Non-abelian Cohomology Research Articles

Crossed modules, non-abelian extensions of associative conformal algebras and wells exact sequences

In this paper, we introduce the notions of crossed modules of associative conformal algebras, two-term strongly homotopy associative conformal algebras, and discuss the relationship between them and the third Hochschild cohomology of associative conformal algebras. We classify the non-abelian extensions by introducing the non-abelian cohomology. We show that non-abelian extensions of an associative conformal algebra can be viewed as Maurer–Cartan elements of a suitable differential graded Lie algebra, and prove that there is a one-one correspondence between the Deligne groupoid of this differential graded Lie algebra and the non-abelian cohomology. For any two associative conformal algebras [Formula: see text] and [Formula: see text], we provide the condition that a pair of automorphisms in Aut(A) × Aut(B) can be extended to an automorphisms of the non-abelian extension algebra of [Formula: see text] by [Formula: see text], and give the fundamental sequence of Wells in the context of associative conformal algebras.

Flux quantization on M5-branes

We highlight the need for global completion of the field content in the M5-brane sigma-model analogous to Dirac’s charge/flux quantization, and we point out that the superspace Bianchi identities on the worldvolume and on its ambient supergravity background constrain the M5’s flux-quantization law to be in a non-abelian cohomology theory rationally equivalent to a twisted form of co-Homotopy. In order to clearly bring out this subtle point we give a streamlined re-derivation of the worldvolume 3-flux via M5 “super-embeddings”. Finally, assuming the flux-quantization law to actually be in co-Homotopy (“Hypothesis H”) we show how this implies Skyrmion-like solitons on general M5-worldvolumes and (abelian) anyonic solitons on the boundaries of “open M5-branes” in heterotic M-theory.

Crossed Modules and Non-Abelian Extensions of Differential Leibniz Conformal Algebras

In this paper, we introduce two-term differential Leib∞-conformal algebras and give characterizations of some particular classes of such two-term differential Leib∞-conformal algebras. Furthermore, we discuss the classification of the non-Abelian extensions in terms of non-Abelian cohomology groups. Finally, we explore the inducibility of pairs of automorphisms and derive the analog Wells exact sequences under the circumstance of differential Leibniz conformal algebras.

Cohomologies, non-abelian extensions and Wells exact sequences of λ-differential pre-Lie algebras

The purpose of the present paper is to study cohomologies and non-abelian extensions of λ-differential pre-Lie algebras. First, we consider representations and cohomologies of λ-differential pre-Lie algebras. Next, we investigate non-abelian extensions and classify the non-abelian extensions in terms of non-abelian cohomology groups. Furthermore, we address the inducibility of a pair of automorphisms on non-abelian extensions and develop the Wells exact sequences in the context of λ-differential pre-Lie algebras. Finally, we discuss these results in the case of abelian extensions of λ-differential pre-Lie algebras.

2-term averaging L∞-algebras and non-abelian extensions of averaging Lie algebras

In recent years, averaging operators on Lie algebras (also called embedding tensors in the physics literature) and associated tensor hierarchies have formed an efficient tool for constructing supergravity and higher gauge theories. A Lie algebra with an averaging operator is called an averaging Lie algebra. In the present paper, we introduce 2-term averaging L∞-algebras and give characterizations of some particular classes of such homotopy algebras. Next, we study non-abelian extensions of an averaging Lie algebra by another averaging Lie algebra. We define the second non-abelian cohomology group to classify the equivalence classes of such non-abelian extensions. Next, given a non-abelian extension of averaging Lie algebras, we show that the obstruction for a pair of averaging Lie algebra automorphisms to be inducible can be seen as the image of a suitable Wells map. Finally, we discuss the Wells short exact sequence in the above context.

Generalizations of quasielliptic curves

We generalize the notion of quasielliptic curves, which have infinitesimal symmetries and exist only in characteristic two and three, to a remarkable hierarchy of regular curves having infinitesimal symmetries, defined in all characteristics and having higher genera. This relies on the study of certain infinitesimal group schemes acting on the affine line and certain compactifications. The group schemes are defined in terms of invertible additive polynomials over rings with nilpotent elements, and the compactification is constructed with the theory of numerical semigroups. The existence of regular twisted forms relies on Brion's recent theory of equivariant normalization. Furthermore, extending results of Serre from the realm of group cohomology, we describe non-abelian cohomology for semidirect products, to compute in special cases the collection of all twisted forms.

Non-abelian cohomology of universal curves in positive characteristic

In this paper, we will compute the non-abelian cohomology of the universal complete curve in positive characteristic. This extends Hain’s result on the non-abelian cohomology of generic curves in characteristic zero to positive characteristics. Furthermore, we will prove that the exact sequence of etale fundamental groups of the universal n n -punctured curve in positive characteristic does not split.

Crossed modules and non-abelian extensions of Rota-Baxter Leibniz algebras

We introduce the notions of crossed modules of Rota-Baxter Leibniz algebras, 2-term Rota-Baxter sh Leibniz algebras, and discuss the relationship between them and the 3-th cohomology group of Rota-Baxter Leibniz algebras. We classify the non-abelian extensions by introducing the non-abelian cohomology. Based on this classification, we study the inducibility of a pair of automorphisms about a non-abelian extension of Rota-Baxter Leibniz algebras, and give the fundamental sequence of Wells in the context of Rota-Baxter Leibniz algebras.

Non-abelian extensions of Rota-Baxter Lie algebras and inducibility of automorphisms

A Rota-Baxter Lie algebra gT is a Lie algebra g equipped with a Rota-Baxter operator T:g→g. In this paper, we consider non-abelian extensions of a Rota-Baxter Lie algebra gT by another Rota-Baxter Lie algebra hS. We define the non-abelian cohomology Hnab2(gT,hS) which classifies equivalence classes of such extensions. Given a non-abelian extension [Display omitted] of Rota-Baxter Lie algebras, we also show that the obstruction for a pair of Rota-Baxter automorphisms in Aut(hS)×Aut(gT) to be induced by an automorphism in Aut(eU) lies in the cohomology group Hnab2(gT,hS). As a byproduct, we obtain the Wells short-exact sequence in the context of Rota-Baxter Lie algebras. Finally, we show how these results fit with abelian extensions of Rota-Baxter Lie algebras.

Some results on factorization of monoids

Factorizations of monoids are studied. Two necessary and sufficient conditions in terms of the so-called descent 1-cocycles for a monoid to be factorized through two submonoids are found. A full classification of those factorizations of a monoid whose one factor is a subgroup of the monoid is obtained. The relationship between monoid factorizations and non-abelian cohomology of monoids is analyzed. Some applications of semi-direct product of monoids are given.

Non-connected Lie groups, twisted equivariant bundles and coverings

Let Gamma be a finite group acting on a Lie group G. We consider a class of group extensions 1 rightarrow G rightarrow hat{G} rightarrow Gamma rightarrow 1 defined by this action and a 2-cocycle of Gamma with values in the centre of G. We establish and study a correspondence between hat{G}-bundles on a manifold and twisted Gamma -equivariant bundles with structure group G on a suitable Galois Gamma -covering of the manifold. We also describe this correspondence in terms of non-abelian cohomology. Our results apply, in particular, to the case of a compact or reductive complex Lie group hat{G}, since such a group is always isomorphic to an extension as above, where G is the connected component of the identity and Gamma is the group of connected components of hat{G}.

Braided anti-flexible bialgebras

We introduce the concept of braided anti-flexible bialgebra and construct cocycle bicrossproduct anti-flexible bialgebras. As an application, we solve the extending problem for anti-flexible bialgebras by using some non-abelian cohomology theory.

Unified products for Jordan algebras. Applications

Given a Jordan algebra A and a vector space V, we describe and classify all Jordan algebras containing A as a subalgebra of codimension dimk(V) in terms of a non-abelian cohomological type object JA(V,A). Any such algebra is isomorphic to a newly introduced object called unified productA♮V. The crossed/twisted product of two Jordan algebras are introduced as special cases of the unified product and the role of the subsequent problem corresponding to each such product is discussed. The non-abelian cohomology Hnab2(V,A) associated to two Jordan algebras A and V which classifies all extensions of V by A is also constructed. Several applications and examples are given: we prove that Hnab2(k,kn) is identified with the set of all matrices D∈Mn(k) satisfying 2D3−3D2+D=0, where we consider the abelian Jordan algebra structure on k and kn.

Non-abelian extensions of pre-Lie algebras

In this paper, we study non-abelian extensions of pre-Lie algebras. First we introduce the non-abelian cohomology for pre-Lie algebras by which we classify non-abelian extensions of pre-Lie algebras. Then we construct the bimultipliers for pre-Lie algebras, which naturally gives rise to a strict Lie 2-algebra. We show that there is a one-to-one correspondence between isomorphic classes of non-abelian extensions of pre-Lie algebras and homotopy classes of homomorphisms from the subadjacent Lie algebra to the strict Lie 2-algebra obtained from the bimultipliers. Finally we classify non-abelian extensions of pre-Lie algebras by Maurer-Cartan elements of a differential graded Lie algebra.

Research on Splitting Isomorphism of Leibniz Algebra and Non-Abelian expansion

In this study, Leibniz algebras and the derivations and properties of Leibniz algebras were given, respectively. The stable automorphism group of explicit splitting extension was calculated via the stable automorphism group of Abelian extension of finite group splitting. Based on the stable automorphism group of the splitting extension studied, the non-Abelian extension and the second order non-Abelian co-homology group of Leibniz algebra were investigated in detail according to the stable automorphism group of the splitting extension.

Extending structures for 3-Lie algebras

The cohomology and deformation theory of 3-Lie algebras are revisited. The theory of extending structures and unified product for 3-Lie algebras are developed. It is proved that the extending structures of 3-Lie algebras can be classified by using some non-abelian cohomology and deformation map theory.

Proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing a field

Let be a regular local ring containing a field. Let be a reductive group scheme over . We prove that a principal -bundle over is trivial if it is trivial over the field of fractions of . In other words, if is the field of fractions of , then the map of the non-Abelian cohomology pointed sets induced by the inclusion of in has trivial kernel. This result was proved in [1] for regular local rings containing an infinite field.

Twisted Weyl Groups of Compact Lie Groups and Nonabelian Cohomology

Given a compact connected Lie group G with an S 1 -module structure and a maximal compact torus T of G S 1 , we define twisted Weyl group W ( G , S 1 , T ) of G associated to S 1 -module and show that two elements of T are δ -conjugate if and only if they are in one W ( G , S 1 , T ) -orbit. Based on this, we prove that the natural map W ( G , S 1 , T ) \ H 1 ( S 1 , T ) → H 1 ( S 1 , G ) is bijective, which reduces the calculation for the nonabelian cohomology H 1 ( S 1 , G ) .

On the Classification of Lie Bialgebras by Cohomological Means

We approach the classification of Lie bialgebra structures on simple Lie algebras from the viewpoint of descent and non-abelian cohomology. We achieve a description of the problem in terms of faithfully flat cohomology over an arbitrary ring over \mathbb{Q} , and solve it for Drinfeld-Jimbo Lie bialgebras over fields of characteristic zero. We consider the classification up to isomorphism, as opposed to equivalence, and treat split and non-split Lie algebras alike. We moreover give a new interpretation of scalar multiples of Lie bialgebras hitherto studied using twisted Belavin-Drinfeld cohomology.

An Analogue of the Coleman\u2013Mandula Theorem for Quantum Field Theory in Curved Spacetimes

The Coleman–Mandula (CM) theorem states that the Poincaré and internal symmetries of a Minkowski spacetime quantum field theory cannot combine nontrivially in an extended symmetry group. We establish an analogous result for quantum field theory in curved spacetimes, assuming local covariance, the timeslice property, a local dynamical form of Lorentz invariance, and additivity. Unlike the CM theorem, our result is valid in dimensions {ngeq 2} and for free or interacting theories. It is formulated for theories defined on a category of all globally hyperbolic spacetimes equipped with a global coframe, on which the restricted Lorentz group acts, and makes use of a general analysis of symmetries induced by the action of a group G on the category of spacetimes. Such symmetries are shown to be canonically associated with a cohomology class in the second degree nonabelian cohomology of G with coefficients in the global gauge group of the theory. Our main result proves that the cohomology class is trivial if G is the universal cover {mathcal{S}} of the restricted Lorentz group. Among other consequences, it follows that the extended symmetry group is a direct product of the global gauge group and {mathcal{S}}, all fields transform in multiplets of {mathcal{S}}, fields of different spin do not mix under the extended group, and the occurrence of noninteger spin is controlled by the centre of the global gauge group. The general analysis is also applied to rigid scale covariance.