Non-commutative Poisson algebra structures on affine Kac-Moody algebras

Non-commutative Poisson algebras are the algebras having an associative algebra structure and a Lie structure together with the Leibniz law. The non-commutative Poisson algebra structures on the infinite-dimensional algebras are studied. We show that these structures are standard on the poset subalgebras of the associative algebra of all endomorphisms of the countable-dimensional vector space These structures on KaceMoody algebras of affine type are determined. It is shown that the associative products on the derived Lie ideals are trivial, and the associative product action of the scaling elements are fully described. @ 1998 Elsevier Science B.V. 1991 Math. Subj. Class.: 17B60; 37B67 Deformation theory of algebras has been widely studied. This theory is originated by Gerstenhaber in a series of paper [3-71, presently the best overview of this subjects is the long paper by Gerstenhaber and Schack [8] or the book by Shnider and Sternberg [ 111. Our interest is the deformation theory of Poisson algebras. It happens many times that the Leibniz law does not hold after deforming a Lie structure without changing an associative algebra structure. What deformations of these associative algebra structures satisfy the Leibniz laws with the deformed Lie algebra structure? We do not restrict ourselves to find such associative algebra structures of commutative ones. This motivates us to propose the problem: For a Lie algebra L with a Lie product [-, -I, find an (not necessarily commutative) associative algebra product, denoted by xy the associative algebra product of x, y in L, satisfying the Leibniz Zaw: [xy,z] = [x,z]y + x[y,z] for x,y,x E L. We call such an algebra having an associative algebra structure and a Lie algebra structure together with the Leibniz law * E-mail: [email protected]. 0022-4049/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved PfI SOO22-4049(96)00141-7 268 F Kubol Journal of Pure and Applied Algebra 126 (1998) 267-286 a non-commutative Poisson algebra. This notion is in the framework of “Leibniz pairs” introduced by Flato et al. [2]. While the most interesting examples of non-commutative Poisson algebras L are probably the infinite-dimensional ones, we have asked the question of what possibilities are allowed for the finite-dimensional ones and we answered in the previous paper [lo] that the associative algebra products of non-commutative Poisson algebra L are trivial, i.e., LL = 0 when L are finite-dimensional semisimple Lie algebras over an algebraically closed field of characteristic zero. The simplicity of Lie structure is crucial in the finite-dimensional case. Unlike finite-dimensional simple Lie algebras (i.e., Kac-Moody algebras of finite type), infinite-dimensional Kac-Moody algebras have the center, hence not simple. In the begining of the study of non-commutative Poisson algebra structures of Kac-Moody algebras, we expected that the associative algebra products were trivial or they were similar to those of “basic nilpotent algebras” introduced in [lo]. But these are not those cases. The scaling element plays an important role and we finally get Theorem. Let g(A) be a Kac-Moody algebra with a Lie bracket [-,-I over the field 62 of the complex numbers corresponding to a generalized Cartan matrix A of afine type, c the canonical central element and d the scaling element of g(A). Let us denote by g’(A) the derived Lie ideal [g(A), g(A)] so that g(A) = g’(A) + Cd. Then the associative algebra structure of the non-commutative Poisson algebra structure on g(A) is given by g’tA)g’tA) = 0 and the associative product action of d is one of the following three types; for s, 1,r E @, (n) dd = SC, g’(A)d = dg’(A) = 0, (s # 0), (1) dd = Id, g’(A)d = 0, dx = lx for any x E g’(A). (r) dd = rd, dg’(A) = 0, xd = rx for any x E g’(A). It is easy to verify that the Kac-Moody algebra g(A) together with the associative algebra structure given above is actually a non-commutative Poisson algebra. For example, to verify the Leibniz law we put y1 = xi + pld, yz = x2 + pzd, y3 = x3 + p3d (xi E g’(A), pi E @). In the case that an associative product is given by (1) in Theorem, we have bl,YZlyS = 0 and then bl, Y2y31 = y2blyY31 = lPZ([xl,x3I + plbhl + mh,dl). Before going into the investigation of the associative algebra structures on KacMoody Lie algebras, we take a look at the Lie structures of some infinite-dimensional associative algebras. For any associative algebra B one can always construct a noncommutative Poisson structure in which we take the Lie product {-, -} to be the scalar multiple of the ordinary associative commutator [-, -I, i.e., by setting {x, y} = p[x, y] = p(xy yx). If B is a direct sum of subalgebras B,(y E r) then one E KubolJournal of Pure and Applied Algebra 126 (1998) 267-286 269 can even take different pr on each of the components. A non-commutative Poisson algebra obtained in this way is called standard or said to have a standard structure. Let M,(C) be the associative algebra of row-finite N x N matrices with coefficients in C consisting of those matrices (xii) for which xij = 0 except finite number of j’s in each i-row. Note that the algebra Moo(C) is isomorphic to the algebra En&(V) of all endomorphisms of a countable-dimensional vector space Y over C. When a subalgebra B of M,(C) contains all the diagonal matrices, define the poset I = Z(B) by setting i 4 j if eij E B (where eij is the matrix with 1 in the (i,j)th place and 0 elsewhere). Let r be the poset determined by reducing I modulo the equivalence relation defined by loops. Such algebras B are called “poset algebras” [8]. Let .Z = C(B) be the nerve of I(B). Then we have Proposition. Let B be a subalgebra of M,(C) containing the diagonal matrices, and C = C(B) the simplical complex associated to B. Suppose that any connected component of C has the property that for any pair of l-faces there exists a polygon which has these two l-faces as its edges. Then any non-commutative Poisson algebra structure on B must be standard. In particular, M,(C) itself has only a standard one. Proof. Let {-, -} be any Lie product on B satisfying the Leibniz law: {ab,c} = a{ b, C} + {a, c}b (a, b, c E B). The Leibniz law leads to the formulas {eii, ejj} = 0, {eii, ejk} = ni( ) e,k e,k w h ere &(ejk) = 0 if i # j and i # k, and ieij, ekI> = -Lk(eij)eijek[ + h(ek,)ek[eij, when eij,ekl E B with i # j, k # 1. Now take an element i + j < k in f(B), and let B(i, j, k) be a subalgebra of B spanned by eii,ejj,ekk,eij,ejk,eik. Note that C(A(i, j, k)) is a 2-simplex. Let US put 1 := A.i(eij). By the equalities {eiiejj,eij} = 0 and Ai(eik)eik = {eii,f?~ejk}, We have Aj(eij) = -2, Ai(eik) = 2 and &(eik) = -1. S&X {fZjj,eik} = {ejj,f?ijejk} = -?tf?ik •k nj(ejk)e&, OIle gets Aj(f?jk) = A and hence &(ejk) = -A. The equality {eij, ejk} = /k& follows from the formulas given above. This shows that for each element i 4 j + k in f(B) we can assign a scalar 1, = n(i, j, k) such that the Lie product { -, -} coincides with a standard Lie product ,?[ -, -1 on B( i, j, k) where [-, -1 is the ordinary associative commutator on M&C). Assume that C = C(B) is connected. If i’ 4 j’ + k’ is another element in f(B), then I(i, j, k) = A(i’, j’, k’) by our assumption on C. Hence, the equality { -, -} = A[--, -1 holds on the subalgebra Bo consisting of the finite linear sums of {eij(eij E A}. Let x = cz, Ci4ei, (possibly infinite sum) be an element of B and put G := {x,ek[}. By the equalities e,,G = {eppX,f?k/} {epp,ekl}X = n(Cpq[f?pq,ek/] [epp,ek/]x), one obtains G = ( cFZp=, epp) G = n[x, f?kl] A [ cy=‘=, epp, ekl]x = A[x, ek/], This shows that our equality { -, -} = A[-, -1 holds on the whole algebra B. 0 270 F Kubo I Journal of‘ Pure and Applied Algebra 126 (1998) 267-286 Note that this theorem is still true for the locally finite associative algebra M = xi j Ceij (finite sum in i,j), which is simple and isomorphic to the direct limit li1$4, (C) of the full algebras Mn(@) of 12 x n matrices. The corresponding result of the finite-dimensional case will be found in [lo, Theorem 21. 1. Non-commutative Poisson algebra structures on Kac-Moody algebras Throughout this paper we let A = (Aij);j=s a generalized Car-tan matrix of rank n, i.e., of affine type. A realization of A is a triple (H,II,II”) such that H is a complex vector space of dimension n + 1, II = (010,. . . , IX,} C H’ and II” = {a:, . . . , CC,“} C H are linearly independent subsets in H* and H, respectively, satisfying (Ri,Uy) := ai = Aji for i, j = 0,. . . , IZ. The Kac-Moody algebra g(A) corresponding to A is an infinite-dimensional Lie algebra generated by H and so-called Chevalley generators {ea, . . . , e,,fo,. . . , f,,} with a table of Lie multiplications [-, -1 : [cc?, ej] = Aijei, [MV,~] = -AqA, [ei,jj] = Sij$‘, (adej)l-AjZ(ei) = 0 for i # j and (adfi)l-Alg(J) = 0 for i # j. We have two decompositions, a root-space decomposition with respect to H; here A is a root system and gU = {X E g(A)l[h,x] = a(h)x for all h E H}, and a triangular decomposition