THE KAC CONSTRUCTION OF THE CENTRE OF U(g) FOR LIE SUPERALGEBRAS

In 1984, Victor Kac [K4] suggested an approach to a description of central elements of a completion of U(g) for any Kac-Moody Lie algebra g. The method is based on a recursive procedure. Each step is reduced to a system of linear equations over a certain subalgebra of meromorphic functions on the Cartan subalgebra. The determinant of the system coincides with the Shapovalov determinant for g. We prove that the Kac approach can also be applied to finite dimensional Lie superalgebras g(A) with Cartan matrix A (as claimed in [K4]) and reproduce for them Sergeev’s description of the centers of U(g) [S2]. In order to prove this, one needs to show that the recursive procedure stops after a finite number of steps. The original paper [K4] does not indicate how to check this fact. Here we give a detailed presentation of the Kac approach and apply it to finite dimensional Lie superalgebras g(A). In particular, we deduce the Kac formulas for the Shapovalov determinants and verify the finiteness of the recursive procedure.