractions ⇐⇒ S is a right denominator set.
2 If this is the case each such pair (ŘS, η̌) is universal and then each ŘS is
isomorphic to the canonical localized algebra RS.
3 Each ŘS is isomorphic to the quotient set RS−1 := (R × S)/ ∼ with respect to
the following generalized equivalence relation ∼ on R × S
(r1, s1) ∼ (r2, s2) ⇔ ∃b1, b2 ∈ R s.t. s1b1 = s2b2 ∈ S and r1b1 = r2b2 ∈ R.
11/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
53. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Theorem: [Ø.Ore] (1931) For (R, S) as before the following is true:
1 ŘS is a right K-algebra of fractions ⇐⇒ S is a right denominator set.
2 If this is the case each such pair (ŘS, η̌) is universal and then each ŘS is
isomorphic to the canonical localized algebra RS.
3 Each ŘS is isomorphic to the quotient set RS−1 := (R × S)/ ∼ with respect to
the following generalized equivalence relation ∼ on R × S
(r1, s1) ∼ (r2, s2) ⇔ ∃b1, b2 ∈ R s.t. s1b1 = s2b2 ∈ S and r1b1 = r2b2 ∈ R.
Note: The proof of this theorem is quite complicated. We can find in [D.S.
Passman] (1980) a more direct proof.
11/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
54. Preamble
Localization
Deformation Quantization
Results
Preliminary
Commutative case
Noncommutative case
Ore conditions
Theorem: [Ø.Ore] (1931) For (R, S) as before the following is true:
1 ŘS is a right K-algebra of fractions ⇐⇒ S is a right denominator set.
2 If this is the case each such pair (ŘS, η̌) is universal and then each ŘS is
isomorphic to the canonical localized algebra RS.
3 Each ŘS is isomorphic to the quotient set RS−1 := (R × S)/ ∼ with respect to
the following generalized equivalence relation ∼ on R × S
(r1, s1) ∼ (r2, s2) ⇔ ∃b1, b2 ∈ R s.t. s1b1 = s2b2 ∈ S and r1b1 = r2b2 ∈ R.
Note: The proof of this theorem is quite complicated. We can find in [D.S.
Passman] (1980) a more direct proof. Moreover, RS−1 carries a canonical unital K-
algebra structure. In terms of the equivalences classes r1s−1
1 and r2s−1
2 we have:
r1s−1
1 + r2s−1
2 = (r1c1 + r2c2)s−1 and (r1s−1
1 )(r2s−1
2 ) = (r1r′)(s2s′)−1 where
s1c1 = s2c2 = s ∈ S (c1 ∈ S and c2 ∈ R) and r2s′ = s1r′ (s′ ∈ S and r′ ∈ R).
11/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
56. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
For a K-vector space V let V [[λ]] = {v =
P∞
i=0 viλi with vi ∈ V } be the
K[[λ]]-module of formal power series. Here K ∈ {R, C}.
13/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
57. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
For a K-vector space V let V [[λ]] = {v =
P∞
i=0 viλi with vi ∈ V } be the
K[[λ]]-module of formal power series. Here K ∈ {R, C}.
For a smooth diff. manifold X we write C∞(X) = C∞(X, K).
13/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
58. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
For a K-vector space V let V [[λ]] = {v =
P∞
i=0 viλi with vi ∈ V } be the
K[[λ]]-module of formal power series. Here K ∈ {R, C}.
For a smooth diff. manifold X we write C∞(X) = C∞(X, K).
Def.: A (formal) star product ∗
on a manifold X is a K[[λ]]-bilinear associative
operation C∞(X)[[λ]] × C∞(X)[[λ]] → C∞(X)[[λ]]
satisfying the following properties for all
f, g ∈ C∞(X):
1 ∗ f = f ∗ 1 = f,
f ∗ g = f · g + O(λ),
f ∗ g =
P∞
k=0 Ck(f, g)λk,
where Ck : C∞(X) ⊗ C∞(X) → C∞(X) are
bidifferential operators.
13/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
59. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
For a K-vector space V let V [[λ]] = {v =
P∞
i=0 viλi with vi ∈ V } be the
K[[λ]]-module of formal power series. Here K ∈ {R, C}.
For a smooth diff. manifold X we write C∞(X) = C∞(X, K).
Def.: A (formal) star product ∗
on a manifold X is a K[[λ]]-bilinear associative
operation C∞(X)[[λ]] × C∞(X)[[λ]] → C∞(X)[[λ]]
satisfying the following properties for all
f, g ∈ C∞(X):
1 ∗ f = f ∗ 1 = f,
f ∗ g = f · g + O(λ),
f ∗ g =
P∞
k=0 Ck(f, g)λk,
where Ck : C∞(X) ⊗ C∞(X) → C∞(X) are
bidifferential operators.
Example in C∞(R2)[[λ]]
In coordinates (x, p) the following
formula defines a star product for
f, g ∈ C∞(R2):
f ∗ g =
∞
X
k=0
λk
k!
∂kf
∂pk
∂kg
∂xk
(Multiplication of diff. operators)
13/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
60. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Deformation Quantization was founded by the seminal article [Bayen, Flato,
Frønsdal, Lichnerowicz, Sternheimer] (1978) and has become a large research area
covering several algebraic theories.
14/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
61. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Deformation Quantization was founded by the seminal article [Bayen, Flato,
Frønsdal, Lichnerowicz, Sternheimer] (1978) and has become a large research area
covering several algebraic theories.
The article by [Kontsevich] (1997) shows that important constructions are
possible, especially for Poisson manifolds. (C1(f, g) − C1(g, f) = {f, g}).
14/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
62. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Deformation Quantization was founded by the seminal article [Bayen, Flato,
Frønsdal, Lichnerowicz, Sternheimer] (1978) and has become a large research area
covering several algebraic theories.
The article by [Kontsevich] (1997) shows that important constructions are
possible, especially for Poisson manifolds. (C1(f, g) − C1(g, f) = {f, g}).
However, this theory does not play an important role in this job.
14/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
65. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Question:
How to relate localization with star-products?
Let ∗ =
P∞
k=0 λkCk be a star-product on a manifold X.
16/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
66. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Question:
How to relate localization with star-products?
Let ∗ =
P∞
k=0 λkCk be a star-product on a manifold X.
We set K = K[[λ]] and R = C∞(X)[[λ]], ∗
.
16/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
67. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Question:
How to relate localization with star-products?
Let ∗ =
P∞
k=0 λkCk be a star-product on a manifold X.
We set K = K[[λ]] and R = C∞(X)[[λ]], ∗
.
Let be Ω ⊂ X a open set and RΩ := C∞(Ω, K)[[λ]].
Let ∗Ω =
P∞
k=0 λkCk|Ω (the restriction of bidifferential operators to open sets is
well-defined!) =⇒ ∗Ω is a star-product on RΩ.
16/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
68. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Question:
How to relate localization with star-products?
Let ∗ =
P∞
k=0 λkCk be a star-product on a manifold X.
We set K = K[[λ]] and R = C∞(X)[[λ]], ∗
.
Let be Ω ⊂ X a open set and RΩ := C∞(Ω, K)[[λ]].
Let ∗Ω =
P∞
k=0 λkCk|Ω (the restriction of bidifferential operators to open sets is
well-defined!) =⇒ ∗Ω is a star-product on RΩ.
It is clear that there is a morphism between unital K-algebras:
ηΩ = η :
R → RΩ
f 7→ f|Ω
16/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
70. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Abstract localization
Consider S := {g ∈ R | ∀ x ∈ Ω : g0(x) ̸= 0} .
1R is in S and ∀g, h ∈ S we have
(g ∗ h)0(x) = g0(x)h0(x) ̸= 0, ∀x ∈ Ω.
17/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
71. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Abstract localization
Consider S := {g ∈ R | ∀ x ∈ Ω : g0(x) ̸= 0} .
1R is in S and ∀g, h ∈ S we have
(g ∗ h)0(x) = g0(x)h0(x) ̸= 0, ∀x ∈ Ω.
⇒ S is a multiplicative subset of the unital
K-algebra R = C∞(X)[[λ]].
17/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
72. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Abstract localization
Consider S := {g ∈ R | ∀ x ∈ Ω : g0(x) ̸= 0} .
1R is in S and ∀g, h ∈ S we have
(g ∗ h)0(x) = g0(x)h0(x) ̸= 0, ∀x ∈ Ω.
⇒ S is a multiplicative subset of the unital
K-algebra R = C∞(X)[[λ]].
Consider then the noncommutative abstract
localization
RS
17/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
73. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Abstract localization
Consider S := {g ∈ R | ∀ x ∈ Ω : g0(x) ̸= 0} .
1R is in S and ∀g, h ∈ S we have
(g ∗ h)0(x) = g0(x)h0(x) ̸= 0, ∀x ∈ Ω.
⇒ S is a multiplicative subset of the unital
K-algebra R = C∞(X)[[λ]].
Consider then the noncommutative abstract
localization
RS
Concrete localization
The space of all formal power series
only defined in Ω already provides
us with the K-algebra
RΩ = (C∞
(Ω)[[λ]], ⋆Ω)
17/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
74. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Abstract localization
Consider S := {g ∈ R | ∀ x ∈ Ω : g0(x) ̸= 0} .
1R is in S and ∀g, h ∈ S we have
(g ∗ h)0(x) = g0(x)h0(x) ̸= 0, ∀x ∈ Ω.
⇒ S is a multiplicative subset of the unital
K-algebra R = C∞(X)[[λ]].
Consider then the noncommutative abstract
localization
RS
Concrete localization
The space of all formal power series
only defined in Ω already provides
us with the K-algebra
RΩ = (C∞
(Ω)[[λ]], ⋆Ω)
Question:
RS
?
∼
= RΩ
Are these algebras isomorphic??
17/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
75. Preamble
Localization
Deformation Quantization
Results
Star products
Localization for star products on open sets
Abstract localization
Consider S := {g ∈ R | ∀ x ∈ Ω : g0(x) ̸= 0} .
1R is in S and ∀g, h ∈ S we have
(g ∗ h)0(x) = g0(x)h0(x) ̸= 0, ∀x ∈ Ω.
⇒ S is a multiplicative subset of the unital
K-algebra R = C∞(X)[[λ]].
Consider then the noncommutative abstract
localization
RS
Concrete localization
The space of all formal power series
only defined in Ω already provides
us with the K-algebra
RΩ = (C∞
(Ω)[[λ]], ⋆Ω)
Question:
RS
?
∼
= RΩ
Are these algebras isomorphic??
Of course, look at next page.
17/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
77. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
One of the results of my thesis is the following:
19/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
78. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
One of the results of my thesis is the following:
Theorem: [H. ARAUJO, M. BORDEMANN, B. HURLE] (2020) Using the previously
fixed notations we get for any open set Ω ⊂ X:
19/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
79. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
One of the results of my thesis is the following:
Theorem: [H. ARAUJO, M. BORDEMANN, B. HURLE] (2020) Using the previously
fixed notations we get for any open set Ω ⊂ X:
1 (RΩ, ∗Ω) together with the restriction morphism η consitutes a right K-algebra of
fractions for (R, S).
19/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
80. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
One of the results of my thesis is the following:
Theorem: [H. ARAUJO, M. BORDEMANN, B. HURLE] (2020) Using the previously
fixed notations we get for any open set Ω ⊂ X:
1 (RΩ, ∗Ω) together with the restriction morphism η consitutes a right K-algebra of
fractions for (R, S).
2 As an immediate consequence we have that S is a right denominator set.
19/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
81. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
One of the results of my thesis is the following:
Theorem: [H. ARAUJO, M. BORDEMANN, B. HURLE] (2020) Using the previously
fixed notations we get for any open set Ω ⊂ X:
1 (RΩ, ∗Ω) together with the restriction morphism η consitutes a right K-algebra of
fractions for (R, S).
2 As an immediate consequence we have that S is a right denominator set.
3 This implies in particular that the algebraic localization RS
∼
= RS−1 of R with
respect to S is isomorphic to the concrete localization RΩ as unital
K-algebras.
19/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
82. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
One of the results of my thesis is the following:
Theorem: [H. ARAUJO, M. BORDEMANN, B. HURLE] (2020) Using the previously
fixed notations we get for any open set Ω ⊂ X:
1 (RΩ, ∗Ω) together with the restriction morphism η consitutes a right K-algebra of
fractions for (R, S).
2 As an immediate consequence we have that S is a right denominator set.
3 This implies in particular that the algebraic localization RS
∼
= RS−1 of R with
respect to S is isomorphic to the concrete localization RΩ as unital
K-algebras.
We don’t directly prove that S is right denominator set. This will follow from the
general theorem of localization.
19/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
83. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
One of the results of my thesis is the following:
Theorem: [H. ARAUJO, M. BORDEMANN, B. HURLE] (2020) Using the previously
fixed notations we get for any open set Ω ⊂ X:
1 (RΩ, ∗Ω) together with the restriction morphism η consitutes a right K-algebra of
fractions for (R, S).
2 As an immediate consequence we have that S is a right denominator set.
3 This implies in particular that the algebraic localization RS
∼
= RS−1 of R with
respect to S is isomorphic to the concrete localization RΩ as unital
K-algebras.
We don’t directly prove that S is right denominator set. This will follow from the
general theorem of localization.
The idea of the proof is to show the three conditions for (RΩ, ∗Ω, η) to be a right
K-algebra of fractions.
19/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
84. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof:
(a) η is S- inverting.
20/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
85. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof:
(a) η is S- inverting.
1 For g ∈ S ⊂ R and γ = η(g) we try to construct ψ ∈ RΩ; γ ∗Ω ψ = 1.
20/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
86. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof:
(a) η is S- inverting.
1 For g ∈ S ⊂ R and γ = η(g) we try to construct ψ ∈ RΩ; γ ∗Ω ψ = 1.
2 We work order by order. For k = 0 it’s easy (x 7→ ψ0(x) = γ0(x)−1
).
20/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
87. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof:
(a) η is S- inverting.
1 For g ∈ S ⊂ R and γ = η(g) we try to construct ψ ∈ RΩ; γ ∗Ω ψ = 1.
2 We work order by order. For k = 0 it’s easy (x 7→ ψ0(x) = γ0(x)−1
).
3 If we have the functions ψ0, . . . , ψk ∈ C∞
(Ω, K) already constructed the next one
ψk+1 is multiplied by γ0 and depends only on ψ0, . . . , ψk and on the γ’s. Indeed,
0 = γ∗Ωψ
k+1
=
k+1
X
l, p, q = 0
l + p + q = k + 1
Cl(γp, ψq) = γ0ψk+1+Fk+1(ψ0, . . . , ψk, γ0, . . . , γk+1)
20/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
88. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof:
(a) η is S- inverting.
1 For g ∈ S ⊂ R and γ = η(g) we try to construct ψ ∈ RΩ; γ ∗Ω ψ = 1.
2 We work order by order. For k = 0 it’s easy (x 7→ ψ0(x) = γ0(x)−1
).
3 If we have the functions ψ0, . . . , ψk ∈ C∞
(Ω, K) already constructed the next one
ψk+1 is multiplied by γ0 and depends only on ψ0, . . . , ψk and on the γ’s. Indeed,
0 = γ∗Ωψ
k+1
=
k+1
X
l, p, q = 0
l + p + q = k + 1
Cl(γp, ψq) = γ0ψk+1+Fk+1(ψ0, . . . , ψk, γ0, . . . , γk+1)
4 Same construction for left inverse. (Associativity =⇒: right inverse = left inverse).
20/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
89. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
We follow steps of Lemma 6.1, p.113, in
J. C. Tougeron(1972) book to prove:
21/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
90. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
We follow steps of Lemma 6.1, p.113, in
J. C. Tougeron(1972) book to prove:
(b) Every ϕ ∈ RΩ is equal to
η(f) ∗Ω η(g)∗Ω−1 for some f ∈ R
and g ∈ S.
21/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
91. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
We follow steps of Lemma 6.1, p.113, in
J. C. Tougeron(1972) book to prove:
(b) Every ϕ ∈ RΩ is equal to
η(f) ∗Ω η(g)∗Ω−1 for some f ∈ R
and g ∈ S.
(c) The kernel of η is equal to the
space of functions f ∈ R such that
there is g ∈ S with f ∗ g = 0.
Tougeron’s Lemma: Let Ω be an open set
of Rn, and (ϕi)i∈N a sequence of smooth
functions Ω → K. Then there is a smooth
function α : Rn → R s. t.
1 α takes only values between 0 and 1.
Moreover α(x) = 0 for all x ̸∈ Ω, and
α(x) 0 for all x ∈ Ω.
2 For each nonnegative integer i the
function ϕ′
i : Rn → K defined by
ϕ′
i(x) :=
ϕi(x)α(x) if x ∈ Ω
0 if x ̸∈ Ω
is
smooth.
21/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
92. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
(Returning to the) Sketch of the proof
(b) Every ϕ ∈ RΩ is equal to η(f) ∗Ω η(g)∗Ω−1 for some f ∈ R and g ∈ S.
22/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
93. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
(Returning to the) Sketch of the proof
(b) Every ϕ ∈ RΩ is equal to η(f) ∗Ω η(g)∗Ω−1 for some f ∈ R and g ∈ S.
To prove that we need some ingredients:
- For a compact set K and non negative integer m define:
pK,m(f) = max{|Dn
f(v)| | n ≤ m, τX(v) ∈ K and h(v, v) ≤ 1}.
- Where pK,m : A → R
- Which will define an exhaustive system of seminorms, hence a locally convex
topological vector space which is known to be metric and sequentially complete,
hence Fréchet.
22/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
94. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
1 Consider the equation η(f) = ϕ ∗Ω η(g) =
P∞
i=0 λi
Pi
k=0 Ck|Ω(ϕi−k, η(g)).
23/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
95. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
1 Consider the equation η(f) = ϕ ∗Ω η(g) =
P∞
i=0 λi
Pi
k=0 Ck|Ω(ϕi−k, η(g)).
2 Ansatz: g (no λs!), g ≥ 0, and g =
P∞
j=0 ϵjgj where ϵj 0 and
23/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
96. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
1 Consider the equation η(f) = ϕ ∗Ω η(g) =
P∞
i=0 λi
Pi
k=0 Ck|Ω(ϕi−k, η(g)).
2 Ansatz: g (no λs!), g ≥ 0, and g =
P∞
j=0 ϵjgj where ϵj 0 and
a. (Kn)n∈N sequence of compact subsets with Kj ⊂ Kj+1 and
S
n∈N Kn = Ω,
23/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
97. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
1 Consider the equation η(f) = ϕ ∗Ω η(g) =
P∞
i=0 λi
Pi
k=0 Ck|Ω(ϕi−k, η(g)).
2 Ansatz: g (no λs!), g ≥ 0, and g =
P∞
j=0 ϵjgj where ϵj 0 and
a. (Kn)n∈N sequence of compact subsets with Kj ⊂ Kj+1 and
S
n∈N Kn = Ω,
b. (gj)j∈N C∞
-functions X → R with gj|Kj = 1 and supp(gj) ⊂ Kj+1,
23/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
98. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
1 Consider the equation η(f) = ϕ ∗Ω η(g) =
P∞
i=0 λi
Pi
k=0 Ck|Ω(ϕi−k, η(g)).
2 Ansatz: g (no λs!), g ≥ 0, and g =
P∞
j=0 ϵjgj where ϵj 0 and
a. (Kn)n∈N sequence of compact subsets with Kj ⊂ Kj+1 and
S
n∈N Kn = Ω,
b. (gj)j∈N C∞
-functions X → R with gj|Kj = 1 and supp(gj) ⊂ Kj+1,
c. For all j ∈ N: ϵjpKj+1,j(gj) 1
2
j
,
d. For all i ≤ j ∈ N: ϵj
Pi
k=0 pKj+1,j (Ck|Ω(ϕi−k, η(gj))) 1
2
j
.
23/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
99. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
1 Consider the equation η(f) = ϕ ∗Ω η(g) =
P∞
i=0 λi
Pi
k=0 Ck|Ω(ϕi−k, η(g)).
2 Ansatz: g (no λs!), g ≥ 0, and g =
P∞
j=0 ϵjgj where ϵj 0 and
a. (Kn)n∈N sequence of compact subsets with Kj ⊂ Kj+1 and
S
n∈N Kn = Ω,
b. (gj)j∈N C∞
-functions X → R with gj|Kj = 1 and supp(gj) ⊂ Kj+1,
c. For all j ∈ N: ϵjpKj+1,j(gj) 1
2
j
,
d. For all i ≤ j ∈ N: ϵj
Pi
k=0 pKj+1,j (Ck|Ω(ϕi−k, η(gj))) 1
2
j
.
3 Then g(N) =
PN
j=0 ϵjgj → g (N → ∞) s.t. ∀ x ∈ Ω : g(x) 0 and g|XΩ = 0,
23/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
100. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Sketch of the proof
1 Consider the equation η(f) = ϕ ∗Ω η(g) =
P∞
i=0 λi
Pi
k=0 Ck|Ω(ϕi−k, η(g)).
2 Ansatz: g (no λs!), g ≥ 0, and g =
P∞
j=0 ϵjgj where ϵj 0 and
a. (Kn)n∈N sequence of compact subsets with Kj ⊂ Kj+1 and
S
n∈N Kn = Ω,
b. (gj)j∈N C∞
-functions X → R with gj|Kj = 1 and supp(gj) ⊂ Kj+1,
c. For all j ∈ N: ϵjpKj+1,j(gj) 1
2
j
,
d. For all i ≤ j ∈ N: ϵj
Pi
k=0 pKj+1,j (Ck|Ω(ϕi−k, η(gj))) 1
2
j
.
3 Then g(N) =
PN
j=0 ϵjgj → g (N → ∞) s.t. ∀ x ∈ Ω : g(x) 0 and g|XΩ = 0,
4 and for each i ∈ N: the sequence of unique fiN : X → K such that
η(fiN ) =
Pi
k=0 Ck|Ω(ϕi−k, η(g(N))) and fiN |XΩ = 0 converges to a smooth
fi : X → C with η(fi) =
Pi
k=0 Ck|Ω(ϕi−k, η(g)) solving the problem.
23/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
101. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
(c) The kernel of η is equal to the space of functions f ∈ R such that there is g ∈ S
with f ∗ g = 0.
24/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
102. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
(c) The kernel of η is equal to the space of functions f ∈ R such that there is g ∈ S
with f ∗ g = 0.
1 Clearly {f ∈ R; f ⋆Ω g = 0, g ∈ S} ⊂ ker(η) because f ∗ g = 0 ⇒ η(f) ∗Ω η(g) = 0
⇒ η(f) = 0 since η(g) is invertible in RΩ.
24/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
103. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
(c) The kernel of η is equal to the space of functions f ∈ R such that there is g ∈ S
with f ∗ g = 0.
1 Clearly {f ∈ R; f ⋆Ω g = 0, g ∈ S} ⊂ ker(η) because f ∗ g = 0 ⇒ η(f) ∗Ω η(g) = 0
⇒ η(f) = 0 since η(g) is invertible in RΩ.
2 Conversely, f ∈ ker(η) ⇒ fi(x) = 0, ∀x ∈ Ω.
24/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
104. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
(c) The kernel of η is equal to the space of functions f ∈ R such that there is g ∈ S
with f ∗ g = 0.
1 Clearly {f ∈ R; f ⋆Ω g = 0, g ∈ S} ⊂ ker(η) because f ∗ g = 0 ⇒ η(f) ∗Ω η(g) = 0
⇒ η(f) = 0 since η(g) is invertible in RΩ.
2 Conversely, f ∈ ker(η) ⇒ fi(x) = 0, ∀x ∈ Ω.
3 Taking g as in the property (b)(fonction aplatisseur), for ϕ0 = 1, ϕi = 0 for i ≥ 1 we
obtain
∀x ∈ X, (f ⋆ g)i = 0.
24/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
105. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Germs
Keeping the privious notation, where R = C∞
(X, K)[[λ]], ∗
is unital K[[λ]]-algebra we consider:
For an open set U ⊂ X we can take the
unital K[[λ]]-algebra
RU = C∞(U, K)[[λ]], ∗U
.
25/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
106. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Germs
Keeping the privious notation, where R = C∞
(X, K)[[λ]], ∗
is unital K[[λ]]-algebra we consider:
For an open set U ⊂ X we can take the
unital K[[λ]]-algebra
RU = C∞(U, K)[[λ]], ∗U
.
For open sets U ⊃ V denote by
ηU
V : RU → RV be the restriction
morphism (ηU for ηX
U ).
The family RU
U∈X
with the restriction
morphisms ηU
V defines a sheaf of
K-algebras over X.
25/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
107. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Germs
Keeping the privious notation, where R = C∞
(X, K)[[λ]], ∗
is unital K[[λ]]-algebra we consider:
For an open set U ⊂ X we can take the
unital K[[λ]]-algebra
RU = C∞(U, K)[[λ]], ∗U
.
For open sets U ⊃ V denote by
ηU
V : RU → RV be the restriction
morphism (ηU for ηX
U ).
The family RU
U∈X
with the restriction
morphisms ηU
V defines a sheaf of
K-algebras over X.
Let x0 ∈ X and Xx0
⊂ X the set of all open
sets containing x0.
25/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
108. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Germs
Keeping the privious notation, where R = C∞
(X, K)[[λ]], ∗
is unital K[[λ]]-algebra we consider:
For an open set U ⊂ X we can take the
unital K[[λ]]-algebra
RU = C∞(U, K)[[λ]], ∗U
.
For open sets U ⊃ V denote by
ηU
V : RU → RV be the restriction
morphism (ηU for ηX
U ).
The family RU
U∈X
with the restriction
morphisms ηU
V defines a sheaf of
K-algebras over X.
Let x0 ∈ X and Xx0
⊂ X the set of all open
sets containing x0.
Considering
R̃x0 = {(U, f); U ∈ Xx0 , f ∈ C∞
(U, K)[[λ]]}
The Stalk at x0 is
Rx0 =
R̃x0
∼
=
S
U∈Xx0
RU
!
∼
25/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
109. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Germs
Keeping the privious notation, where R = C∞
(X, K)[[λ]], ∗
is unital K[[λ]]-algebra we consider:
For an open set U ⊂ X we can take the
unital K[[λ]]-algebra
RU = C∞(U, K)[[λ]], ∗U
.
For open sets U ⊃ V denote by
ηU
V : RU → RV be the restriction
morphism (ηU for ηX
U ).
The family RU
U∈X
with the restriction
morphisms ηU
V defines a sheaf of
K-algebras over X.
Let x0 ∈ X and Xx0
⊂ X the set of all open
sets containing x0.
Considering
R̃x0 = {(U, f); U ∈ Xx0 , f ∈ C∞
(U, K)[[λ]]}
The Stalk at x0 is
Rx0 =
R̃x0
∼
=
S
U∈Xx0
RU
!
∼
25/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
110. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
The relation ∼x0 defined by
(U, f) ∼x0 (V, g) ⇔ ηU
W (f) = ηV
W (g)
for W ∈ Xx0
; W ⊂ U ∩ V.
is an equivalence relation.
26/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
111. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
The relation ∼x0 defined by
(U, f) ∼x0 (V, g) ⇔ ηU
W (f) = ηV
W (g)
for W ∈ Xx0
; W ⊂ U ∩ V.
is an equivalence relation.
Denoting by Rx0 the quotient set
R̃x0 / ∼x0 and by ηU
x0
: RU → Rx0 the
restriction of the canonical projection
R̃x0 → Rx0 to RU ⊂ R̃x0 .
26/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
112. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
The relation ∼x0 defined by
(U, f) ∼x0 (V, g) ⇔ ηU
W (f) = ηV
W (g)
for W ∈ Xx0
; W ⊂ U ∩ V.
is an equivalence relation.
Denoting by Rx0 the quotient set
R̃x0 / ∼x0 and by ηU
x0
: RU → Rx0 the
restriction of the canonical projection
R̃x0 → Rx0 to RU ⊂ R̃x0 .
We get a unital associative K-algebra denoted
by Rx0 , ∗x0
26/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
113. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
The relation ∼x0 defined by
(U, f) ∼x0 (V, g) ⇔ ηU
W (f) = ηV
W (g)
for W ∈ Xx0
; W ⊂ U ∩ V.
is an equivalence relation.
Denoting by Rx0 the quotient set
R̃x0 / ∼x0 and by ηU
x0
: RU → Rx0 the
restriction of the canonical projection
R̃x0 → Rx0 to RU ⊂ R̃x0 .
We get a unital associative K-algebra denoted
by Rx0 , ∗x0
Consider S = S(x0) = {g ∈ R | g0(x0) ̸= 0}
and
26/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
114. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
The relation ∼x0 defined by
(U, f) ∼x0 (V, g) ⇔ ηU
W (f) = ηV
W (g)
for W ∈ Xx0
; W ⊂ U ∩ V.
is an equivalence relation.
Denoting by Rx0 the quotient set
R̃x0 / ∼x0 and by ηU
x0
: RU → Rx0 the
restriction of the canonical projection
R̃x0 → Rx0 to RU ⊂ R̃x0 .
We get a unital associative K-algebra denoted
by Rx0 , ∗x0
Consider S = S(x0) = {g ∈ R | g0(x0) ̸= 0}
and
I = Ix0 = {g ∈ R | g0(x0) = 0}
26/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
115. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
The relation ∼x0 defined by
(U, f) ∼x0 (V, g) ⇔ ηU
W (f) = ηV
W (g)
for W ∈ Xx0
; W ⊂ U ∩ V.
is an equivalence relation.
Denoting by Rx0 the quotient set
R̃x0 / ∼x0 and by ηU
x0
: RU → Rx0 the
restriction of the canonical projection
R̃x0 → Rx0 to RU ⊂ R̃x0 .
We get a unital associative K-algebra denoted
by Rx0 , ∗x0
Consider S = S(x0) = {g ∈ R | g0(x0) ̸= 0}
and
I = Ix0 = {g ∈ R | g0(x0) = 0}
S = R I is a multiplicative subset and Ix0
maximal ideal of R.
26/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
116. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
The relation ∼x0 defined by
(U, f) ∼x0 (V, g) ⇔ ηU
W (f) = ηV
W (g)
for W ∈ Xx0
; W ⊂ U ∩ V.
is an equivalence relation.
Denoting by Rx0 the quotient set
R̃x0 / ∼x0 and by ηU
x0
: RU → Rx0 the
restriction of the canonical projection
R̃x0 → Rx0 to RU ⊂ R̃x0 .
We get a unital associative K-algebra denoted
by Rx0 , ∗x0
Consider S = S(x0) = {g ∈ R | g0(x0) ̸= 0}
and
I = Ix0 = {g ∈ R | g0(x0) = 0}
S = R I is a multiplicative subset and Ix0
maximal ideal of R.
Finally we present the same result as before for
germs:
26/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
117. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Theorem: Using the previously fixed notations we get for any point x0 ∈ X:
1 (Rx0 , ∗x0 ) together with the morphism ηx0 : R → Rx0 consitutes a right
K-algebra of fractions for (R, S(x0)).
2 As an immediate consequence we have that S(x0) is a right denominator set.
3 This implies in particular that the algebraic localization RS−1 of R with
respect to S = S(x0) is isomorphic to the concrete stalk Rx0 as unital
K-algebras.
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118. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Example
The following example provides a non-Ore subset which is a subset of an Ore
subset.
28/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
119. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Example
The following example provides a non-Ore subset which is a subset of an Ore
subset. Consider C∞(R2, R) with the standard star product ∗ given by the formula
f ∗ g =
∞
X
k=0
λk
k!
∂kf
∂pk
∂kg
∂xk
.
Let R = C∞(R2, R)[[λ]], and let Ω ⊂ R2 be the open set of all (x, p) ∈ R2 where
p ̸= 0. Then,
The subset S = {1, p, p2, p3, . . .} ⊂ R is a multiplicative subset of (R, ∗) which is
contained in the Ore subset SΩ but which is neither right nor left Ore.
For instance, for r = (x, p) 7→ ex and s = (x, p) 7→ p we can not find r′, s′ such
that, r′ ∗ s = s′ ∗ r
28/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
120. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Question: Localization commutes with deformation?
Proposition: Let A be a commutative
unital K-algebra and a differential star
product ∗ =
P∞
i=0 λiCi on R := A[[λ]].
For any multiplicative subset S0 ⊂ A there
exists a unique star product ∗S0 on
AS0 [[λ]] such that the numerator map η
canonically extended as a K[[λ]]-linear map
(also denoted η) A[[λ]] → AS0 [[λ]] is a
morphism of unital K[[λ]]-algebras.
29/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
121. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Question: Localization commutes with deformation?
Proposition: Let A be a commutative
unital K-algebra and a differential star
product ∗ =
P∞
i=0 λiCi on R := A[[λ]].
For any multiplicative subset S0 ⊂ A there
exists a unique star product ∗S0 on
AS0 [[λ]] such that the numerator map η
canonically extended as a K[[λ]]-linear map
(also denoted η) A[[λ]] → AS0 [[λ]] is a
morphism of unital K[[λ]]-algebras.
With the above structures A, S0, ∗ consider
the subset S = S0 + λR ⊂ R = A[[λ]].
The subset S = S0 + λR is a multiplicative
subset of the algebra (R, ∗)
Moreover, its image under η consists of
invertible elements of the K[[λ]]-algebra
AS0 [[λ]], ∗S0
.
It follows that there is a canonical
morphism
Φ :
A[[λ]]
S
∗S
→ AS0 [[λ]], ∗S0
.
29/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
122. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Thanks for your attention!!!
30/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
123. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Thanks for your attention!!!
Danke für Ihre Aufmerksamkeit!!!
Obrigado pela sua atenção!!!
Merci de votre attention!!!
30/31 Hamilton ARAUJO - Université de Haute-Alsace (Mulhouse, FR) Localization and Lie Rinehart algebras in deformation quantization
124. Preamble
Localization
Deformation Quantization
Results
Localization for star products in open sets
Germs
A non Ore example
Bibliography
Araujo, H., Bordemann, M., Hurle, B.: Noncommutative localization in smooth
deformation quantization. Preprint, ArXiv:2010.15701 2020.
Bayen, F., Flato, M., Frønsdal, C., Licherowicz, A., Sternheimer, D.: Deformation
theory and quantization. I, II. Annals of Phys. 111, 61-110, 111-151 (1978).
Lam, T.Y.: Lectures on Modules and Rings. Springer Verlag, Berlin, 1999.
Mac Lane, S.: Categories for the Working Mathematician. 2nd ed., Springer, New
York, 1998.
Škoda, Z.: Noncommutative localization in noncommutative geometry,
arXiv:math/0403276v2, 2005.
Tougeron, J.-C.: Idéaux des fonctions différentiables, Springer Verlag, Berlin, 1972.
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