Normal bases of PI-algebras

What is it about?

The paper contains a survey of problems and results concerning normal bases of affine PI-algebras. The authors consider various algorithmic questions inspired by Shirshov's lemma and Shirshov's height theorem, which provided a solution to Kurosh's problem on the local finiteness of algebraic algebras. They point out connections between these notions and certain algebraic properties. Specifically, the relationship of an essential Shirshov base and an essential height to the representability and the Gelʹfand-Kirillov dimension of affine PI-algebras, respectively, and the relationship of a power vector description to the algorithmic solvability of the isomorphism problem for subalgebras of the matrix algebra over the ring of polynomials. More attention is paid to problems and results for monomial algebras. In particular, the authors give the representability criterion for a monomial algebra in terms of normal bases.

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