Lie-Algebra Centers via De-Categorification

Let g be a Lie algebra over an algebraically closed field k of characteristic zero. Define the universal grading group C(g) as having one generator g_ρ for each irreducible g-representation ρ, one relation g_π = g_ρ^-1 whenever π is weakly contained in the dual representation ρ^∗ (i.e. the kernel of π in the enveloping algebra U(g) contains that of ρ^∗), and one relation g_ρ = g_ρ′g_ρ′′ whenever ρ is weakly contained in ρ^' ⨂ ρ^''. The main result is that attaching to an irreducible representation its central character gives an isomorphism between C(g) and the dual ℑ^∗ of the center ℑ ≤ g when g is (a) finite-dimensional solvable; (b) finite-dimensional semisimple. The group C(g) is also trivial when the enveloping algebra U(g) has a faithful irreducible representation (which happens for instance for various infinite-dimensional algebras of interest, such as sl(∞), o(∞) and sp(∞)). These are analogues of a result of Müger’s for compact groups and a number of results by the author on locally compact groups, and provide further evidence for the pervasiveness of such center-reconstruction phenomena.