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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 Ogievetsky

Balkan Workshop dedicated to Prof. Julius Wess


                  Todor Popov   DJ&CG
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
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
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
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
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
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
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
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
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
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
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
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
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
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

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

  • 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 Ogievetsky Balkan Workshop dedicated to Prof. Julius Wess Todor Popov DJ&CG
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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