Graded Identities for the Adjoont Representation of $sl_2$

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 $\mathfrak{sl}_2(K)$.

Theorems & Definitions (31)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof