Short $\mathrm{SL}_2$-structures on simple Lie algebras

Abstract

In Vinberg's works certain non-Abelian gradings of simple Lie algebras were introduced and investigated, namely, short $\mathrm{SO}_3$- and $\mathrm{SL}_3$-structures. We investigate a different kind of these, short $\mathrm{SL}_2$-structures. The main results refer to the one-to-one correspondence between such structures and certain special Jordan algebras. Bibliography: 8 titles.