K-automorphisms of a weak-crossed product F-algebra over a Galois extension K/F

Abstract

We compute the K-automorphism group of AK=K⊗FAf of a weak crossed product algebra Af for a weak 2-cocycle f over a Galois extension K/F with Galois group G. The K-automorphism group of AK decomposes into its unipotent part and the reductive part Hˆ. Automorphisms in Hˆ are computed via their restriction to semi-simple part of AK. There is a strong relationship between Hˆ and the lower-subtractive relation (≤) induced by f on G. We introduce a subgroup of Hˆ, namely Λ, which contains interesting combinatorial information of ≤. We also present a duality on lower subtractive relations which simplifies the computation of the automorphism group. For the Weak Bruhat order of a Coxeter system, which is an important example of a lower-subtractive relation, it is shown that the automorphism group of the corresponding idempotent algebra is related to the diagram automorphisms of the associated graph.