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 Λn = K⌊x1, . . . , xn⌋ be the Grassmann algebra over a commutative ring K with 12 ∈ K, and δ be a skew K-derivation of Λn. It is proved that δ is a unique sum δ = δ ev + δ of an even and odd skew derivation. Explicit formulae are given for δ and δ via the elements δ(x1), . . . , δ(xn). It is proved that the set of all even skew derivations of Λn coincides with the set of all the inner skew derivations. Similar results are proved for derivations of Λn. In particular, DerK(Λn) is a faithful but not simple AutK(Λn)module (where K is reduced and n ≥ 2). All differential and skew differential ideals of Λn are found. It is proved that the set of generic normal elements of Λn that are not units forms a single AutK(Λn)-orbit (namely, AutK(Λn)x1) if n is even and two orbits (namely, AutK(Λn)x1 and AutK(Λn)(x1 + x2 · · · xn)) if n is odd.