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

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.

The proof of Theorem 4.7 is incomplete