ИСТИНА

By the length of a finite system of generators for a finite-dimensional algebra over an arbitrary field we mean the least positive integer k such that the products of length not exceeding k span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. The length evaluation can be a difficult problem, since, for example, the length of the full matrix algebra is still unknown (Paz’s Problem, 1984). Paz conjectured that the length of any generating set for the algebra of n by n matrices is at most 2n−2. In this talk we will show that this conjecture holds under the assumption that the generating set contains a nonderogatory matrix or a matrix with minimal polynomial of degree n−1. We will also present linear bounds for the length of generating sets that include a matrix with some restrictions on its Jordan normal form. This talk is based on a joint work with Alexander Guterman (Moscow State University), Thomas Laffey and Helena Smigoc (University College Dublin).