Short $SL_2$-structures on simple Lie algebras

TL;DR

This paper explores short $SL_2$-structures on simple Lie algebras, establishing a correspondence with certain Jordan algebras, expanding the understanding of non-abelian gradings in Lie theory.

Contribution

It introduces and studies short $SL_2$-structures on simple Lie algebras and links them to specific Jordan algebras, a new perspective in Lie algebra gradings.

Findings

Establishes a one-to-one correspondence between short $SL_2$-structures and special Jordan algebras.

Extends the framework of non-abelian gradings in simple Lie algebras.

Provides new classifications of Lie algebra structures based on $SL_2$-gradings.

Abstract

Throughout the papers of E.B. Vinberg some non-abelian gradings of simple Lie algebras were introduced and investigated, namely short $SO_3 -$ and $SL_3$ - structures. We study another kind of them -- short $SL_2$ - structures. The main results relate to one-to-one corresponding between such structures and some special Jordan algebras.