Interplay between $\mathbb{Z}_2$-gradings and automorphisms in the Grassmann algebra: a survey

TL;DR

This survey explores the relationship between $Z_2$-gradings and automorphisms in the infinite-dimensional Grassmann algebra, highlighting recent results, examples, and a conjecture on classifying superalgebra structures.

Contribution

It compiles recent findings on automorphisms of order two and their induced superalgebra structures, including concrete examples and a new conjecture.

Findings

Automorphisms of order two induce superalgebra structures.

Existence of non-homogeneous automorphisms in the Grassmann algebra.

A conjecture on classifying superalgebra structures.

Abstract

Let $F$ be a field of characteristic different from two, and let $E$ be the Grassmann algebra of an infinite-dimensional $F$-vector space $L$. In this paper, we survey recent results concerning automorphisms of order two of $E$ and the corresponding induced superalgebra structures. We also present concrete examples of non-homogeneous automorphisms of $E$, as discussed in \cite{agpkauto, anagomes}. The paper concludes with a conjecture regarding the classification of the superalgebra structures of $E$.