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 $E_{ can}$.
Categories: math.RA