Parity duality of super $r$-matrices via $\mathcal O$-operators and pre-Lie superalgebras

TL;DR

This paper explores the duality and hierarchy of super $r$-matrices through $ ext{O}$-operators and pre-Lie superalgebras, revealing a parity duality between even and odd structures in superalgebra theory.

Contribution

It generalizes the classical correspondence between $r$-matrices and $ ext{O}$-operators to the super case and establishes a parity duality linking even and odd $ ext{O}$-operators and super $r$-matrices.

Findings

Established a duality between even and odd $ ext{O}$-operators.

Constructed an infinite hierarchy of super $r$-matrices.

Connected pre-Lie superalgebras to parity pairs of $ ext{O}$-operators and super $r$-matrices.

Abstract

This paper studies super $r$-matrices and operator forms of the super classical Yang-Baxter equation. First by a unified treatment, the classical correspondence between $r$-matrices and $\mathcal{O}$-operators is generalized to a correspondence between homogeneous super $r$-matrices and homogeneous $\mathcal{O}$-operators. Next, by a parity reverse of Lie superalgebra representations, a duality is established between the even and the odd $\mathcal{O}$-operators, giving rise to a parity duality among the induced super $r$-matrices. Thus any homogeneous $\OO$-operator or any homogeneous super $r$-matrix with certain supersymmetry produces a parity pair of super $r$-matrices, and generates an infinite tree hierarchy of homogeneous super $r$-matrices. Finally, a pre-Lie superalgebra naturally defines a parity pair of $\mathcal{O}$-operators, and thus a parity pair of super $r$-matrices.