Non-abelian Cohomology Research Articles

Non-abelian cohomology via parity quasi-complexes

The homotopy category of parity quasi-complexes is introduced. The homotopy structure is compatible with the nonabelian homology of parity quasi-complexes. Parity contracting homotopies are defined, determining the parity free resolutions in a canonical way, enabling the nonabelian bar construction. In this way, the even/odd grouping of the simplicial maps in the cocycle conditions of nonabelian cohomology is explained.

Density of monodromy actions on non-abelian cohomology

In this paper we study the monodromy action on the first Betti and de Rham non-Abelian cohomology arising from a family of smooth curves. We describe sufficient conditions for the existence of a Zariski-dense monodromy orbit. In particular, we show that for a Lefschetz pencil of sufficiently high degree the monodromy action is dense.

Lie Groupoids, Gerbes, and Non-Abelian Cohomology

Lie Groupoids, Gerbes, and Non-Abelian Cohomology

Higher non-abelian cohomology of groups

The first non-abelian cohomology of groups introduced by Guin is extended to any dimensions and for a substantially wider class of coefficients called G-partially crossed P-modules. The first and the second non-abelian cohomologies of groups are described in terms of torsors and extensions of groups respectively. Higher non-abelian cohomology pointed sets are described in terms of cotriple right derived functors of the group of derivations with respect to the first contravariant variable. For any short exact coefficient sequence a long exact cohomology sequence is obtained extending the well-known exact cohomology sequences and higher cohomology of groups with coefficients in any G-group is introduced.

Non-abelian Cohomology and Rational Points

Using non-abelian cohomology we introduce new obstructions to the Hasse principle. In particular, we generalize the classical descent formalism to principal homogeneous spaces under noncommutative algebraic groups and give explicit examples of application.

Categorification and Group Extensions

We review several known categorification procedures, and introduce a functorial categorification of group extensions (Section 4.1) with applications to non-Abelian group cohomology (Section 4.2). The obstruction to the existence of group extensions (Section 4.2.4, Equation (9)) is interpreted as a “coboundary” condition (Proposition 4.5).

Stacks and the homotopy theory of simplicial sheaves

Stacks are described as sheaves of groupoids G satisfying an eective descent condition, or equivalently such that the clas- sifying object BG satisÞes descent. The set of simplicial sheaf homotopy classes (E;BG) is identiÞed with equivalence classes of acyclic homotopy colimits Þbred over BG, generalizing the classical relation between torsors and non-abelian cohomology. Group actions give rise to quotient stacks, which appear as pa- rameter spaces for the separable transfer construction in special cases.

Twisted forms of finite étale extensions and separable polynomials

Examples of twisted forms of finite étale extensions and separable polynomials are calculated using Mayer‐Vietoris sequences for non‐abelian cohomology.

Non-abelian cohomology of abelian Anosov actions

We develop a new technique for calculating the first cohomology of certain classes of actions of higher-rank abelian groups (${\mathbb Z}^k$ and ${\mathbb R}^k$, $k\ge 2$) with values in a linear Lie group. In this paper we consider the discrete-time case. Our results apply to cocycles of different regularity, from Hölder to smooth and real-analytic. The main conclusion is that the corresponding cohomology trivializes, i.e. that any cocycle from a given class is cohomologous to a constant cocycle. The principal novel feature of our method is its geometric character; no global information about the action based on harmonic analysis is used. The method can be developed to apply to cocycles with values in certain infinite dimensional groups and to rigidity problems.

Extensions galoisiennes non commutatives :normalité cohomologie non abélienne

We study various types of noncommutative Galois extensions and present some examples. We state a criterion which decides whether a given automorphism of a Galois extension belongs to the corresponding Galois group or not. We then compute Borovoi’s nonabelian cohomology sets (in degree 0, 1, 2) of the Galois group with coefficients in a crossed module associated to the Galois extension.

Cohomology of cofibred categorical groups

In this paper we study and interpret a certain non-abelian cohomology H i( B, G) , 0≤ i≤2, of a small category B with coefficients in a B -cofibred categorical group G . The work is developed in a purely abstract setting, but several examples that indicate a very close connection with algebraic and topological problems are discussed explicitly. For instance, we obtain two new interpretations for the Brauer group of a Galois extension of commutative rings: an algebraic one in terms of equivalence classes of torsors over the Galois group and a topological one in terms of homotopy classes of cross-sections for a fibration over an Eilenberg–MacLane space of type K( G,1).

Non-Abelian Cohomology with Coefficients in Crossed Bimodules

Abstract When the coefficients are crossed bimodules, Guin's non-abelian cohomology [Guin, C. R. Acad. Sci. Paris 301: 337–340, 1985], [Guin, J. Pure Appl. Algebra 50: 109–137, 1988] is extended in dimensions 1 and 2, and a nine-term exact cohomology sequence is obtained.

Non-Abelian Cohomology of Groups

Abstract Following Guin's approach to non-abelian cohomology [Guin, Pure Appl. Algebra 50: 109–137, 1988] and, using the notion of a crossed bimodule, a second pointed set of cohomology is defined with coefficients in a crossed module, and Guin's six-term exact cohomology sequence is extended to a nine-term exact sequence of cohomology up to dimension 2.

Noncommutataive Descent and Non-Abelian Cohomology

Noncommutataive Descent and Non-Abelian Cohomology

Non-Abelian Cohomology with Coefficients in Crossed Bimodules

When the coecients are crossed bimodules, Guin's non- abelian cohomology (2), (3) is extended in dimensions 1 and 2, and a nine-term exact cohomology sequence is obtained.

Weak c*-Hopf algebras: the coassociative symmetry of non-integral dimensions

By allowing the coproduct to be non-unital and weakening the counit and antipode axioms of a C∗-Hopf algebra too, we obtain a selfdual set of axioms describing a coassociative quantum group, that we call a weak C∗-Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. It is the same structure that can be obtained by replacing the multiplicative unitary of Baaj and Skandalis with a partial isometry. The algebraic properties, the existence of the Haar measure and representation theory are briefly discussed. An algorithm is explained how to construct examples (in particular ones with non-integral dimensions) from non-Abelian cohomology.

Non-Abelian Cohomology of Groups

Following Guin's approach to non-abelian cohomology (4) and, using the notion of a crossed bimodule, a second pointed set of cohomology is deÞned with coecients in a crossed module, and Guin's six-term exact cohomology sequence is extended to a nine-term exact sequence of cohomology up to dimension 2.

Noncommutative descent and non-Abelian cohomology

Noncommutative descent and non-Abelian cohomology

Non-Abelian cohomology and Fermion flows over ?2-Graded *-algebras

We extend to the case of Fermion flows the description in terms of Lue's non-Abelian cohomology of the generators of quantum stochastic flows.

Finite Group Actions on Siegel Modular Spaces

The theory of nonabelian cohomology is used to show that the set of fixed points of a finite group acting on a Siegel modular space is a union of Shimura varieties