We propose a method of quantization of certain Lie bialgebra structures on
the polynomial Lie algebras related to quasi-trigonometric solutions of the
classical Yang-Baxter equation. The method is based on so-called affinization
of certain seaweed algebras and their quantum analogues.Comment: 9 pages, LaTe

Samsonov, M.

Stolin, A.

Yermolova-Magnusson, J.

English

arXiv

Julia Y.-MAGNUSSON

Maxim SAMSONOV

Alexander STOLIN

Crossref

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Classical quasi-trigonometric r−matrices of
Cremmer-Gervais type and their quantization
Yermolova-Magnusson J. ∗
Onsala Space Observatory
Chalmers University of Technology
Göteborg, Sweden
Samsonov M. †
St. Petersburg State University
Institute of Physics
St. Petersburg, Russia
Stolin A. ‡
Department of Mathematics
University of Göteborg
Göteborg, Sweden
Abstract
We propose a method of quantization of certain Lie bialgebra struc-
tures on the polynomial Lie algebras related to quasi-trigonometric so-
lutions of the classical Yang-Baxter equation. The method is based
on so-called affinization of certain seaweed algebras and their quantum
analogues.
1 Introduction
The aim of this paper is to propose a method of quantizing certain Lie bial-
gebras structures on polynomial Lie algebras. The subject was introduced by
V.G. Drinfeld in the beginning of 90th, when in his paper [3] he posed the
following problem: can any Lie bialgebra be quantized? Couple of years later,
two mathematicians Etingof and Kazhdan came up with the positive answer,
while the problem of finding explicit quantization formulas remained open.
∗e-mail: julia@oso.chalmers.se
†e-mail: samsonov@pink.phys.spbu.ru
‡e-mail: astolin@math.chalmers.se
1
First important contribution to the complete solution of the problem were
presented in [11], [4] and [5], where the authors successfully quantized a list of
all quasi-triangular Lie bialgebra structures on finite dimensional simple Lie
algebras also known as the Belavin-Drinfeld list.
The infinite-dimensional cases were first brought up in [7]. However, a real
break-through came with [8] and [12]and was achieved due to construction of
a deformed versions of Yangians Y (sln) and quantum affine algebras Uq(ŝln)
for n = 2, 3.
In the present paper we consider the case Uq(ŝln) for n = 4. Our solution
is based on a q−version of so-called seaweed algebras.
A seaweed subalgebra of sln is defined as an intersection of two parabolic
subalgebras one of which containing the Borel subalgebra B+ and another
B−. The case when both parabolic subalgebras are maximal was treated in
connection with the study of rational solutions of the classical Yang-Baxter
equation. The results of the study, as well as a complete answer to the question;
when such an algebra is Frobenius, can be found in [13]. Later on in [2] it was
found out when an arbitrary seaweed algebra is Frobenius.
As a subject of this study we choose a sl4(C[t]) Lie algebra which Lie
bialgebra structure we are going to quantize. Its coalgebra is defined by a
quasi-trigonometric solution of the classical Yang-Baxter equation, this type
of solutions was first introduced in [6]. We will begin with a brief summery of
the results obtained in this paper.
Let g be a simple Lie algebra and let Ω ∈ g⊗ g be the quadratic Casimir
element. We say that a solution X(z, t) of the classical Yang-Baxter equation
is quasi-trigonometric if it is of the form:
X(z, t) =
tΩ
z − t
+ p(z, t),
where p(z, t) is a polynomial with coefficients in g⊗g. Any quasi-trigonometric
solution of the classical Yang-Baxter equation defines a Lie bialgebra structure
on g[t] and the corresponding Lie cobracket on g[t] is given by the formula
{A(t) ∈ g[t]} → {[X(z, t), A(t)⊗ 1 + 1⊗ A(z)] ∈ g[t]⊗ g[z]}.
It was proved in [6] that for g = sln there is a one-to-one correspondence
between quasi-trigonometric r−matrices and Lagrangian subalgebras of g⊕ g
transversal to a certain Lagrangian subalgebra of g⊕ g defined by a maximal
parabolic subalgebra of g. Here we define Lagrangian with respect to the
following symmetric non-degenerate invariant bilinear form on g⊕ g:
Q((a, b), (c, d)) = K(a, c)−K(b, d),
where K is the Killing form.
We are going to quantize a quasi-trigonometric r−matrix with coefficients
in sl4 ⊗ sl4 defined by the following Lagrangian subalgebra W of sl4 ⊕ sl4:
W = {(X, Y ) ∈ sl4 ⊕ sl4 : X = Ad(T )Y }.
2
Here
T =










0 1 0 0
1 0 1 0
0 0 1 0
0 1 0 1










.
2 Preliminaries
We denote by ŝln, n > 2, the following affine Lie algebra with the set of simple
affine roots {α0, α1, . . . , αn−1}, defined by the generators {e±αi , hαi}0≤i≤n−1
and the relations
[hαi , e±αj ] = ±(αi, αj)e±αj , [eαi , e−αj ] = δijhαi ; (1)
e2±αie±αj − 2 e±αie±αje±αi + e±αje
2
±αi
= 0, if |i− j| = 1; (2)
e±αie±αj − e±αje±αi = 0, if |i− j| 6= 1. (3)
Here {(αi, αj)} is the affine Cartan matrix of the root system A
(1)
n−1 that is:
{(αi, αj)} = 2 if i = j, {(αi, αj)} = −1 if |i − j| = 1, n − 1 and 0 otherwise.
The root α0 + α1 + . . . + αn−1 is called imaginary and is denoted by δ, the
corresponding Cartan element hδ =
∑
i hαi is central in ŝln. It is customary to
exclude α0 from the defining relations substituting it by δ − α1 − . . .− αn−1.
The universal enveloping algebra U(ŝln) has the following quantum ana-
logue denoted by Uq(ŝln), which is a Hopf algebra. Formally it is defined by
the same set of generators as U(ŝln), however the defining relations are being
q−deformed.
[hαi , e±αj ] = ±(αi, αj)e±αj , [eαi , e−αj ] = δij
qhαi − q−hαi
q − q−1
; (4)
e2±αie±αj − (q + q
−1) e±αie±αje±αi + e±αje
2
±αi
= 0, if |i− j| = 1; (5)
e±αie±αj − e±αje±αi = 0, if |i− j| 6= 1. (6)
A Hopf algebra structure is defined uniquely by the following values of the
coproduct on the Chevalley generators
∆(hαi) = hαi ⊗ 1 + 1⊗ hαi ; (7)
∆(eαi) = q
−hαi ⊗ eαi + eαi ⊗ 1, ∆(e−αi) = e−αi ⊗ q
hαi + 1⊗ e−αi ; (8)
S(hαi) = −hαi , S(eαi) = −q
hαieαi , S(e−αi) = −e−αiq
−hαi ; (9)
3
ε(hαi) = 0, ε(eαi) = 0, ε(e−αi) = 0. (10)
Following [9, 10], we fix a normal ordering
α1 ≺ α1 + α2 ≺ · · · ≺ α1 + α2 + . . .+ αn−1 ≺ α2 ≺ · · · ≺ αn−1 ≺
≺ . . . ≺ δ ≺ . . . ≺ δ − αn−1 ≺ δ − αn−1 − αn−2 ≺ δ − α1 − . . . ≺ αn−1.
Let α, β and γ be pairwise non-collinear roots taken so that there are no other
roots α′ and β ′ with the property γ = α′ + β ′. Then, if e±α and e±β have
already been constructed, one sets
eγ = eαeβ − q
(α,β)eβeα, e−γ = e−βe−α − q
−(β,α)e−αe−β.
Consider a seaweed Lie algebra SWn := P
+
1 ∩ P
−
n−1 which is defined as the
intersection of the maximal parabolic subalgebras P±i , obtained by omitting
the generators corresponding to the roots containing ±αi. Classically, we have
an embedding of the restricted seaweed Lie algebra ι : SW ′n →֒ ŝln−1, where
SW ′n := (SWn\{hα1 , hαn−1})
⋃
{p =
n−1
∑
k=1
(n− 2k)
n
hαk}.
We show that this embedding can be quantized in the sense that it can be
lifted to an embedding:
ιq : Uq(SW
′
n) →֒ U
Kn−1
q (ŝln−1) (11)
Here the coalgebra structure on Uq(SW
′
n) is obtained from the standard (see
above) coalgebra structure on Uq(sln) by twisting it by a certain element Kn ∈
Uq(sln)
⊗2, which will be described below.
The element Kn will be chosen as follows: Kn = q
r0 , where r0 is the
Cartan part of the so-called Cremmer-Gervais r−matrix. The Cremmer-
Gervais r−matrices correspond to the shift-by-one Belavin-Drinfeld triples:
τ : Γ1 → Γ2, where Γi = {αi, . . . , αn+i−2}, i = 1, 2. To explain this in more
details let us recall classification of quasi-triangular r−matrices for a simple
Lie algebra g. First of all the quasi-triangular r−matrices are solutions of the
system
r12 + r21 = Ω (12)
[r12, r13] + [r12, r23] + [r13, r23] = 0 (13)
where Ω is the quadratic Casimir element in g⊗ g.
Belavin and Drinfeld proved that any solution of this system is defined by
a triple (Γ1,Γ2, τ), where Γ1,Γ2 are subdiagrams of the Dynkin diagram of g
and τ is an isometry between these two subdiagrams. Further, each Γi defines
a reductive subalgebra of g, and τ extends to an isometry (with respect to
the corresponding restrictions of the Killing form) between the obtained two
reductive subalgebras of g. The following property of τ should be satisfied:
τk(α) 6∈ Γ1 for any α ∈ Γ1 and some k. Let Ω0 be the Cartan part of Ω. Then
one can construct a quasi-triangular r−matrix according to the following
4
Theorem 1 (Belavin-Drinfeld). Let r0 ∈ h⊗ h satisfies the systems
r120 + r
21
0 = Ω0, (14)
(τ(α)⊗ 1 + 1⊗ α)(r0) = 0 (15)
for any α ∈ Γ1. Then the tensor
r = r0 +
∑
α>0
X−α ⊗Xα +
∑
α,β>0,α≻β
X−α ∧Xβ
satisfies (12),(13). Moreover, any solution of (12),(13) is of the above form,
for a suitable triangular decomposition of g and suitable choice of a basis {Xα}.
3 Quantization of Uq(SW
′
5) and its affinization
We define SW5 ⊂ sl5 is the following seaweed Lie subalgebra:






∗ 0 0 0 0
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗
0 0 0 0 ∗






.
Consider the restricted seaweed algebra
SW ′5 := (SW5\{hα1 , hα4})
⋃
{p5 :=
3
5
hα1 +
1
5
hα2 −
1
5
hα3 −
3
5
hα4}.
Proposition 1. There exists an embedding: ι(5) : SW ′5 →֒ ŝl4.
Proof. We define ι(5) by its values on the generators of SW ′5 letting (to avoid
ambiguities, we mark the generators of the affine algebra ŝl4 by hat, for in-
stance: êα).
ι(5)(p5) =
3
4
hα1 +
2
4
hα2 +
1
4
hα3 , ι
(5)(eα4) = êδ−α1−α2−α3 ,
ι(5)(e−α1) = ê−α1 , ι
(5)(eα2) = êα2 ,
ι(5)(eα3) = êα3 , ι
(5)(hα2) = ĥα2 ,
ι(5)(hα3) = ĥα3 .
and fixing the normal ordering in ŝl4 so that α1 ≺ α2 ≺ α3 ≺ δ−α1−α2−α3.
The proof will be finished once we will verify that the defining relations of
SW ′5 are being preserved by ι
(5). This is straightforward.
5
We define Uq(SW
′
5) as a subalgebra of Uq(sl5) generated by hα2 , hα3 , p5,
e−α1 , eα4 , e±α2 , e±α3 . Unfortunately, the subalgebra defined as Uq(SW
′
5) is not
a Hopf subalgebra of Uq(sl5) with respect to the standard comultiplication in
Uq(sl5). However, twisting the standard comultiplication by K5 = q
r0(5), where
r0(5) is the Cartan part of the Cremmer-Gervais r−matrix for sl5, we obtain
the following result:
Proposition 2. Let K5 = q
r0(5) where
r0(5) =
hα1 ⊗ (
2
5
hα1 +
3
5
hα2 +
3
5
hα3 +
2
5
hα4) + hα2 ⊗ (
3
5
hα2 +
4
5
hα3 +
3
5
hα4)+
+hα3 ⊗ (−
1
5
hα1 +
3
5
hα3 +
3
5
hα4) + hα4 ⊗ (−
1
5
hα1 −
1
5
hα2 +
2
5
hα4)
is the Cartan part of the Cremmer-Gervais r−matrix. Then the Hopf alge-
bra UK5q (sl5) induces a coalgebraic structure on the quantum seaweed algebra
U(SW ′5) ⊂ U
K5
q (sl5).
Proof. Let us consider UK5q (sl5) and check that the elements corresponding to
the generators of Uq(SW
′
5) define a Hopf subalgebra in U
K5
q (sl5). In order to
prove this, we have to show that the following relations hold for 1 ≤ i ≤ 4:
(αi ⊗ id)(r0(5)) ∈ Uq(SW
′
5) (16)
−hαi + (id⊗ αi)(r0(5))) ∈ Uq(SW
′
5) (17)
(id⊗ αi)(r0(5)) ∈ Uq(SW
′
5) (18)
hαi + (αi ⊗ id)(r0(5)) ∈ Uq(SW
′
5). (19)
Taking into account that hα2 , hα3 ∈ Uq(SW
′
5) and formulas (14)-(15), we see
that using the explicit formula for r0(5) it is sufficient to prove that
(id⊗ α1)(r0(5)) =
1
5
hα1 −
3
5
hα2 −
2
5
hα3 −
1
5
hα4 ∈ Uq(SW
′
5). (20)
It can be checked directly and this finishes the proof.
In what follows we need to introduce an element K̂4 ∈ Uq(ŝl4)
⊗2, which is
a four-dimensional analogue of the element K5 ∈ Uq(sl5)
⊗2:
K̂4 = q
r̂0(4), where
r̂0(4) =
ĥα1 ⊗ (
3
8
ĥα1 +
1
2
ĥα2 +
3
8
ĥα3) + ĥα2 ⊗ (
1
2
ĥα2 +
1
2
ĥα3) + ĥα3 ⊗ (−
1
8
ĥα1 +
3
8
ĥα3).
Theorem 2. There is an embedding of Hopf algebras:
ι(5)q : Uq(SW
′
5) →֒ U
K̂4
q (ŝl4),
where ι
(5)
q is defined by the same formulas on the quantum generators as ι(5)
was defined on the classical generators in Proposition 1.
6
Proof. By the properties (12)-(13) it is sufficient to verify that
ι(5)q ((id⊗ α1)(r0(5))) = (id⊗ α1)(r̂0(4)) (21)
ι(5)q ((α4 ⊗ id)(r0(5))) = −((α1 + α2 + α3)⊗ id)(r̂0(4)) (22)
(21) is equivalent to the following equality
ι(5)q
(
1
5
hα1 −
3
5
hα2 −
2
5
hα3 −
1
5
hα4
)
=
1
4
ĥα1 −
1
2
ĥα2 −
1
4
ĥα3
while (22) can be rewritten as follows:
ι(5)q
(
−
1
5
hα1 −
2
5
hα2 −
2
3
hα3 +
1
5
hα4
)
= −
1
4
ĥα1 −
1
2
ĥα2 −
3
4
ĥα3 .
4 Quantization of quasi-trigonometric r− ma-
trices
Following [4, 5], we can construct the Cremmer-Gervais twist in Uq(sl5) as the
following product:
FCG5 = F
(3)F (2)F (1) · K5
where
F (k) =
∏
α1βα3
expq2((q
−1 − q)e′τk(β) ⊗ e
′
−β),
expq(x) :=
∑
n≥0
xn
(n)q!
, (n)q! ≡ (1)q(2)q . . . (n)q, (k)q ≡ (1− q
k)/(1− q)
and
e′αk = q
(αk⊗id)(r0(5))eαk , e
′
−αk
= q−(id⊗αk)(r(0)(5))e−αk .
The elements e′±αk have the following coproducts after twisting by K5:
K5∆(e
′
αk
)K−15 = e
′
αk
⊗ q2(αk⊗id)(r0(5)) + 1⊗ e′αk
K5∆(e
′
−αk
)K−15 = q
−2(id⊗αk)(r0(5)) ⊗ e′−αk + e
′
−αk
⊗ 1.
Corollary 1. F ′CG5 = FCG5K
−1
5 restricts to a twist of Uq(SW
′
5) leading to an
affine twist in Uq(ŝl4) via embedding ι
(5)
q :
F̂CG4 = (ι
(5)
q ⊗ ι
(5)
q )(FCG5 · K
−1
5 ) · K̂4. (23)
Proof. Since the element K5 is a twist for Uq(sl5) we see that:
K125 (∆⊗ id)(K5) = K
23
5 (id⊗∆)(K5),
then, by the chain property of the twists, F ′CG5 defines a twist for U
K5
q (sl5),
and it can be restricted to a twist for Uq(SW
′
5) leading to (23) if we use the
same argument relating Uq(ŝl4) and U
K̂4
q (ŝl4) via twisting by K̂4.
7
5 Discussion
It follows from Proposition 1 that in the present paper we have quantized the
following quasi-trigonometric r−matrix in sl4:
X(z, t) =
tΩ
z − t
+ rDJ+
SK((e23 + e34)⊗ e21 + e24 ⊗ e31 + e34 ⊗ e32 − r0(4)+
(z − t)(e21 ⊗ e41 + e41 ⊗ e21 + e31 ⊗ e31)+
ze31 ⊗ e42 − te42 ⊗ e31 + ze41 ⊗ e32 − te32 ⊗ e41.
Here eij are matrix units, SK(a⊗ b) = a⊗ b− b⊗ a and rDJ is the Drinfeld-
Jimbo r−matrix for sl4 In terms of Theorem 1 it is defined by the following
data:
Γ1 = Γ2 = ∅, r0 =
Ω0
2
The corresponding quantum algebra can be obtained by twisting the stan-
dard quantum affine algebra Uq(ŝl4) by the element F̂CG4. Then the quan-
tum R−matrix F̂21CG4Raff F̂
−1
CG4
quantizes X(z, t). The universal quantum
R−matrix Raff was constructed in [10].
Using methods of [6] one can prove that this r−matrix is gauge equiv-
alent to that of defined by the Lagrangian subalgebra W ⊂ sl4 ⊕ sl4 from
Introduction. Extra terms of X(z, t),
X(z, t)−
tΩ
z − t
− rDJ ,
define a classical twist for the polynomial Lie bialgebra sl4[t].
As a conclusion, we’ve to mention that the results presented in this paper
support the conjecture made in [6], namely:
every classical twist gives rise to a quantum twist.
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