# Lie super-bialgebra structures on a class of generalized super $W$-algebra $\mathfrak{L}$

## TL;DR

This paper classifies all Lie super-bialgebra structures on a specific generalized super $W$-algebra, showing they are all triangular coboundary by analyzing its cohomology.

## Contribution

It proves the triviality of the first cohomology group for the algebra, leading to a complete classification of its Lie super-bialgebra structures as triangular coboundary.

## Key findings

- All Lie super-bialgebra structures are triangular coboundary.
- The first cohomology group with coefficients in the adjoint tensor module is trivial.
- Provides a cohomological approach to classifying super-bialgebra structures.

## Abstract

In this paper, Lie super-bialgebra structures on a class of generalized super $W$-algebra $\mathfrak{L}$ are investigated. By proving the first cohomology group of $\mathfrak{L}$ with coefficients in its adjoint tensor module is trivial, namely, $H^1(\mathfrak{L},\mathfrak{L}\otimes {\mathfrak{L}})=0$, we obtain that all Lie super-bialgebra structures on $\mathfrak{L}$ are triangular coboundary.