The Homology of Heisenberg Lie Algebras Over Fields of Characteristic Two

Abstract

The generating function of the Betti numbers of the Heisenberg Lie algebra over a field of characteristic 2 is calculated using discrete Morse theory. Introduction The Heisenberg Lie algebra of dimension 2n + 1, denoted by f)n, is the vector space with basis B = {z, x', . . . ,xn, yi, . . . yn} where the only non-zero Lie products of basis elements are [xi,Vi] = -'yi,Xi) = z. In this paper the Betti numbers of the homology of f)n over a field of characteristic 2 are computed with the aid of algebraic discrete Morse theory from [2] . The notation from [2] will be freely used. The main result Theorem 1. The generating function of the Betti numbers of the Heisenberg Lie algebra over a field k of characteristic 2 is given by e^h.«-..«*(i+'3)('+f;;+(')(2t). When k has characteristic 0, Santharoubane [1] has shown that dimfci/?(f,n,A;)=(2^-^.2^2), i<n (the need for the ground field to have characteristic 0 is not explicitly mentioned). Let us first recall the construction of the Chevalley-Eilenberg complex V of f)n, whose homology is the homology of f)n: the complex V is given by