Automorphisms and superalgebra structures on the Grassmann algebra

TL;DR

This paper investigates the superalgebra structures on the infinite-dimensional Grassmann algebra over a field of characteristic zero, classifying gradings and polynomial identities, and constructing new non-homogeneous gradings isomorphic to known cases.

Contribution

It classifies superalgebra structures on the Grassmann algebra, linking automorphisms of order 2 to gradings, and constructs new gradings isomorphic to known types.

Findings

Polynomial identities of graded structures match those of typical cases.

Many non-homogeneous gradings are isomorphic to standard cases.

A new grading with a single generator is constructed, isomorphic to the natural grading.

Abstract

Let $F$ be a field of characteristic zero and let $E$ be the Grassmann algebra of an infinite dimensional $F$-vector space $L$. In this paper we study the superalgebra structures (that is the $\mathbb{Z}_{2}$-gradings) that the algebra $E$ admits. By using the duality between superalgebras and automorphisms of order $2$ we prove that in many cases the $\mathbb{Z}_{2}$-graded polynomial identities for such structures coincide with the $\mathbb{Z}_{2}$-graded polynomial identities of the "typical" cases $E_{\infty}$, $E_{k^\ast}$ and $E_{k}$ where the vector space $L$ is homogeneous. Recall that these cases were completely described by Di Vincenzo and Da Silva in \cite{disil}. Moreover we exhibit a wide range of non-homogeneous $\mathbb{Z}_{2}$-gradings on $E$ that are $\mathbb{Z}_{2}$-isomorphic to $E_{\infty}$, $E_{k^\ast}$ and $E_{k}$. In particular we construct a $\mathbb{Z}_{2}$-grading on $E$ with only one homogeneous generator in $L$ which is $\mathbb{Z}_{2}$-isomorphic to the natural $\mathbb{Z}_{2}$-grading on $E$, here denoted by $E_{can}$.