Maximal T-spaces of the free associative algebra over a finite field

TL;DR

This paper constructs an infinite sequence of proper T-spaces in the free associative algebra over a finite field, including the first known example of a maximal T-space that is not a T-ideal.

Contribution

It introduces the first explicit examples of maximal T-spaces in free associative algebras over finite fields that are not T-ideals, expanding understanding of their structure.

Findings

Constructed an infinite sequence of proper T-spaces with finite codimension.

Proved the first T-space in the sequence is maximal and not a T-ideal.

Provided explicit bases for these T-spaces and their complements.

Abstract

In earlier work, it was established that for any finite field k, the free associative k-algebra on one generator x, denoted by k[x]_0, had infinitely many maximal T-spaces, but exactly two maximal $ideals (each of which is a maximal T-space). However, aside from these two T-ideals, no examples of maximal T-spaces of k[c]_0 have been identified. This paper presents, for each finite field k, an infinite sequence of proper T-spaces of k[x]_0 (no one of which is a T-ideal), each of finite codimension, and for each one, both a linear basis for the T-space itself and a linear basis for a complementary linear subspace are provided. Morever, it is proven that the first T-space in the sequence is a maximal T-space of k[x]_0, thereby providing the first example of a maximal T-space of k[x]_0 that is not a maximal T-ideal.