Poisson Yang-Baxter equations and $\mathcal{O}$-operators of Poisson superalgebras

TL;DR

This paper explores the relationship between $ ext{O}$-operators and solutions to the Poisson Yang-Baxter equation in Poisson superalgebras, revealing new connections and conditions for solutions.

Contribution

It establishes a link between $ ext{O}$-operators and skew-symmetric solutions of the PYBE, including cases involving symplectic forms and semi-direct products.

Findings

Skew-symmetric solutions of PYBE correspond to $ ext{O}$-operators on Poisson superalgebras.

Non-degenerate solutions relate to symplectic forms.

General $ ext{O}$-operators can produce PYBE solutions in semi-direct products.

Abstract

We investigate connections between $\mathcal {O}$-operators of Poisson superalgebras and skew-symmetric solutions of the Poisson Yang-Baxter equation (PYBE). We prove that a skew-symmetric solution of the PYBE on a Poisson superalgebra can be interpreted as an $\mathcal {O}$-operator associated to the co-regular 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 PYBE in certain semi-direct product of Poisson superalgebras.