Degree estimate for commutators

Abstract

Let K 〈 X 〉 be a finite generated free associative algebra over a field K of characteristic zero and let f, g ∈ K 〈 X 〉 generate their centralizer respectively. Assume that the f and g are algebraically independent, the leading homogeneous components of f and g are algebraically dependent and deg (f) {does not divide} deg (g), deg (g) {does not divide} deg (f). In this article, we construct a counterexample to a conjecture of Jie-Tai Yu that deg ([f, g]) > min {deg (f), deg (g)}, which is closely related to the study of the structure of the automorphism group of K 〈 X 〉. We also obtain a counterexample to another related conjecture of Makar-Limanov and Jie-Tai Yu stated in terms of Malcev-Neumann formal power series. In view of the counterexamples we formulate two open problems concerning degree estimate for commutators in view of the study of the structure of the automorphism group of K 〈 X 〉. The counterexamples in this article are constructed by applying the description of the free algebra K 〈 X 〉 considered as a bimodule of K [u] where u is a monomial which is not a power of another monomial and the solution the equation [um, s] = [un, r] with unknowns r, s ∈ K 〈 X 〉. The newly discovered description and the solution of the equation in this article are closely related to the combinatorial and computational aspects of free associative algebras, hence have their own independent interests. © 2009 Elsevier Inc. All rights reserved.