On the identical relations of associative and Lie algebras equipped with an action
This article deals with identities of algebras with actions. The very general type of action is as follows: Let \(R\) and \(A\) be associative \(k\)-algebras, with \(R\rightarrow \mathrm{End}_k(A)\). Then \(A\) is a generalized \(R\)-algebra if for each \(r\in R\) there exists two finite sets of pairs \(r^+=(r_{1,i}^+,r_{2,i}^+)\) and \(r^-=(r_{1,j}^-,r_{2,j}^-)\) such that for all \(r\in R\) and \(a_1, a_2\in A\) \[ r(a_1a_2)=\sum_i r_{1,i}^+(a_1)r_{2,i}^+(a_2)+ \sum_j r_{1,j}^-(a_2)r_{2,j}^-(a_1). \] The main theorem concerns actions with two properties: The first is that the image of \(R\) in \(\mathrm{End}_k (A)\) is finite dimensional, say of dimension~\(m\); the second is that \(A\) satisfies an identity of the form \[ x_1\cdots x_d=\sum_{\sigma\ne1}\sum r_1(x_{\sigma(1)}) \cdots r_d(x_{\sigma(d)}), \] where the inner sum ranges over a finite set of \(d\)-tuples of elements of \(R\). In this case the authors' prove that the algebra \(A\) must satisfy an ordinary polynomial identity. Moreover, they bound the degree of the minimal identity as an explicit function of \(m\) and \(d\).
scientific article; zbMATH DE number 7622329
scientific article