Differential Polynomial Identities of Upper Triangular Matrices Under the Action of the Two-Dimensional Metabelian Lie Algebra

Onofrio M Di Vincenzo,Vincenzo Nardozza

doi:10.1007/s10468-020-10014-w

Abstract

We study the differential polynomial identities of the algebra UTm(F) under the derivation action of the two dimensional metabelian Lie algebra, obtaining a generating set of the TL-ideal they constitute. Then we determine the Sn-structure of their proper multilinear spaces and, for the minimal cases m = 2, 3, their exact differential codimension sequence.

Highlights

Differential polynomial identities constitute a natural and direct generalization of the notion of polynomial identities of an algebra. They take into account the identical relations holding for an algebra whose structure is enriched by the action of a Lie algebra of derivations, and constitute a trending topic within the PI-Theory of associative algebras
Presented by: Michel Van den Bergh Partially supported by GNSAGA 2019
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Summary

Introduction

Differential polynomial identities constitute a natural and direct generalization of the notion of polynomial identities of an algebra. One of the main tasks in studying the differential polynomial identities of an algebra is to find a generating set for. . The unitary subalgebra of generated by the commutators of any length will be denoted , and we will call its elements -proper polynomials. We need just polynomials which are proper with respect to ordinary letters, that is elements of the subalgebra of generated by commutators and differential letters. . Let us fix the notation: Definition 3.4 Let I denote the set constituted by the following differential polynomials: 11 where all indeterminates belong to. Just the case of a product of 1 commutators provides something similar: Lemma 3.8 Let mod , and 1 standard commutators in 1 involve an -letter. We define to be and the secondary letters of in Clearly, is designed to have so that it evalutates to

Let and assume that

We may relate the -structure of to the one of inductively: indeed while