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Codimension growth for weak polynomial identities, and non-integrality of the PI exponent

Part of: Rings with polynomial identity Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  20 July 2020

David Levi da Silva Macedo  and
Plamen Koshlukov
David Levi da Silva Macedo
Affiliation:
Department of Mathematics, State University of Campinas, 651 Sérgio Buarque de Holanda, Campinas, SP13083-859, Brazil (david.levi@ufrpe.br; plamen@unicamp.br)
Plamen Koshlukov
Affiliation:
Department of Mathematics, State University of Campinas, 651 Sérgio Buarque de Holanda, Campinas, SP13083-859, Brazil (david.levi@ufrpe.br; plamen@unicamp.br)
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Abstract

Let K be a field of characteristic zero. In this paper, we study the polynomial identities of representations of Lie algebras, also called weak identities, or identities of pairs. These identities are determined by pairs of the form (A, L) where A is an associative enveloping algebra for the Lie algebra L. Then a weak identity of (A, L) (or an identity for the representation of L associated to A) is an associative polynomial which vanishes when evaluated on elements of L⊆ A. One of the most influential results in the area of PI algebras was the theory developed by Kemer. A crucial role in it was played by the construction of the Grassmann envelope of an associative algebra and the close relation of the identities of the algebra and its Grassmann envelope. Here we consider varieties of pairs. We prove that under some restrictions one can develop a theory similar to that of Kemer's in the study of identities of representations of Lie algebras. As a consequence, we establish that in the case when K is algebraically closed, if a variety of pairs does not contain pairs corresponding to representations of sl2(K), and if the variety is generated by a pair where the associative algebra is PI then it is soluble. As another consequence of the methods used to obtain the above result, and applying ideas from papers by Giambruno and Zaicev, we were able to construct a pair (A, L) such that its PI exponent (if it exists) cannot be an integer. We recall that the PI exponent exists and is an integer whenever A is an associative (a theorem by Giambruno and Zaicev), or a finite-dimensional Lie algebra (Zaicev). Gordienko also proved that the PI exponent exists and is an integer for finite-dimensional representations of Lie algebras.

Keywords

Weak identitiesCodimensionsCodimension growthNon-integer PI exponent

MSC classification

Primary: 17B01: Identities, free Lie (super)algebras
Secondary: 17B30: Solvable, nilpotent (super)algebras 17B70: Graded Lie (super)algebras 16R10: $T$-ideals, identities, varieties of rings and algebras
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 4 , November 2020 , pp. 929 - 949
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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