Lengths of finite dimensional representations of PBW algebras

Let Σ be a set of n × n matrices with entries from a field, for n > 1, and let c(Σ) be the maximum length of products in Σ necessary to linearly span the algebra it generates. Bounds for c(Σ) 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.