Preprint 09-2015

Lengths of quasi-commutative pairs of matrices

Author(s) : Alexander E. Guterman , Olga V. Markova , Volker Mehrmann

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 09-2015

MSC 2000

15A30 Algebraic systems of matrices

16S50 Endomorphism rings; matrix rings

Abstract :
In this paper we discuss some partial solutions of the length conjecture which describes the length of a generating system for matrix algebras. We consider mainly the algebras generated by two matrices which are quasi-commuting. It is shown that in this case the length function is linearly bounded. We also analyze which particular natural numbers can be realized as the lengths of certain special generating sets and prove that for commuting or product-nilpotent pairs all possible numbers are realizable, however there are non-realizable values between lower and upper bounds for the other quasi-commuting pairs. In conclusion we also present several related open problems.

Keywords : Finite-dimensional algebras, Lengths of sets and algebras, Quasi-commuting matrices