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

TL;DR

This paper characterizes the $Z$-graded identities of the Lie algebras $U_1$ and $W_1$ over fields of characteristic two, providing bases for these identities and proving their non-finite basis property.

Contribution

It offers explicit bases for the graded identities of $U_1$ and $W_1$ and demonstrates that these identities cannot be finitely generated.

Findings

Provided bases for the graded identities of $U_1$ and $W_1$

Proved that these identities do not admit finite bases

Enhanced understanding of Lie algebra identities in characteristic two

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.