Casimir Elements for Certain Polynomial Current Lie Algebras

We consider the polynomial current Lie algebra gl(n)[x] corresponding to the general linear Lie algebra gl(n), and its factor-algebra gm by the idealPk m gl(n)xk. We construct two families of algebraically independent generators of the center of the universal enveloping algebra U(gm) by using the quantum determinant and the quantum contraction for the Yangian of level m. 0. Introduction Let g be a nite-dimensional complex Lie algebra. Denote by ' the canonical isomorphism ' : S(g) ! grU(g) of the symmetric algebra S(g) to the graded algebra grU(g) associated with the universal enveloping algebra U(g). The restriction of ' to the subalgebra I(g) of g-invariants in S(g) yields an isomorphism ' : I(g)! gr Z(g); (0.1) where Z(g) denotes the center of U(g). If the Lie algebra g is reductive, each of the algebras I(g) and Z(g) admits a family of algebraically independent generators (see, e.g., Dixmier [Di], Ch. 7.3, 7.4). For some non-reductive Lie algebras g an analogous property still takes place. A class of such Lie algebras was investigated by Rais and Tauvel in [RT]. In that paper one considers the polynomial current Lie algebra g[x] = g C[x] corresponding to a semi-simple complex Lie algebra g, and, given a positive integer m, one de nes the factor-algebra gm of g[x] by the ideal X k mgx k: One of the main results of [RT] is a construction of a family of algebraically independent generators of the algebra I(gm) of gm-invariants in S(gm). In the present paper we study the Lie algebra gm corresponding to the complex reductive Lie algebra g = gl(n), that is, gm is the factor-algebra of gl(n)[x] by the ideal Im = X k mgl(n)x k. (Note that the construction of [RT] can be easily transfered to this case as well.) Our aim is to construct families of algebraically independent generators of the center Z(gm) of the universal enveloping algebra U(gm) (that is, `to quantize' the construction of [RT]). We give explicit expressions for the generators in terms of the basis elements of gm. Using isomorphism (0.1) we thus obtain a family of algebraically independent generators of the algebra I(gm). The main results are formulated in Section 1. The proofs are based on some properties of the algebra Ym(n) = Ym(gl(n)) called the Yangian of level m for the Lie algebra gl(n) (see [C], [Dr]). First, we prove a Poincar e{Birkho {Witt-type theorem for the algebra Ym(n). Then 2 we show that Ym(n) admits a ltration such that the corresponding graded algebra is isomorphic to U(gm). Further, we use the fact (see, e.g., [MNO]) that the coe cients of both the quantum determinant qdet T (u) 2 Ym(n)[u] and the quantum contraction z(u) 2 Ym(n)[u] belong to the center of the algebra Ym(n). So, taking the images of these coe cients in the graded algebra grYm(n) ' U(gm) we get two families of central elements in U(gm). Finally, using the results of [G] and [RT] we apply an analogue of the Harish-Chandra homomorphism for the algebra U(gm) to prove that the images of the coe cients of the quantum determinant are algebraically independent and generate the center of U(gm). To prove this property for the images of the coe cients of the quantum contraction we apply the quantum Liouville formula [MNO]. As a corollary, we obtain that the coe cients of the quantum determinant, as well as those of the quantum contraction, are algebraically independent generators of the center of Ym(n). This work can be regarded as a generalization of [M] where an analogous approach was applied to the construction of Casimir elements and computing their Harish-Chandra images for the classical Lie algebras of series A{D. I am grateful to Grigori Olshanski who drew my attention to the papers [RT] and [G]. I would also like to thank him for many helpful discussions. 1. Construction of Casimir elements Denote by Eij , (i; j = 1; . . . ; n) the standard basis of the general linear Lie algebra gl(n). Then the elements E(k) ij := Eijxk with 1 i; j n and 0 k m 1 form a basis of the Lie algebra gm = gl(n)[x]=Im; see Introduction. Let u be a formal variable. To construct the rst family of Casimir elements, introduce the following U(gm)-valued polynomials in u: Eij(u) := ijum + (E(0) ij m(j 1) ij)um 1 + E(1) ij um 2 + + E(m 1) ij ; where i; j = 1; . . . ; n, and de ne the \determinant" of the noncommutative matrix E(u) =