A lower bound for faithful representations of nilpotent Lie algebra

L. CAGLIERO,; N. ROJAS

LINEAR AND MULTILINEAR ALGEBRA

TAYLOR & FRANCIS LTD

Lugar: Londres; Año: 2014

0308-1087

Given a finite dimensional nilpotent Lie algebra $ $, let  $mu( )$ (resp.  $mu_{nil}( )$) be the minimal dimension of $V$ such that $ $ admits a faithful representation (resp. nilrepresentation) on $V$.In this paper we present a lower bound for  $mu_{nil}( )$ for a $p$-step nilpotent Lie algebra $ $ 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 $ $.We present some estimates of this minimum which leads to a very explicitlower bound for $mu_{nil}( )$ that involves the dimensions of $ $ and its center.This bound allows us to obtain $mu( )$ for some families of nilpotent Lie algebras.