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

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