# Block Degeneracy for Graded Lie Superalgebras of Cartan Type

## TL;DR

This paper demonstrates that certain graded Cartan type Lie superalgebras over algebraically closed fields of positive characteristic have a category of restricted supermodules consisting of a single block, simplifying their representation theory.

## Contribution

It introduces a class of Lie superalgebras with a unified block structure for their restricted supermodules, extending to graded Cartan types W, S, and H under specific conditions.

## Key findings

- Category of restricted supermodules is of one block for the introduced Lie superalgebras.
- For p > 3, graded Cartan type Lie superalgebras of types W, S, H also have a single block category.
- Simplifies the representation theory of these Lie superalgebras.

## Abstract

Let $\mathbb{k}$ be an algebraically closed field of characteristic $ p>0. $ In this short note, we illustrate a class of Lie superalgebras over $ \mathbb{k} $ such that the category of restricted supermodules is of one block. As an application, if $ p>3 $ and $ \mathfrak{g} $ is a graded restricted Cartan type Lie superalgebra of type W, S and H, then the category of restricted $ \mathfrak{g} $ supermodules is of one block.