Limit T-subalgebras in free associative algebras

Abstract

Let F〈X〉 be the free unitary associative algebra over a field F on a free generating set X. An unitary subalgebra R of F〈X〉 is called a T-subalgebra if R is closed under all endomorphisms of F〈X〉. A T-subalgebra R⁎ in F〈X〉 is limit if every larger T-subalgebra W⫌R⁎ is finitely generated (as a T-subalgebra) but R⁎ itself is not. It follows easily from Zorn's lemma that if a T-subalgebra R is not finitely generated then it is contained in some limit T-subalgebra R⁎. In this sense limit T-subalgebras form a “border” between those T-subalgebras which are finitely generated and those which are not. In the present article we give the first example of a limit T-subalgebra in F〈X〉, where F is an infinite field of characteristic p>2 and |X|≥4. Note that, by Shchigolev's result, over a field F of characteristic 0 every T-subalgebra in F〈X〉 is finitely generated; hence, over such a field limit T-subalgebras in F〈X〉 do not exist.