Basic Noncommutative Galois Theory

Time-ordered perturbation theory on noncommutative spacetime: basic rules

Assuming the S-matrix on noncommutative (NC) spacetime can still be developped perturbatively in terms of the time-ordered exponential of the interaction Lagrangian, we investigate the perturbation theory of NC field theory. We first work out with care some typical Green functions starting from the usual concepts of time-ordering and commutation relations for free fields. The results are found ...

Finitary Galois Extensions over Noncommutative Bases

We study Galois extensions M (co-)H ⊂ M for H-(co)module algebras M if H is a Frobenius Hopf algebroid. The relation between the action and coaction pictures is analogous to that found in Hopf-Galois theory for finite dimensional Hopf algebras over fields. So we obtain generalizations of various classical theorems of Kreimer-Takeuchi, Doi-Takeuchi and Cohen-FischmanMontgomery. We find that the ...

Galois Theory

Galois Theory

Remark 0.1 (Notation). |G| denotes the order of a finite group G. [E : F ] denotes the degree of a field extension E/F. We write H ≤ G to mean that H is a subgroup of G, and N G to mean that N is a normal subgroup of G. If E/F and K/F are two field extensions, then when we say that K/F is contained in E/F , we mean via a homomorphism that fixes F. We assume the following basic facts in this set...

Galois Theory

Proposition 1.3. Let φ be an automorphism of a field extension K/F , and f(x) ∈ F [x]. Let α1, . . . , αn be the roots of f(x) lying in K. Then φ permutes the set {α1, . . . , αn}. If also the set of αi generate K over F , then two automorphisms φ1, φ2 of K/F which agree on all the αi are equal. Thus, in this case we have an inclusion of Aut(K/F ) as a subgroup of Sym({α1, . . . , αn}) ∼= Sn. P...