On the identical relations of associative and Lie algebras equipped with an action

Abstract

Let R be a unitary associative algebra over a field. We call an algebra A a generalized R-algebra when A is endowed with an R-module action with the property that, for each r∈R, there exists finitely many elements r+=(r1+,r2+)∈R2 and r−=(r1−,r2−)∈R2 such that, for all a1,a2∈A,r⋅(a1a2)=∑r+(r1+⋅a1)(r2+⋅a2)+∑r−(r2−⋅a2)(r1−⋅a1). Suppose an associative generalized R-algebra A satisfies an identical relation of the formx1⋯xd−∑1≠σ∈Sd∑r(r1⋅xσ(1))⋯(rd⋅xσ(d))≡0, where Sd denotes the symmetric group of degree d and the inner sum runs over finitely many r=(r1,…,rd)∈Rd. We prove: if the algebra of endomorphisms on A defined by the action of R is m-dimensional, then A satisfies a classical polynomial identity of degree bounded by an explicit function of d and m only. We also prove the analogous result holds when A is a Lie algebra, thus extending a collection of results in associative and Lie PI-theory.