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$\mathbb{Z}$-graded identities of the Lie algebras $U_1$ in characteristic 2

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

Published online by Cambridge University Press:  08 March 2022

CLAUDEMIR FIDELIS  and
PLAMEN KOSHLUKOV
CLAUDEMIR FIDELIS
Affiliation:
Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, 785 Aprígio Veloso, Bodocongó, P.O.Box: 10044, Campina Grande, PB, 58429-970, Brazil Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, SP, 05508-090, Brazil. e-mail: claudemir@mat.ufcg.edu.br
PLAMEN KOSHLUKOV
Affiliation:
Department of Mathematics, UNICAMP, 651 Sergio Buarque de Holanda, 13083-859 Campinas, SP, Brazil. e-mail: plamen@unicamp.br
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Abstract

Let K be any field of characteristic two and let $U_1$ and $W_1$ be the Lie algebras of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$ and of the polynomial ring K[t], respectively. The algebras $U_1$ and $W_1$ are equipped with natural $\mathbb{Z}$-gradings. In this paper, we provide bases for the graded identities of $U_1$ and $W_1$, and we prove that they do not admit any finite basis.

MSC classification

Primary: 16R10: $T$-ideals, identities, varieties of rings and algebras
Secondary: 17B01: Identities, free Lie (super)algebras 17B65: Infinite-dimensional Lie (super)algebras 17B66: Lie algebras of vector fields and related (super) algebras 17B70: Graded Lie (super)algebras

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 174 , Issue 1 , January 2023 , pp. 49 - 58
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

Supported by FAPESP grant No. 2019/12498-0.

‡

Partially supported by FAPESP grant No. 2018/23690-6 and by CNPq grant No. 302238/2019-0.

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