Automorphisms and superalgebra structures on the Grassmann algebra

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 $E_{can}$.

19 pages