odimension growth for polynomial identities of representations of Lie algebra

Author(s)

da Silva Macedo, David Levi ^[1] ; Koshlukov, Plamen ^[2]

Affiliation

^[1] Univ Fed Rural Pernambuco, Dom Manoel de Medeiros Str, BR-52171900 Recife, PE - Brazil

^[2] Univ Estadual Campinas, 651 Sergio Buarque de Holanda, BR-13083859 Campinas, SP - Brazil

• Document type: Journal article
• Source: Mathematische Nachrichten; v. 295, n. 2, p. 281-308, FEB 2022.
• Web of Science Citations: 0

Abstract

Let K be a field of characteristic zero. We study the asymptotic behavior of the codimensions for polynomial identities of representations of Lie algebras, also called weak identities. These identities are related to pairs of the form (A,L)\$(A,L)\$ where A is an associative enveloping algebra for the Lie algebra L. We obtain a characterization of ideals of weak identities with polynomial growth of the codimensions in terms of their cocharacter sequence. Recall that such a characterization was obtained by Kemer in {[}12] for associative algebras and by Benediktovich and Zalesskii in {[}2] for Lie algebras. We prove that the pairs (UT2,UT2(-))\$\textbackslash{}Big (UT\_2,UT\_2<\^{}>[(-)]\textbackslash{}Big )\$, (E,E(-))\$\textbackslash{}big (E,E<\^{}>[(-)]\textbackslash{}big )\$ and (M2,sl2)\$\textbackslash{}big (M\_2,sl\_2\textbackslash{}big )\$ generate varieties of pairs of almost polynomial growth. Here E denotes the infinite dimensional Grassmann algebra with 1. Also UT2\$UT\_2\$ is the associative subalgebra of M-2 (the 2 x 2 matrices over the field K) consisting of upper triangular matrices and sl2\$sl\_2\$ is the Lie subalgebra of M2(-)\$M\_2<\^{}>[(-)]\$ of the traceless matrices. (AU)

• FAPESP's process: 18/23690-6 - Structures, representations, and applications of algebraic systems
• Grantee: Ivan Chestakov
• Support Opportunities: Research Projects - Thematic Grants