This paper primarily concerns Galois cohomology groups associated to Galois representations over a complete local ring R. The underlying Galois module and the corresponding cohomology groups which we consider are discrete R-modules. Under certain hypotheses, we prove that the first cohomology group is an almost divisible Rmodule. We also consider the subgroup of locally trivial elements in the ...

The aim of this paper is to introduce the quantum analogues of torsors for a compact quantum group and to investigate their relations with representation theory. Let A be a Hopf algebra over a field k. A theorem of Ulbrich asserts that there is an equivalence of categories between neutral fibre functors on the category of finitedimensional A-comodules and A-Galois extensions of k. We give the c...

We give sufficient conditions for a differential equation to have a given semisimple group as its Galois group. For any group G with G0 = G1 · . . . · Gr where each Gi is a simple group of type Al, Cl, Dl, E6 or E7, we construct a differential equation over C(x) having Galois group G.

Much work has gone into matrix representations of Galois groups, but there is a whole new class of naturally occurring representations that have as yet gone almost unnoticed. In fact, it is well-known in various areas of mathematics that the main sources of totally disconnected groups are matrix groups over local fields AND automorphism groups of locally finite trees [12]. It is perhaps surpris...

We study the action of the absolute Galois group Gal(Q̄/Q) on the algebraic fundamental group of P1C\{0, 1,∞}. We define a class of functions called Belyi-extending maps, and using them we are able to (1) give new Galois invariants of Grothendieck’s dessins d’enfants and (2) give explicit relations on the fσ in terms of which the Galois action on π̂1(P 1 C \ {0, 1,∞}) is known.

There is a very beautiful correspondence between branched covers of the Riemann sphere P1 and subgroups of the fundamental group π1(P1 − {branch points}), exactly analogous to the correspondence between subfields of an algebraic extension E/F and subgroups of the Galois group Gal(E/F ). This paper explores the concept of a Hecke algebra, which in this context is a generalization of the Galois g...

It is shown that Morita equivalence preserves quasi-Koszulity, and a finite-dimensional K-algebra A is quasi-Koszul if and only if the skew group algebra A * G is, where G is a finite group satisfying charK ∤ |G|. It follows from these results that a finite-dimensional K-algebra A is quasi-Koszul if and only if the smash product A#G * is, where G is a finite group satisfying charK ∤ |G|. Furthe...

Let K be a field and f(X) be a separable polynomial in K[X]. The Galois group of f(X) over K permutes the roots of f(X) in a splitting field, and labeling the roots as r1, . . . , rn provides an embedding of the Galois group into Sn. We recall without proof two theorems about this embedding. Theorem 1.1. Let f(X) ∈ K[X] be a separable polynomial of degree n. (a) If f(X) is irreducible in K[X] t...

We give sufficient conditions for a differential equation to have a given semisimple group as its Galois group. For any group G = H × G, H finite and G = ∏Gi where each Gi is SLni or Sp2ni , ni ≥ 2, we construct a differential equation over C(x) having Galois group G.