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 $[x_1{x}_2]x_3=0$, $(x_1^2,x_2,x_3)=2(x_1,x_2,x_1{x}_3)$, $x_1(x_1^2{x}_2)=x_1^2(x_1{x}_2)$; 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.

22 pages; v2: Example 4 corrected; v3: final version, to appear in Comm. Algebra

References in corpus (2)

• Formal distribution algebras and conformal algebras
• Special identities for quasi-Jordan algebras