Some remarks on the center of the universal enveloping algebra of a classical simple Lie algebra

Abstract

Let L be a finite-dimensional simple Lie algebra over an algebraically closed field K of characteristic zero, let U be its universal enveloping algebra, and let Z be the center of U. If / is the dimension of a Cartan subalgebra H of L, then it is known that Z is a polynomial ring in / independent variables. In this paper a set of / algebraically independent generators of Z is produced rather explicitly for the classical algebras of type A, B, C, D by casewise considerations. It is straightforward to show that generating Z is equivalent to generating the L-invariants It in the symmetric algebra SL* of L*. In addition, there is a homomorphism from 5L* onto SH which embeds 7* into the Weyl-invariants Iw. Due to Chevalley this embedding is also a surjection. For the classical simple Lie algebras the action of the Weyl group W on SH* is describable in a sufficiently convenient fashion so as to permit easy construction of generators of Iw. It is shown here that certain generating sets of Iw can be explicitly lifted back to It via trace functions on the first fundamental representation of L. As a result of this construction of the generators of Iw and the lifting process, the following well-known results are proven rather directly for the classical algebras: 1. It = Iw (Chevalley), and 2. Z and Iw are polynomial rings in / algebraically independent variables. The center Z of U plays a fundamental role in the finitedimensional representation theory of L. Since any irreducible representation is determined up to isomorphism by its character, if zu , z, were generators of Z and if M and N were non-isomorphic irreducible L-modules, then for some i one must have (z,)M^ (z,)N (due to Schur's lemma they are scalars). The central element (z{ - (Z,)N)/((£*)M - (^)N) would act as one on M and zero on N. For any list of pairwise non-isomorphic irreducible L-modules one could thus find a central element acting as one on one of them, and as zero on the rest. Such elements could be used to isolate the isotypic components in a reducible