Lengths of finite dimensional representations of PBW algebras

Let be a set of n× n matrices with entries from a field, for n > 1, and let c( ) be the maximum length of products in necessary to linearly span the algebra it generates. Bounds for c( ) have been given by Paz and Pappacena, and Paz conjectures a bound of 2n− 2 for any set of matrices. In this paper we present a proof of Paz’s conjecture for sets of matrices obeying a modified Poincaré–Birkhoff–Witt (PBW) property, applicable to finite dimensional representations of Lie algebras and quantum groups. A representation of the quantum plane establishes the sharpness of this bound, and we prove a bound of 2n− 3 for sets of matrices with this modified PBW property which do not generate the full algebra of all n× n matrices. This bound of 2n− 3 also holds for representations of Lie algebras, although we do not know whether it is sharp in this case. © 2004 Elsevier Inc. All rights reserved. AMS classification: 15A30