FAITHFUL REPRESENTATIONS OF MINIMAL DIMENSION OF CURRENT HEISENBERG LIE ALGEBRAS

Given a Lie algebra g over a field of characteristic zero k, let mu(g) = min{dim pi : p is a faithful representation of g}. Let h(m) be the Heisenberg Lie algebra of dimension 2m + 1 over k and let k[t] be the polynomial algebra in one variable. Given m is an element of N and p is an element of k[t], let h(m,p) = h(m) circle times k[t]/(p) be the current Lie algebra associated to h(m) and k[ t]/( p), where ( p) is the principal ideal in k[ t] generated by p. In this paper we prove that mu(h(m,p)) = m deg p + [2 root deg p]. We also prove a result that gives information about the structure of a commuting family of operators on a finite dimensional vector space. From it is derived the well-known theorem of Schur on maximal abelian subalgebras of gl(n, k).