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# T. Popov - Drinfeld-Jimbo and Cremmer-Gervais Quantum Lie Algebras

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The SEENET-MTP Workshop JW2011
Scientific and Human Legacy of Julius Wess
27-28 August 2011, Donji Milanovac, Serbia

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### T. Popov - Drinfeld-Jimbo and Cremmer-Gervais Quantum Lie Algebras

1. 1. Drinfeld-Jimbo and Cremmer-Gervais Quantum Lie Algebras Todor Popov Institute for Nuclear Research and Nuclear Energy Bulgarian Academy of Sciences, Sofia August 28, 2011 joint work with Oleg OgievetskyBalkan Workshop dedicated to Prof. Julius Wess Todor Popov DJ&CG
2. 2. Quantum Space toward Quantum Lie Algebra xi xj − xj xi = 0 Commutative space Deformation kl xi xj − σij xk xl = 0 Quantum space Lie Extension kl k xi xj − σij xk xl = Cij xk Quantum Lie algebra What are the possible Quantum Lie Algebras compatible with a given braiding σ? Todor Popov DJCG
3. 3. Woronowicz bicovariant diﬀerential calculus Woronowicz developed NC diﬀ. geometry on a quantum group. Hopf algebra A of “ functions on the quantum group” (A, ∆, S) Bicovariant bimodule Γ over A of “diﬀerential forms” ∆L : Γ → A ⊗ Γ ∆R : Γ → Γ ⊗ A ∆L (ω i ) := 1 ⊗ ω i ∆R (ω i ) := ω j ⊗ rji The diﬀerential is d :A→Γ da = (χi ∗ a)ω i , χi ∈ A ∀a ∈ A Left and the right actions on Γ are related by elements fji ∈ A ω i b = (fji ∗ b)ω j := b(1) fji (b(2) )ω j . χi ∈ A form a basis of left-invariant vector ﬁelds. The Leibniz rule implies the coproduct on (A , ∆ , S ) ∆ χi = χj ⊗ fi j + 1 ⊗ χi . Todor Popov DJCG
4. 4. Algebra W and Universal Enveloping Algebra U(L) ij j σ is natural braiding σkl = fl i (rk ) σ : Γ ⊗A Γ → Γ ⊗A Γ σ1 σ2 σ1 = σ2 σ1 σ2 and Cij are the “structure constants” Cij = χj (rik ) k k A bicovariant diﬀerential calculus is determined by the algebra W generated by χi and fji on A with relations kl k χi χj − σij χk χl = Cij χk , σij fka fl b = fi k fjl σkl , kl ab σij χk fl a + Cij fl a = fi k fjl Ckl + fi a χj , χi fja = σij fka χl kl l a kl The associative algebra with generators χi ∈ L kl k χi χj − σij χk χl = Cij χk (1) U(L) := T (L)/(im(id ⊗2 − σ − C )) . (2) L → T (L) U(L) Todor Popov DJCG
5. 5. Quantum Lie Algebra (L, σ, C ) (of Hecke type) a vector space L endowed with a braiding σ : L ⊗ L → L ⊗ L (we take σ to be of Hecke type) σ12 σ23 σ12 = σ23 σ12 σ23 1)(σ + q −2 1 = 0 (σ − 1 1) and a bracket C : L ⊗ L → L such that i) q-antisymmetry C σ = −q −2 C ii) braided Jacobi identities (C12 = C ⊗ id and C23 = id ⊗ C ) C C23 = C C12 + C C23 σ12 σ C12 = C23 σ12 σ23 σ C12 σ23 + σ C23 = C12 σ23 σ12 + C23 σ12 . Todor Popov DJCG
6. 6. ICE R-matrix Ice Ansatz: the only non-vanishing entries of an R-matrix ˆ ij Sij = 0 ˆ ji Sij = 0 ˆ ii Sii = 0 Lemma (Ogievetsky) ˆ Let V be a vector space. Any solution S ∈ End(V ⊗ V ) within the ICE ansatz ˆ kl Sij = aij δil δjk + bij δik δjl is amenable to the Drinfeld-Jimbo R-matrix with entries aij and bij aij = pij q −2θij , bij = 1 − q −2θij , (3) depending on the parameters pij , such that pij pji = 1( pi,i = 1). 1 i j θij = 0 otherwise Todor Popov DJCG
7. 7. Drinfeld-Jimbo Quantum Lie Algebra Theorem ˆ Let S be a standard Drinfeld-Jimbo Yang-Baxter solution (3) ˆ which is non-unitary S 2 = 1 and semisimple, q 2 = −1. 1 The standard quantum space associated to σ = S isˆ kl xi xj = σij xk xl ⇔ xi xj − pij q 2 xj xi = 0 i j . The quadratic-linear algebra kl ˜k xi xj − σij xk xl = Cij xk is a nontrivial quantum Lie algebra iﬀ the parameters are subject to the restrictions ˜k p1j = 1 and Cij = c(δi1 δjk − δj1 δik ) . Todor Popov DJCG
8. 8. RIME R-matrix i) an accumulation of granular ice tufts on the windward sides of exposed objects that is formed from supercooled fog or cloud and built out directly against the wind ii) variant of RHYME ICE ˆ kl Rij = 0 ⇒ {k, l} = {i, j} (4) RIME ˆ kl Rij = 0 ⇒ {k, l} ⊂ {i, j} (5) RIME R-matrix o / Cremmer-Gervais R-matrix ICE R-matrix o / Drinfeld-Jimbo R-matrix Todor Popov DJCG
9. 9. strict RIME R-matrix ICE ˆ kl Rij = 0 ⇔ {k, l} = {i, j} (6) RIME ˆ kl Rij = 0 ⇔ {k, l} ⊂ {i, j} (7) Lemma (OP) ˆ Let V be a vector space. Any solution R ∈ End(V ⊗ V ) of Yang-Baxter equation within the “strict RIME” ansatz reads ˆ kl Rij = (1 − βji )δil δjk + βij δik δjl − βij δik δil + βji δjk δjl , βii = 0 where the parameters βij satisfy βij + βji = βjk + βkj =: β and βij βjk = (βjk − βji )βik . ˆ The “strict RIME” R is of Hecke type with eigenvalues 1 and β − 1 ˆ ˆ R 2 = β R + (1 − β)1 ⊗ 1 . 1 1 (8) Todor Popov DJCG
10. 10. RIME Quantum Lie Algebra Theorem ˆ Let R be a “strict RIME” solution (3) of the Yang-Baxter equation ˆ ˆ unitary R (β = 0) or non-unitary R, (β = 0) The relations of the RIME quantum space associated to σ = R ˆ ˆ kl xi xj = Rij xk xl ⇔ xi xj − xj xi + (βij xi + βji xj )(xi − xj ) = 0 . The quadratic-linear algebra kl k xi xj − σij xk xl = Cij xk is a quantum Lie algebra iﬀ the structure constants are given by Cij = c(δik − δjk ) . k (9) Todor Popov DJCG
11. 11. RIME and “boundary” Cremmer-Gervais Lemma (OP) ˆ i) unitary R(1/µij ), with β = 0 and parameters given by 1 1 βij = := µij µ i − µj so called boundary Cremmer-Gervais,   (RbCG )ij = δli δk +  ˆ kl j −  δs+1 δ j i k+l−s . (10) k≤sl l≤sk ˆ it provides a quantization RbCG := P RbCG = 1 + r of the 1 Gerstenhaber-Giaquinto classical r -matrix i j r= es+1 ∧ ei+j−s . (11) i≤sj Todor Popov DJCG
12. 12. RIME and Cremmer-Gervais Lemma (OP) ˆ ii) non-unitary R(1/[µij ]q−2 ), with β = 0 given by 1 1 − q −2x βij = [x]q−2 := [µij ]q−2 1 − q −2 If we substitute φi = q 2µi and q −2 = 1 − β then one has alternative parametrization βij = φβφi j denoted by R(φ, β). i −φ ˆ Equivalent to Cremmer-Gervais R-matrices for the value p = 1   (RCG ,p )ij = p k−l δli δk + (1 − q −2 )  ˆ kl j −  p k−s δs δ j i k+l−s . k≤sl l≤sk Todor Popov DJCG
13. 13. Cremmer-Gervais basis In both cases the change of the basis to the Cremmer-Gervais matrices ˆ X ˆ ˆ R(µ) −→ RbCG = X (µ) ⊗ X (µ) R(µ) X −1 (µ) ⊗ X −1 (µ) ˆ X ˆ ˆ R(φ, β) −→ RCG ,1 = X (φ) ⊗ X (φ)R(φ, β)X −1 (φ) ⊗ X −1 (φ) is provided by the following transformation matrix n n ∂ei (α) Xij (α) = k ek (α)t := (1 + tαi ) (12) ∂χj k=0 i=1 where ek (α) stand for the elementary symmetric polynomials in variables αi . Todor Popov DJCG
14. 14. Cremmer-Gervais Quantum Lie Algebra Lemma The structure constants of the RIME quantum Lie algebras LCG ,1 and LbCG in the Cremmer-Gervais basis coincide with the structure constants of the “standard” quantum Lie algebra LDJ X C −→ C = (X ⊗ X )C X −1 , ˜ ˜k Cij = c(δi1 δjk − δj1 δik ) . πICE LCG ,1 /L HH ± DJ,1 HHQ HH b HH Q± πICE H\$ LbCG / Lcl Todor Popov DJCG
15. 15. References S. L. Woronowicz, Diﬀerential calculus on quantum matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1989) 125–170. P. Aschieri and L. Castellani, An Introduction to Noncommutative Diﬀerential Geometry on Quantum Groups, Int. J. Mod. Phys. A 8 (1993) 1667–1706. arXiv : hep-th/9207084 (OP) O. Ogievetsky and T. Popov, R-matrices in Rime; Advances in Theoretical and Mathematical Physics 14 (2010), 439–506. arXiv : 0704.1947 [math.QA] O. Ogievetsky, T. Popov , Cremmer-Gervais quantum Lie algebra, Fortsch. Phys. 57 (2009) 654–658. arXiv : 0905.0882v1 [math-ph] Todor Popov DJCG