We give an intepretation of the Cremmer-Gervais r-matrices for sl(n) in terms
of actions of elements in the rational and trigonometric Cherednik algebras of
type GL2 on certain subspaces of their polynomial representations. This is used
to compute the nilpotency index of the Jordanian r-matrices, thus answering a
question of Gerstenhaber and Giaquinto. We also give an interpretation of the
Cremmer-Gervais quantization in terms of the corresponding double affine Hecke
algebra.Comment: 6 page

A. Stolin

A.A. Belavin

E. Cremmer

Garrett Johnson

I. Cherednik

M. Gerstenhaber

P. Etingof

P.P. Kulish

R. Endelman

S.M. Khoroshkin

T. Suzuki

T.J. Hodges

V. Chari

Y. Berest

English

arXiv

We give an interpretation of the Cremmer–Gervais r-matrices for 
                  
                    
                  
                  $${\mathfrak{sl}_n}$$
                 in terms of actions of elements in the rational and trigonometric Cherednik algebras of type GL
                2 on certain subspaces of their polynomial representations. This is used to compute the nilpotency index of the Jordanian r-matrices, thus answering a question of Gerstenhaber and Giaquinto. We also give an interpretation of the Cremmer–Gervais quantization in terms of the corresponding double affine Hecke algebra

Johnson, Garrett

eScholarship - University of California

Cremmer–Gervais r-Matrices and the Cherednik Algebras of Type GL
              2

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Letters in Mathematical Physics

Cremmer–Gervais r-Matrices and the Cherednik Algebras of Type GL 2

Springer - Publisher Connector

DOI 10.1007/s11005-010-0421-5
Lett Math Phys (2010) 94:115–122
Cremmer–Gervais r -Matrices and the Cherednik
Algebras of Type GL2
GARRETT JOHNSON
Department of Mathematics, University of California, Santa Barbara,
CA 93106, USA. e-mail: johnson@math.ucsb.edu
Received: 5 November 2009 / Revised: 20 August 2010 / Accepted: 21 August 2010
Published online: 6 September 2010 – © The Author(s) 2010. This article is published with
open access at Springerlink.com
Abstract. We give an interpretation of the Cremmer–Gervais r -matrices for sln in terms of
actions of elements in the rational and trigonometric Cherednik algebras of type GL2 on
certain subspaces of their polynomial representations. This is used to compute the nilpo-
tency index of the Jordanian r -matrices, thus answering a question of Gerstenhaber and
Giaquinto. We also give an interpretation of the Cremmer–Gervais quantization in terms
of the corresponding double affine Hecke algebra.
Mathematics Subject Classification (2000). 16T25, 20C08.
Keywords. double affine Hecke algebra, rational Cherednik algebra, Cremmer–Gervais
R-matrix, Jordanian R-matrix, classical Yang–Baxter equation, quantum Yang–Baxter
equation.
1. Introduction
Let sln denote the Lie algebra of traceless n ×n matrices having entries in a field
k. Let V denote the vector representation of sln and let r ∈ sln ∧ sln ⊂ End(V ⊗ V )
be a skew-symmetric linear operator. Define r12 := r ⊗ 1, r23 := 1⊗ r , and r13 :=
P23r12P23 where P23 is the permutation operator on V ⊗3: P23(u ⊗ v ⊗ w) = u ⊗
w⊗v. An important class of operators which arise in studying Lie bialgebras and
Poisson–Lie groups are those satisfying the modified classical Yang–Baxter
equation (MCYBE)
[r12, r13]+ [r12, r23]+ [r13, r23]−λ(P123 − P213)=0
for some λ∈k (see [3,8] for more details). The solutions to the MCYBE are called
classical r -matrices and fall into two classes: those satisfying the MCYBE for λ
nonzero (resp. zero) are called quasitriangular (resp. triangular).
In the early 1980s, Belavin and Drinfel’d successfully classified all quasitrian-
gular r -matrices in the case when k is the field of complex numbers [1]. This
classification gives us a solution space which we view as a disjoint union of quasi-
projective subvarieties of P(sln ∧ sln). In contrast, the triangular r -matrices are
116 GARRETT JOHNSON
more mysterious as there is not a constructive classification of them (except in the
smaller cases of sl2 and sl3, see [3, Chapter 3]), only a homological interpreta-
tion exists (due to the work of Stolin [15]: for details see [3, Section 3.1.D.] and
[8, Section 3.5]), and there are currently few known examples.
In the paper [9], Gerstenhaber and Giaquinto investigate the behavior along
the boundaries of the aforementioned quasi-projective varieties and show that the
boundary points are all triangular r -matrices. In the same paper, they construct
the most general class of known examples of triangular r -matrices, the so-called
generalized Jordanian r -matrices rJ,n (see [7,9]). They prove that the Jordanian
r -matrices lie on the boundary of the component corresponding to the quasitrian-
gular Cremmer–Gervais r -matrices (discussed in [6,7,9,11–13]) and conjecture that
r3J,n =0.
In Theorem 4.3, we prove that the nilpotency index of rJ,n is quite different
than conjectured. We do this by first interpreting the quantum Cremmer–Gervais
R-matrix R in terms of the double affine Hecke algebra (DAHA) HHq,t of type
GL2 (Theorem 2.2) and the classical Cremmer–Gervais r -matrix in terms of the
degenerate DAHA HH′c (Theorem 3.4). Using Suzuki’s embedding [16] of the ratio-
nal Cherednik algebra HH′′−c into HH′c, we give a simple interpretation of both the
Cremmer–Gervais and Jordanian r -matrices as operators on the polynomial repre-
sentation of HH′′−c. Using the relations in HH′′−c, we find that the nilpotency index
of rJ,n is n when n is odd, and 2n − 1 when n is even. The conceptual difference
between the two cases has a representation theoretic origin: the polynomial rep-
resentation of the rational Cherednik algebra HH′′c of type A1 is reducible if and
only if the deformation parameter c has the form n/2 for an odd positive integer n
(a special case of Dunkl’s theorem [2]).
2. The Yang–Baxter Equations and the Double Affine Hecke Algebra
Let k be a field of characteristic 0 and let K = k(q, t1/2). We begin with a
DEFINITION 2.1. (see [4,5]) The double affine Hecke algebra HHq,t of type GL2
is the associative K -algebra with generators X±11 , X
±1
2 ,Y
±1
1 ,Y
±1
2 , T and relations
X j X−1j = X−1j X j =Y j Y −1j =Y −1j Y j =1,
(T − t1/2)(T + t−1/2)=0, T X1T = X2, T Y2T =Y1,
Y −12 X1Y2X
−1
1 = T 2, Y1Y2X j =q X j Y1Y2,
Y j X1X2 =q X1X2Y j , [Y1,Y2]=0, [X1, X2]=0,
for j =1,2.
The K -vector space K [X±11 , X±12 ] can be made into a HHq,t -module, called the
polynomial representation, defined as follows. Let (12) act on K [X±11 , X±12 ] by
swapping variables and let S = 1−(12)X1−X2 . For integers a,b define a,b. f (X1, X2) :=
CREMMER–GERVAIS R-MATRICES AND THE CHEREDNIK ALGEBRAS 117
f (qa X1,qb X2). The double affine Hecke algebra HHq,t acts faithfully on K [X±11 ,
X±12 ] via
T → t1/2(12)− (t1/2 − t−1/2)X2S,
Y1 → T 0,1(12), Y2 →0,1(12)T −1,
and the X ’s act via multiplication. For an operator R ∈EndK (K [X±11 , X±12 ]), let
R12, R13, R23 denote the corresponding operators on K [X±11 , X±12 , X±13 ]. In [10],
Gerstenhaber and Giaquinto introduced the modified quantum Yang–Baxter equa-
tion (MQYBE). An operator is called a modified quantum R-matrix if it satisfies
the MQYBE
R12R13R23 − R23R13R12 −λ (P123R12 − P213R23)=0
for some scalar λ. Here, Pi jk denotes the permutation on the variables Xi → X j →
Xk → Xi . Furthermore, R is called unitary if R(12)R(12)=1.
The classical analogue of the MQYBE is called the modified classical Yang–Bax-
ter equation (MCYBE). An operator r is called a classical r -matrix if it satisfies the
MCYBE
[r12, r13]+ [r12, r23]+ [r13, r23]−λ(P123 − P213)=0
for some scalar λ (for more details, see [3]). Furthermore, r is called skew-
symmetric if (12)r(12)=−r . For shorthand, we will denote the left hand sides of
the above equations by MQYBEλ(R) and MCYBEλ(r), respectively.
THEOREM 2.2. The operator R = (12)Y2(12)Y −12 is unitary and satisfies the mod-
ified quantum Yang–Baxter equation
R12R13R23 − R23R13R12 =
(
1− t−1
)2
(P123R12 − P213R23) .
Proof. It is obvious that R is unitary. To show R satisfies the MQYBE, we first
set v = (X1 + X2)S. This operator satisfies MQYBE1(v)= 0, MCYBE1(v)= 0, and
v2 = 0. Therefore, it follows that MQYBEλ2(exp(λv))= 0 for all scalars λ. Setting
F =0,−1 and Rˆ = exp((1− t−1)v), we have R = F−121 RˆF12. Furthermore
(i) F12F13F23 = F23F13F12,
(ii) Rˆ12F23F13 = F13F23 Rˆ12,
(iii) Rˆ23F12F13 = F13F12 Rˆ23.
So R is obtained by twisting Rˆ by F and hence satisfies the MQYBE (cf. [14,
Thm. 1]).
118 GARRETT JOHNSON
3. Semiclassical Limit
Setting q = eh and t = ech , we view K as a subfield of k[[c, h]] and the above for-
mulae for the Y ’s in the polynomial representation become
Y1 →1+h
(
X1
∂
∂X1
+ cX2S + c2
)
+ O(h2)
Y2 →1+h
(
X2
∂
∂X2
− cX2S − c2
)
+ O(h2)
Define yi as the coefficient of h in Yi in the above expressions. The operators y1,
y2 obey commutation relations which motivate the following
DEFINITION 3.1. (see Cherednik [5]) The degenerate (or trigonometric) double
affine Hecke algebra HH′c of type GL2 is the k(c)-algebra having generators y1, y2,
X±11 , X
±1
2 , and (12) and relations
(12)2 =1, [X1, X2]=0, (12)X1(12)= X2,
(12)y1 −y2(12)= c, [y1,y2]=0,
[yi , X j ]=
{
Xi + c(12)X1 if i = j
−c(12)X1 if i 	= j
Remark 3.2. If MQYBEλ(1+ hr + O(h2)) = 0, then λ is of the form h2 + O(h3)
for some scalar  and MCYBE(r)=0. Furthermore, if 1+ hr + O(h2) is unitary,
then r is skew-symmetric.
One can readily obtain the following
LEMMA 3.3. R =1+h (y1 −y2 − c(12))+ O(h2)
Using Remark 3.2 together with Lemma 3.3, we have
COROLLARY 3.4. R, y1, y2 as above
(i) r :=y1 −y2 − c(12) is skew-symmetric
(ii) MCYBEc2(r)=0.
In this situation, r is called the semiclassical limit of R. Since r is homogeneous,
it follows that for any natural number n, we can restrict the action of r to the
subspace k(c)n,n of k(c)[X±1 , X±2 ] spanned by the monomials Xa1 Xb2 with 0≤a,b ≤
n −1. Doing this yields
rn =2
∑
1≤k<l≤n
(k − l − c)ekk ∧ ell+2c
∑
1≤k<l≤n
ekl ∧ elk+4c
∑
1≤i<k< j≤n
ei+ j−k, j ∧ eki .
Here, ei j ∧ ekl is the operator
Xa1 X
b
2 →
1
2
(
δ j,a+1δl,b+1Xi−11 X
k−1
2 − δl,a+1δ j,b+1Xk−11 Xi−12
)
.
CREMMER–GERVAIS R-MATRICES AND THE CHEREDNIK ALGEBRAS 119
Remark 3.5. The formula for rn above suggests that we can view it as being in
gln ∧gln . Thus, we have a one-parameter family of solutions to the MCYBE over
gln . Setting c =− n2 is the only instance that rn will be in sln ∧ sln .
As mentioned in the introduction, the skew-symmetric solutions to the MCYBE
[r12, r13]+ [r12, r23]+ [r13, r23]−λ(P123 − P213)=0
fall into two classes: those solutions satisfying the MCYBE with λ nonzero (resp.
zero) are called quasitriangular (resp. triangular) r -matrices. The quasitriangular r -
matrices (over C) were classified in the early 1980’s by Belavin and Drinfel’d using
combinatorial objects on the Dynkin graph, called BD-triples [1]. The classifica-
tion tells us that the solution space of quasitriangular r -matrices may be viewed as
a disjoint union of quasi-projective subvarieties of P(sln ∧sln): the aforementioned
subvarieties are indexed by the BD-triples. In this case rn corresponds to the maxi-
mal BD-triple obtained by deleting an extremal node. It is the so-called Cremmer–
Gervais r-matrix (discussed in [6,7,9,11–13]).
THEOREM 3.6. When c =− n2 , rn is the Cremmer–Gervais r -matrix.
Proof. Apply the Lie algebra automorphism ei j → −en+1− j,n+1−i of gln to the
formula for rn above, then multiply the result by − 1n , and finally set c =− n2 ; we
obtain the same formula for the Cremmer—Gervais r -matrix rCG as it appears
in [9].
4. Connections with the Rational Cherednik Algebra
We begin this section by recalling a
DEFINITION 4.1. (see Cherednik [5]) The rational Cherednik algebra (over k(c))
HH ′′c of type GL2 has generators (12), x1, x2, u1, u2 and relations
(12)2 =1, (12)x1(12)=x2, (12)u1(12)=u2,
[x1,x2]=0, [u1,u2]=0,
[ui ,x j ]=
{
1− c(12) if i = j
c(12) if i 	= j
The polynomial representation k(c)[x1,x2] of HH′′c is defined where (12) per-
mutes the variables, x1 and x2 act via multiplication, and the ui act by the Dunkl
operators
ui → ∂
∂xi
+ c(−1)i S.
120 GARRETT JOHNSON
In [16], Suzuki shows that there is an algebra embedding ψ :HH′′−c →HH′c defined
on generators by
(12) → (12)
x1 → X1
x2 → X2
u1 → X−11
(
y1 + c2 − c(12)
)
u2 → X−12
(
y2 + c2
)
Using this algebra embedding, we see that the Cremmer–Gervais r -matrix has
an interpretation in the rational Cherednik algebra. Here, rn corresponds to x1u1−
x2u2 ∈HH′′−c.
In [9], Gerstenhaber and Giaquinto provide the largest known class of exam-
ples of triangular r -matrices, the so-called generalized Jordanian r -matrices (also
discussed in [7]). They demonstrate that the Jordanian r -matrices lie on the bound-
ary of the orbit SLn .rn , where SLn acts via the adjoint action. One can translate
this into the setting of the rational Cherednik algebra. Here,
eτ ·ad(u1+u2).(x1u1 −x2u2)=x1u1 −x2u2 + τ(u1 −u2).
Therefore, we have the following
COROLLARY 4.2. u1 − u2 ∈HH′′−c is a boundary solution to the MCYBE (in par-
ticular, MCYBE0(u1 − u2)= 0). Restricting its action to the linear subspace k(c)n,n
and setting c =−n/2 corresponds to the Jordanian r -matrix rJ,n .
As an operator on the polynomial representation of HH′′c ,
u1 −u2 = ∂
∂x1
− ∂
∂x2
−2cS.
This gives us a one-parameter family of triangular r -matrices.
In [9], Gerstenhaber and Giaquinto conjecture that r3J,n = 0. We use the above
interpretation to show that this is not true and to compute the nilpotency index.
THEOREM 4.3. (Gerstenhaber–Giaquinto Conjecture) The nilpotency index of
rJ,n ∈HH′′n/2 is n when n is odd and 2n −1 when n is even.
Proof. Set x = 12 (x1 − x2), x′ = 12 (x1 + x2). We have [u1 − u2,x′] = 0 and (u1 −
u2)x
m = (m − n[[m]])xm−1. Here [[m]] = 1 if m is odd and [[m]] = 0 otherwise.
Observe that
xi1x
j
2 = (−1) jxi+ j + (−1) j (i − j)xi+ j−1x′ + · · · .
One computes that in the case when n is odd, we have
(u1 −u2)n−1xn−12 = (n −1)(−2)(n −3)(−4) · · · (3−n)(2)(1−n) 	=0
CREMMER–GERVAIS R-MATRICES AND THE CHEREDNIK ALGEBRAS 121
and for all 0≤ i, j ≤ n − 1, (u1 − u2)n(xi1x j2) = 0. In the case when n is even, one
computes
(u1 −u2)2(n−1)(xn−11 xn−12 )= (2n −2)(n −3)(2n −4)(n −5) · · · (3−n)(2)(1−n) 	=0
and for all 1≤ i, j ≤n −1, (u1 −u2)2n−1(xi1x j2)=0.
Remarks. For n2, the Cremmer–Gervais r -matrix rn is not nilpotent except when
c =−1/2 and n =2. One can see this by viewing rn as the operator
X1
∂
∂X1
− X2 ∂
∂X2
+ c (X1 + X2) S
and verifying r2n (X1 − X2)= (1+2c)(X1 − X2) and r2n (X21 − X22)=4(1+c)(X21 − X22).
So in this particular case, nilpotency is only a boundary condition.
The conceptual difference between the even and odd cases in Theorem 4.3 has
a representation theoretic origin: the polynomial representation of the rational
Cherednik algebra HH′′c of type A1 is reducible if and only if the deformation
parameter c has the form n/2 for an odd positive integer n (a special case of
Dunkl’s theorem [2]).
Acknowledgements
I am grateful to Milen Yakimov for many helpful discussions. The author was par-
tially supported by NSF grant DMS-0701107.
Open Access This article is distributed under the terms of the Creative Commons Attri-
bution Noncommercial License which permits any noncommercial use, distribution, and
reproduction in any medium, provided the original author(s) and source are credited.
References
1. Belavin, A.A., Drinfel’d, V.G.: Solutions of the classical Yang–Baxter equations for
simple Lie algebras. Funct. Anal. Appl. 16, 159–180 (1982)
2. Berest, Y., Etingof, P., Ginzburg, V.: Finite dimensional representations of rational
Cherednik algebras. In. Math. Res. Notices 19, 1053–1090 (2003)
3. Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press,
New York (1994)
4. Cherednik, I.: Double affine Hecke algebras, Knizhnik–Zamolodchikov equations, and
Macdonald operators. Int. Math. Res. Notices 9, 171–180 (1992)
5. Cherednik, I.: Double Affine Hecke Algebras. London Mathematical Society Lecture
Note Series (2005)
6. Cremmer, E., Gervais, J.L.: The quantum group structure associated with non-linearly
extended Virasoro algebras. Comm. Math. Phys. 134, 619–632 (1990)
7. Endelman, R., Hodges, T.J.: Generalized Jordanian R-matrices of Cremmer–Gervais
Type. Lett. Math. Phys. 32, 225–237 (2000)
122 GARRETT JOHNSON
8. Etingof, P., Schiffman, O.: Lectures on Quantum Groups. International Press, Inc.,
Boston (1998)
9. Gerstenhaber, M., Giaquinto, A.: Boundary Solutions of the classical Yang–Baxter
equation. Lett. Math. Phys. 40, 337–353 (1997)
10. Gerstenhaber, M., Giaquinto, A.: Boundary Solutions of the quantum Yang–Baxter
equation and solutions in three dimensions. Lett. Math. Phys. 44, 131–141 (1998)
11. Hodges, T.J.: On the Cremmer–Gervais quantizations of SL(n). Int. Math. Res.
Notices 10, 465–481 (1995)
12. Hodges, T.J.: The Cremmer–Gervais solution of the Yang–Baxter equation. Proc. Am.
Math. Soc. 127(6), 1819–1826 (1999)
13. Khoroshkin, S.M., Pop, I.I., Samsonov, M.E., Stolin, A.A., Tolstoy, V.N.: On some
Lie bialgebra structures on polynomial algebras and their quantization. Comm. Math.
Phys. 282(3), 625–662 (2008)
14. Kulish, P.P., Mudrov, A.I.: On twisting solutions to the Yang–Baxter equation. Czecho-
slov. J. Phys. 50(1), 115–122 (2000)
15. Stolin, A.: On rational solutions of Yang–Baxter equation for sln . Math. Scand. 69,
57–80 (1991)
16. Suzuki, T.: Rational and trigonometric degeneration of the double affine Hecke algebra
of type A. Int. Math. Res. Not. 37, 2249–2262 (2005)


Cremmer-Gervais r-matrices and the Cherednik Algebras of type GL2

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