Derivations and skew derivations of the Grassmann algebras

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