Graded Identities for the Adjoont Representation of $sl_2$

TL;DR

This paper characterizes the finite basis of graded identities for the adjoint representation of the Lie algebra sl_2 over a field of characteristic zero, focusing on the canonical Z_2-grading.

Contribution

It provides the first explicit description of a finite basis for the Z_2-graded identities of the adjoint representation of sl_2.

Findings

Finite basis for Z_2-graded identities of sl_2's adjoint representation

Explicit description of identities for the pair (M_3(K), sl_2(K))

Analysis of the canonical grading on sl_2 and induced grading on M_3(K)

Abstract

Let $K$ be a field of characteristic zero and let $\mathfrak{sl}_2 (K)$ be the 3-dimensional simple Lie algebra over $K$. In this paper we describe a finite basis for the $\mathbb{Z}_2$-graded identities of the adjoint representation of $\mathfrak{sl}_2 (K)$, or equivalently, the $\mathbb{Z}_2$-graded identities for the pair $(M_3(K), \mathfrak{sl}_2 (K))$. We work with the canonical grading on $\mathfrak{sl}_2 (K)$ and the only nontrivial $\mathbb{Z}_2$-grading of the associative algebra $M_3(K)$ induced by that on $\mathfrak{sl}_2(K)$.