# Supersymmetric elements in divided powers algebras

## TL;DR

This paper introduces supersymmetric elements in divided powers algebras, characterizes them through linear equations, and determines generators for specific cases, advancing the understanding of invariants in Lie superalgebras.

## Contribution

It defines supersymmetric elements in divided powers algebras and provides explicit generators for particular cases, extending prior work on invariants of Lie superalgebras.

## Key findings

- Characterization of supersymmetric elements via linear equations
- Generators determined for cases n=0, n=1, m≤2, n=2
- Extension of invariants description in divided powers algebras

## Abstract

Description of adjoint invariants of general Linear Lie superalgebras $\mathfrak{gl}(m|n)$ by Kantor and Trishin is given in terms of supersymmetric polynomials. Later, generators of invariants of the adjoint action of the general linear supergroup $GL(m|n)$ and generators of supersymmetric polynomials were determined over fields of positive characteristic. In this paper, we introduce the concept of supersymmetric elements in the divided powers algebra $Div[x_1, \ldots, x_m,y_1, \ldots, y_n]$, and give a characterization of supersymmetric elements via a system of linear equations. Then we determine generators of supersymmetric elements for divided powers algebras in the cases when $n=0$, $n=1$, and $m\leq 2, n=2$.