DERIVATIONS AND SKEW DERIVATIONS OF THE GRASSMANN ALGEBRAS

Abstract

Surprisingly, skew derivations rather than ordinary derivations are more basic (important) object in study of the Grassmann algebras. Let Λ_n = K ⌊x₁, …, x_n⌋ be the Grassmann algebra over a commutative ring K with ½ ∈ K, and δ be a skew K-derivation of Λ_n. It is proved that δ is a unique sum δ = δ ^ev + δ ^od of an even and odd skew derivation. Explicit formulae are given for δ^ev and δ^od via the elements δ (x₁), …, δ (x_n). It is proved that the set of all even skew derivations of Λ_n coincides with the set of all the inner skew derivations. Similar results are proved for derivations of Λ_n. In particular, Der _K(Λ_n) is a faithful but not simple Aut _K(Λ_n)-module (where K is reduced and n ≥ 2). All differential and skew differential ideals of Λ_n are found. It is proved that the set of generic normal elements of Λ_n that are not units forms a single Aut _K(Λ_n)-orbit (namely, Aut _K(Λ_n)x₁) if n is even and two orbits (namely, Aut _K(Λ_n)x₁ and Aut _K(Λ_n)(x₁ + x₂ ⋯ x_n)) if n is odd.