FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2008, VOLUME 14, NUMBER 5, PAGES 171-184

**On the Kurosh problem in varieties of algebras**

D. I. Piontkovski

Abstract

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We consider a couple of versions of the 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 polylinear elements of arbitrarily 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 each such clone may be assumed to be finite-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 omitted.
At the same time, the general relations of operads, algebras, and
varieties are widely discussed.

Location: http://mech.math.msu.su/~fpm/eng/k08/k085/k08512h.htm

Last modified: May 6, 2009