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arXiv · 2608.22385

Leibniz algebras and their connection to Jordan pair disystems

Esther García·Miguel Gómez Lozano·Rubén Muñoz Alcázar·Guillermo Vera de Salas
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Abstract

In this paper we present a generalization of Jordan pairs, called Jordan pair disystem, which appear naturally as the wings of a Leibniz algebra with a finite $\mathbb{Z}$-grading; conversely, we give the Tits-Kantor-Koecher construction for Jordan pair disystems, obtaining a Leibniz algebra with a short $\mathbb{Z}$-grading. This construction extends the construction given by Gubarev and Kolesnikov for Jordan dialgebras (arXiv:0907.1740). We introduce homotopes for Jordan pair disystems at elements and we show that we obtain again a Jordan dialgebra. Moreover, we introduce abelian inner ideals and their kernels for Leibniz algebras, and prove that the subquotient of a Leibniz algebra with respect to an abelian inner ideal is a Jordan pair disystem. This notion of subquotient extends the construction of Jordan dialgebras at $Q$-Jordan elements given by Felipe and Vel\'asquez.

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Esther García, Miguel Gómez Lozano, Rubén Muñoz Alcázar, Guillermo Vera de Salas. 2026-08-23. Leibniz algebras and their connection to Jordan pair disystems. https://arxiv.org/abs/2608.22385

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