Minimal faithful representations of the free 2-step nilpotent Lie algebra of the rank r

摘要

Given a finite dimensional Lie algebra g, let z(g) denote the center of g and let mu(g) be the minimal possible dimension for a faithful representation of g. In this paper we obtain mu(L-r,L-2), where L-r,L-k is the free k-step nilpotent Lie algebra of rank r. In particular we prove that mu(L-r,L-2) = inverted right perpendicular root 2r(r - 1) inverted left perpendicular + 2 r >= 4. It turns out that mu(L-r,L-2) similar to mu(z((L-r,L-2)) similar to 2 root dimL(r,2 )(as r -> infinity()) we present some evidence that this could be true for L-r,L-k for any k. This is considerably lower than the known bounds for mu(L-r,L-k), which are (for fixed k) polynomial in dim L-r,L-k. (C) 2020 Elsevier Inc. All rights reserved.