Parameter-dependent solutions of the classical
Yang-Baxter equation on sl(n,C).
M. Cahen
V. De Smedt
Abstract
For any integersnandm(m≥4) such thatn+mis odd we exhibit trian-gular solutions of the classical Yang-Baxter equation onsl((n+ 1)(m+ 2),C)
parametrized by points of a quotient of complex projective spaceP⌊n2⌋(C) by the action of the symmetric groupSym(⌊n+12 ⌋) and we prove that no two of these solutions are isomorphic.
1
Introduction
The motivation for this work is to exhibit solutions of the classical Yang-Baxter equations depending on a large number of parameters, Such solutions lead, by a construction indicated by Drinfeld [1], to quantum groups. We hope that these parameter-dependent quantum groups may have interesting geometrical applications [3].
2
The classical Yang-Baxter equation.
LetG be a finite-dimensional Lie algebra overK ( = Ror C ); an elementR ∈ ∧2G
is said to be a solution of the classical Yang-Baxter equation iff
[R, R] = 0
where [, ] :∧2G ⊗ ∧2G → ∧3G is the Schouten bracket, defined on bivectors by
Received by the editors November 1995. Communicated by Y. F´elix.
1991Mathematics Subject Classification : 16W30, 17Bxx.
Key words and phrases : Classical Yang Baxter equation, Poisson Lie Groups.
518 M. Cahen – V. De Smedt
[X1∧X2, Y1∧Y2] := [X1, Y1]∧X2∧Y2 − [X1, Y2]∧X2∧Y1
−[X2, Y1]∧X1 ∧Y2 + [X2, Y2]∧X1∧Y1
Lemma 2.1. (see [2] corollary 2.2.4 and proposition 2.2.6).There is a bijective
correspondence between the solutionsR of the classical Yang-Baxter equation on G and the symplectic subalgebras (H, ω) of G; i.e. ω belongs to ∧2H⋆, the radical of ω is zero and ω is a 2-cocycle ofH with values inK for the trivial representation of
H on K.
3
A family of symplectic nilpotent Lie algebras.
LetGbe the complex vector space generated by the elementsx, y, ei,j; 0≤i≤n,0≤
complex numbersai is different from zero. Define the elementω of ∧2G by ω(x, y) := 1 and ω(ei,j, ei′,j′) = (−1)i+j δi′,n−i δj′,m−j
have even multipliciy if n is odd and if n is even there is exactly one λi which has
Parameter-dependent solutions of the classical Yang-Baxter equation on sl(n,C).519
ofP⌊n2⌋(C) with homogeneous coordinates xi (o≤i≤ ⌊n
2⌋) is given by the standard
action on the ⌊n+1
2 ⌋first coordinates. Hence
Proposition 3.1. The isomorphism class of the nilpotent algebra G(a),(a :=
(a0, . . . , an) ∈ Cn+1 \ {0}) is determined by a point of complex projective space
4
A family of solutions of the classical Yang-Baxter equation for
sl((n+1)(m+2),C).
transported symplectic form ω of G(a). The corresponding solutions of the classical
Yang-Baxter equation are given by
Observe that a bialgebra structure on sl(N,C) determines uniquely a element r ∈
Λ2(sl(N,C)). If r is a solution of the classical Yang-Baxter equation (cf
2) it is well known that there exists a unique minimal subalgebraHofsl(N,C) such
that r∈Λ2(H); we shall call H the support of r.
Proposition 4.1. LetRa the solution of the classical Yang Baxter equation given
by formula (1)
1. LetK :=⌊n+2
2 ⌋. The support ofRaare isomorphic if and only if the projection
of(a0, . . . , aK−1)∈CK\ {0}on PK−1(C)/Sym(K)are identical. In particular
if the projections are distinct, the corresponding bialgebras are non isomorphic.
2. LetΩbe the open dense subset ofCK defined as follows. Ifφ :CK →CK(K−1) :
(a0, . . . , aK−1)→((1−aaij)|i6=j)thenΩ =φ−1(CK(K−1)\(∪i,j{(z0, . . . zK−1)|
zi = zj} ∪i{(z0, . . . zK−1) | zi = 0}). Let a, a′ ∈ Ω then the bialgebra struc-tures definied by Ra an d Ra′ are isomorphic if and only if the projection of
(a0, . . . , aK−1)∈CK \ {0} on (CK \ {0})/Sym(K)are identical.
5
Aknowledgement
520 M. Cahen – V. De Smedt
References
[1] Drinfeld, V. G.: On constant quasiclassical solution of the Yang Baxter quantum equation. Soviet. Math. Dokl. 28, pp. 667-671 (1983)
[2] Chari, V., Pressley, A.: A guide of Quantum Groups. Cambridge: University Press, 1994
[3] Turaev, V.G., The Yang Baxter equation and invariant of links. Invent. Math.
92 pp. 527-553 (1988)
