On Kurosh problem in varieties of algebras

Abstract

We consider a couple of versions of classical Kurosh problem (whether there is an infinite-dimensional algebraic algebra?) for varieties of linear multioperator algebras over a field. We show that, given an arbitrary signature, there is a variety of algebras of this signature such that the free algebra of the variety contains multilinear elements of arbitrary large degree, while the clone of every such element satisfies some nontrivial identity. If, in addition, the number of binary operations is at least 2, then one can guarantee that each such clone is finitely-dimensional. Our approach is the following: we translate the problem to the language of operads and then apply usual homological constructions, in order to adopt Golod's solution of the original Kurosh problem. The paper is expository, so that some proofs are omited. At the same time, the general relations of operads, algebras, and varieties are widely discussed.