Abstract:
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.
Citation:
D. I. Piontkovski, “On the Kurosh problem in varieties of algebras”, Fundam. Prikl. Mat., 14:5 (2008), 171–184; J. Math. Sci., 163:6 (2009), 743–750
\Bibitem{Pio08}
\by D.~I.~Piontkovski
\paper On the Kurosh problem in varieties of algebras
\jour Fundam. Prikl. Mat.
\yr 2008
\vol 14
\issue 5
\pages 171--184
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\jour J. Math. Sci.
\yr 2009
\vol 163
\issue 6
\pages 743--750
\crossref{https://doi.org/10.1007/s10958-009-9711-9}
\elib{https://elibrary.ru/item.asp?id=15299454}
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Linking options:
https://www.mathnet.ru/eng/fpm1149
https://www.mathnet.ru/eng/fpm/v14/i5/p171
This publication is cited in the following 10 articles:
A. M. Elishev, A. Ya. Belov, F. Razavinia, Yu Jie-Tai, Wenchao Zhang, “Polynomial automorphisms, quantization, and Jacobian conjecture related problems. I. Introduction”, Geometriya, mekhanika i differentsialnye uravneniya, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 213, VINITI RAN, M., 2022, 110–144
A. M. Elishev, A. Ya. Belov, F. Razavinia, Yu Jie-Tai, Wenchao Zhang, “Polynomial automorphisms, quantization, and Jacobian conjecture related problems. II. Quantization proof of Bergman's centralizer theorem”, Algebra, geometriya i kombinatorika, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 214, VINITI RAN, M., 2022, 107–126
A. M. Elishev, A. Ya. Belov, F. Razavinia, Yu Dzhi-Tai, Venchao Zheng, “Polinomialnye avtomorfizmy, kvantovanie i zadachi vokrug gipotezy Yakobiana. III. Avtomorfizmy, topologiya popolneniya i approksimatsiya”, Algebra, geometriya i kombinatorika, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 215, VINITI RAN, M., 2022, 95–128
A. M. Elishev, A. Ya. Belov, F. Razavinia, Yu Dzhi-Tai, Venchao Zheng, “Polinomialnye avtomorfizmy, kvantovanie i zadachi vokrug gipotezy Yakobiana. IV. Approksimatsii polinomialnymi simplektomorfizmami”, Algebra, geometriya, differentsialnye uravneniya, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 216, VINITI RAN, M., 2022, 153–171
A. M. Elishev, A. Ya. Belov, F. Razavinia, Yu Dzhi-Tai, Venchao Zheng, “Polinomialnye avtomorfizmy, kvantovanie i zadachi vokrug gipotezy Yakobiana. V. Gipoteza Yakobiana i problemy tipa Shpekhta i Bernsaida”, Algebra, geometriya, differentsialnye uravneniya, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 217, VINITI RAN, M., 2022, 107–137
Alexei Kanel-Belov, Alexei Chilikov, Ilya Ivanov-Pogodaev, Sergey Malev, Eugeny Plotkin, Jie-Tai Yu, Wenchao Zhang, “Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry”, Mathematics, 8:10 (2020), 1694
Ilya Ivanov-Pogodaev, Sergey Malev, “Finite Gröbner basis algebras with unsolvable nilpotency problem and zero divisors problem”, Journal of Algebra, 508 (2018), 575
Dmitri Piontkovski, Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation, 2017, 373
Alexei Belov, Leonid Bokut, Louis Rowen, Jie-Tai Yu, Springer Proceedings in Mathematics & Statistics, 79, Automorphisms in Birational and Affine Geometry, 2014, 249
Aljadeff E., Kanel-Belov A., “Representability and Specht problem for G-graded algebras”, Adv. Math., 225:5 (2010), 2391–2428
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