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

Abstract

Let F〈X〉 be the free unitary associative algebra over a field F on the set X={x1,x2,…}. 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≩V is finitely generated (as a T-subspace) but V itself is not. Recently Brandão 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 Rk (k⩾1) in the algebra F〈X〉 over an infinite field F of characteristic p>2. For each k⩾1, the limit T-subspace Rk arises from the central polynomials in 2k variables of the Grassmann algebra G.