Skew-symmetric solutions of the classical Yang-Baxter equation and $\mathcal{O}$-operators of Malcev algebras

Abstract

We study connections between skew-symmetric solutions of the classical Yang-Baxter equation (CYBE) and $\mathcal{O}$-operators of Malcev algebras. We prove that a skew-symmetric solution of the CYBE on a Malcev algebra can be interpreted as an $\mathcal{O}$-operator associated to the coadjoint representation. We show that this connection can be enhanced with symplectic forms when considering non-degenerate skew-symmetric solutions. We also show that $\mathcal{O}$-operators associated to a general representation could give skew-symmetric solutions of the CYBE on certain semi-direct products of Malcev algebras. We reveal the relationship between invertible $\mathcal{O}$-operators and compatible pre-Malcev algebra structures on a Malcev algebra. We finally obtain several analogous results on connections between the CYBE and $\mathcal{O}$-operators in the case of pre-Malcev algebras.