# Central elements in the distribution algebra of a general linear supergroup and supersymmetric elements

## TL;DR

This paper studies the center of the distribution algebra of the general linear supergroup over fields of positive characteristic, focusing on supersymmetric elements and their generators within Frobenius kernels.

## Contribution

It introduces the concept of supersymmetric elements in the distribution algebra and describes a minimal generating set for these elements over Frobenius kernels.

## Key findings

- Images of central elements are supersymmetric
- Supersymmetric elements form an algebra with a minimal generating set
- Provides structural insights into the distribution algebra of supergroups

## Abstract

In this paper we investigate the image of the center $Z$ of the distribution algebra $Dist(GL(m|n))$ of the general linear supergroup over a ground field of positive characteristic under the Harish-Chandra morphism $h:Z \to Dist(T)$ obtained by the restriction of the natural map $Dist(GL(m|n))\to Dist(T)$. We define supersymmetric elements in $Dist(T)$ and show that each image $h(c)$ for $c\in Z$ is supersymmetric. The central part of the paper is devoted to a description of a minimal set of generators of the algebra of supersymmetric elements over Frobenius kernels $T_r$.