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

2020/06/08 by Leandro Cagliero, Cagliero, Leandro, Nadina Rojas +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2006.04676

Abstract

Given a finite dimensional Lie algebra mathfrakg, let\n mathfrakz( mathfrakg) denote the center of mathfrakg and let\n\μ( mathfrakg) be the minimal possible dimension for a faithful\nrepresentation of mathfrakg. In this paper we obtain\n\μ(\L_r,2), where \L_r,k is the free k-step\nnilpotent Lie algebra of rank r. In particular we prove that\n\μ(\L_r,2)= \lceil \√(2r(r-1)) \rceil + 2 for r\n\≥ 4. It turns out that \μ(\L_r,2)\n\∼\μ\( mathfrakz(\L_r,2)\) \∼\n2\√\dim\L_r,2 (as r\→\∞) and we present some evidence\nthat this could be true for \L_r,k for any k, this is\nconsiderably lower than the known bounds for \μ(\L_r,k), which\nare (for fixed k) polynomial in \dim\L_r,k.\n

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