# On Hom-Lie superbialgebras

## TL;DR

This paper extends the theory of Hom-Lie bialgebras to the superalgebra setting, introducing new constructions, structures, and deformations specific to Hom-Lie superbialgebras.

## Contribution

It generalizes Hom-Lie bialgebras to the $ ext{Z}_2$-graded case and develops new methods for constructing and analyzing Hom-Lie superbialgebras.

## Key findings

- Introduced methods for constructing Hom-Lie superbialgebras
- Defined and related Matched pairs and Manin supertriples
- Studied coboundary, triangular structures, and deformations

## Abstract

The purpose of this paper is to generalize to $\mathbb{Z}_2$-graded case the study of Hom-Lie bialgebras which were discussed first by D. Yau, then by C. Bai and Y. Sheng. We provide different ways for constructing Hom-Lie superbialgebras. Also we define Matched pairs, Manin supertriples and discuss their relationships. Moreover, we study coboundary and triangular Hom-Lie bialgebras, as well as infinitesimal deformations of the cobracket.