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

Abstract

Given a finite dimensional Lie algebra g, let z(g) denote the center of g and let µ(g) be the minimal possible dimension for a faithful representation of g. In this paper we obtain µ(Lr,2), where L r,k is the free k-step nilpotent Lie algebra of rank r. In particular we prove that µ(Lr,2) = 2r(r − 1) + 2 for r ≥ 4. It turns out that µ(Lr,2) ∼ µ z(Lr,2) ∼ 2 dim Lr,2 (as r → ∞) and we present some evidence that this could be true for L r,k for any k, this is considerably lower than the known bounds for µ(L r,k ), which are (for fixed k) polynomial in dim L r,k .