Representations of short $SL_2$-structures

The well-known Tits-Kantor-Koeher construction makes it possible to contruct a Lie algebra of a special kind from a Jordan algebra. In 2005, I. Kantor and G. Shpiz proposed a way to connect the representation theory of Jordan algebras with the representation theory of Lie algebras. In this article, we use an alternative way to confirm this connection. Namely, we use an analogue of so-called $S$ structures on Lie modules for $S= SL_2$. The aim of our work is to prove a one-to-one correspondence between representations of Lie algebras with $SL_2$-structure and special representations of Jordan algebras.