Derivations and skew derivations of the Grassmann algebras

TL;DR

This paper studies skew derivations of Grassmann algebras, providing explicit formulas, classifying even and odd derivations, and analyzing their algebraic structure and automorphism orbits.

Contribution

It introduces explicit formulas for skew derivations of Grassmann algebras and classifies their structure, including even and odd derivations and their relation to inner derivations.

Findings

Every skew derivation decomposes uniquely into even and odd parts.

The set of even skew derivations coincides with inner skew derivations.

The automorphism group acts transitively on certain normal elements, forming one or two orbits depending on parity.

Abstract

Surprisingly, skew derivations rather than ordinary derivations are more basic (important) object in study of the Grassmann algebras. Let $\L_n = K\lfloor x_1, ..., x_n\rfloor$ be the Grassmann algebra over a commutative ring $K$ with ${1/2}\in K$, and $\d$ be a skew $K$-derivation of $\L_n$. It is proved that $\d$ is a unique sum $\d = \d^{ev} +\d^{od}$ of an even and odd skew derivation. Explicit formulae are given for $\d^{ev}$ and $\d^{od}$ via the elements $\d (x_1), ..., \d (x_n)$. It is proved that the set of all even skew derivations of $\L_n$ coincides with the set of all the inner skew derivations. Similar results are proved for derivations of $\L_n$. In particular, $\Der_K(\L_n)$ is a faithful but not simple $\Aut_K(\L_n)$-module (where $K$ is reduced and $n\geq 2$). All differential and skew differential ideals of $\L_n$ are found. It is proved that the set of generic normal elements of $\L_n$ that are not units forms a single $\Aut_K(\L_n)$-orbit (namely, $\Aut_K(\L_n)x_1$) if $n$ is even and two orbits (namely, $\Aut_K(\L_n)x_1$ and $\Aut_K(\L_n)(x_1+x_2... x_n)$) if $n$ is odd.