Algebras and semigroups of locally subexponential growth

Abstract

We prove that a countable dimensional associative algebra (resp. a countable semigroup) of locally subexponential growth is M ∞ -embeddable as a left ideal in a finitely generated algebra (resp. semigroup) of subexponential growth. Moreover, we provide bounds for the growth of the finitely generated algebra (resp. semigroup). The proof is based on a new construction of matrix wreath product of algebras.

Embeddings in Lie algebras of subexponential growth

“…The key role in their construction was played by wreath products with the infinite cyclic group. In [2], we used matrix wreath products of algebras to embed an arbitrary countable dimensional associative algebra that is locally of subexponential growth into a finitely generated associative algebra of subexponential growth. In this paper, we obtain a similar result for Lie algebras.…”

“…By the lemma, the algebra U is embeddable in a countable dimensional algebra A, that is locally of subexponential growth and the image of U is contained in the subspace [A, A]. By the Theorem 2 from [2], the algebra A is embeddable in a finitely generated algebra B of subexponential growth. Without loss of generality, we will assume that B ∋ 1.…”