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On the Kurosh problem in varieties of algebras
Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 5, pp. 171-184
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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. 
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D. I. Piontkovski. On the Kurosh problem in varieties of algebras. Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 5, pp. 171-184. http://geodesic.mathdoc.fr/item/FPM_2008_14_5_a11/

[1] Golod E. S., “O nil-algebrakh i finitno-approksimiruemykh $p$-gruppakh”, Izv. AN SSSR. Ser. mat., 28:2 (1964), 273–276 | MR | Zbl

[2] Golod E. S., Shafarevich I. R., “O bashne polei klassov”, Izv. AN SSSR. Ser. mat., 28:2 (1964), 261–272 | MR | Zbl

[3] Kurosh A. G., Obschaya algebra, Lektsii 1969–1970 uchebnogo goda, Nauka, M., 1974 | MR | Zbl

[4] Ufnarovskii V. A., “Kombinatornye i asimptoticheskie metody v algebre”, Itogi nauki i tekhn. Ser. Sovr. mat. i ee pril., 57, VINITI, M., 1990, 5–177 | MR

[5] Fresse B., “Koszul duality of operads and homology of partition posets”, Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory, Papers from the Int. Conf. on Algebraic Topology (Northwestern Univ., Evanston, IL, USA, March 24–28, 2002), Contemp. Math., 346, ed. P. Goerss, Amer. Math. Soc., Providence, 2004, 115–215 | MR | Zbl

[6] Giambruno A., Zaicev M., Polynomial Identities and Asymptotic Methods, Math. Surveys Monographs, 122, Amer. Math. Soc., Providence, 2005 | MR | Zbl

[7] Ginzburg V., Kapranov M., “Koszul duality for operads”, Duke Math. J., 76:1 (1994), 203–272 ; “Erratum”, 80:1 (1995), 293 | DOI | MR | Zbl | DOI | MR | Zbl

[8] Kanel-Belov A., Rowen L. H., Computational Aspects of Polynomial Identities, Research Notes Math., 9, Peters, Wellesley, 2005 | MR | Zbl

[9] Markl M., Shnider S., Stasheff J., Operads in Algebra, Topology and Physics, Math. Surveys Monographs, 96, Amer. Math. Soc., Providence, 2002 | MR | Zbl

[10] Piontkovski D., “Burnside problem for varieties of algebras and operads” (to appear)