The Tits--Kantor--Koecher construction for Jordan dialgebras

Abstract

We study a noncommutative generalization of Jordan algebras called Jordan dialgebras. These are algebras that satisfy the identities $[x_1 x_2]x_3= 0$, $(x_1^2,x_2,x_3)=2(x_1,x_2,x_1x_3)$, $x_1(x_1^2 x_2)=x_1^2(x_1 x_2)$; they are related with Jordan algebras in the same way as Leibniz algebras are related to Lie algebras. We present an analogue of the Tits---Kantor---Koecher construction for Jordan dialgebras that provides an embedding of such an algebra into Leibniz algebra.