This document summarizes work on Drinfeld-Jimbo and Cremmer-Gervais quantum Lie algebras. It describes how quantum spaces arise from braided deformations of commutative spaces, and how bicovariant differential calculi on quantum groups lead to quantum Lie algebras. It presents the Drinfeld-Jimbo and Cremmer-Gervais R-matrices, and shows how they give rise to quantum Lie algebra structures through their associated braidings. It also establishes relationships between Drinfeld-Jimbo, Cremmer-Gervais, and "strict RIME" quantum Lie algebras through changes of basis.
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