On the Kurosh problem in varieties of algebras

被引：0

作者：

Piontkovski D.I. ^{[1
]}

机构：

[1] Department of High Mathematics for Economics, State University Higher School of Economics, Moscow 101990

来源：

Journal of Mathematical Sciences | 2009年 / 163卷 / 6期

基金：

俄罗斯基础研究基金会;

关键词：

Associative Algebra; Formal Power Series; Polynomial Identity; Free Algebra; Free Resolution;

D O I：

10.1007/s10958-009-9711-9

中图分类号：

学科分类号：

摘要：

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 cast the problem in the language of operads and then apply the usual homological constructions in order to adopt Golod's solution to the original Kurosh problem. This paper is expository, so that some proofs are omitted. At the same time, the general relations of operads, algebras, and varieties are widely discussed. © 2009 Springer Science+Business Media, Inc.

引用

页码：743 / 750

页数：7

共 10 条

[1] 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, 346, pp. 115-215, (2004)
[2] Giambruno A., Zaicev M., Polynomial Identities and Asymptotic Methods, Math. Surveys Monographs, 122, (2005)
[3] Ginzburg V., Kapranov M., Koszul duality for operads, Duke Math. J., 76, 1, pp. 203-272, (1994)
[4] Golod E.S., On nil algebras and finitely approximable groups, Izv. Akad. Nauk SSSR, Ser. Mat., 28, pp. 273-276, (1964)
[5] Golod E.S., Shafarevich I.R., On the class field tower, Izv. Akad. Nauk SSSR, Ser. Mat., 28, pp. 261-272, (1964)
[6] Kanel-Belov A., Rowen L.H., Computational Aspects of Polynomial Identities, Research Notes Math., 9, (2005)
[7] Kurosh A.G., General Algebra. Lectures of 1969-1970 Academic Year, (1974)
[8] Markl M., Shnider S., Stasheff J., Operads in Algebra, Topology and Physics, Math. Surveys Monographs, 96, (2002)
[9] Piontkovski D., Burnside Problem for Varieties of Algebras and Operads
[10] Ufnarovsky V.A., Combinatorial and asymptotical methods in algebra, Algebra VI, Encycl. Math. Sci., pp. 1-196, (1995)

← 1 →