A lower bound for faithful representations of nilpotent Lie algebras

2014/07/01 by Cagliero, Leandro, Rojas, Nadina
#17B10 #17B30 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1407.0226

Abstract

In this paper we present a lower bound for the minimal dimension μ(\mathfrakn) of a faithful representation of a finite dimensional p-step nilpotent Lie algebra \mathfrakn over a field of characteristic zero. Our bound is given as the minimum of a quadratically constrained linear optimization problem, it works for arbitrary p and takes into account a given filtration of \mathfrakn. We present some estimates of this minimum which leads to a very explicit lower bound for μ(\mathfrakn) that involves the dimensions of \mathfrakn and its center. This bound allows us to obtain μ(\mathfrakn) for some families of nilpotent Lie algebras.

Related

• Computing faithful representations for nilpotent Lie algebras
• Faithful Representations of Minimal Dimension of 6-dimensional nilpotent Lie algebras
• Minimal faithful representations of the free 2-step nilpotent Lie\n algebra of the rank r
• Faithful Lie algebra modules and quotients of the universal enveloping algebra
• Minimal algebras and 2-step nilpotent Lie algebras in dimension 7