Limit T-subspaces and the central polynomials in n variables of the Grassmann algebra

TL;DR

This paper investigates the structure of limit T-subspaces in free associative algebras, disproving a previous conjecture by constructing infinitely many such subspaces related to the Grassmann algebra's central polynomials.

Contribution

The authors demonstrate that, contrary to prior beliefs, there are infinitely many limit T-subspaces in F<X> associated with the Grassmann algebra, not just the unique one previously conjectured.

Findings

Constructed infinitely many limit T-subspaces R_k for each k ≥ 1

Showed each R_k arises from central polynomials in 2k variables

Disproved the conjecture of uniqueness of the limit T-subspace C(G)

Abstract

Let F<X> be the free unitary associative algebra over a field F on the set X = {x_1, x_2, ...}. A vector subspace V of F<X> is called a T-subspace (or a T-space) if V is closed under all endomorphisms of F<X>. A T-subspace V in F<X> is limit if every larger T-subspace W \gneqq V is finitely generated (as a T-subspace) but V itself is not. Recently Brand\~ao Jr., Koshlukov, Krasilnikov and Silva have proved that over an infinite field F of characteristic p>2 the T-subspace C(G) of the central polynomials of the infinite dimensional Grassmann algebra G is a limit T-subspace. They conjectured that this limit T-subspace in F<X> is unique, that is, there are no limit T-subspaces in F<X> other than C(G). In the present article we prove that this is not the case. We construct infinitely many limit T-subspaces R_k (k \ge 1) in the algebra F<X> over an infinite field F of characteristic p>2. For each k \ge 1, the limit T-subspace R_k arises from the central polynomials in 2k variables of the Grassmann algebra G.