On cohomology theory for topological groups

We construct some new cohomology theories for topological groups and Lie groups and study some of its basic properties. For example, we introduce a cohomology theory based on measurable cochains which are continuous in a neighbourhood of the identity. We show that if G and A are locally compact and second countable, then the second cohomology group based on locally continuous measurable cochains as above parametrizes the collection of locally split extensions of G by A.     

On Finite-Valued Cohomology Theories

In this paper, we describe cohomology theories constructed on the category of pairs of topological spaces and continuous maps and based on finite-valued cochains with coefficients in abelian group.     

Poincaré duality for p-adic Lie groups

We establish Poincaré duality for continuous group cohomology of p-adic Lie groups with rational coefficients and compare integral structures under this duality.     

Dirac Cohomology for Lie Superalgebras

Dirac cohomology is a new tool to study representations of semisimple Lie groups and Lie algebras. The aim of this paper is to define a Dirac operator for a Lie superalgebra of Riemannian type and show that this Dirac operator has similar nature as the one for semisimple Lie algebras. As a consequence, we show how to determine the infinitesimal character of a representation by the infinitesimal character of its Dirac cohomology.     

Deformation quantization of gerbes

This is the first in a series of articles devoted to deformation quantization of gerbes. We introduce basic definitions, interpret deformations of a given stack as Maurer-Cartan elements of a differential graded Lie algebra (DGLA), and classify deformations of a given gerbe in terms of Maurer-Cartan elements of the DGLA of Hochschild cochains twisted by the cohomology class of the gerbe. We also classify all deformations of a given gerbe on a symplectic manifold, as well as provide a deformation-theoretic interpretation of the first Rozansky-Witten class.     

Cohomology of biquotients

Biquotients are non-homogeneous quotient spaces of Lie groups. Using the Serre spectral sequence and the method of Borel, we compute the cohomology algebra of these spaces in cases where the Lie group cohomology is not too complicated. Among these are the biquotients which are known to carry a metric of positive curvature.     

Low-order cohomology of semi-simple lie groups with applications to continuous tensor products and current groups

In this survey article we given an introduction to first-order and second-order continuous cohomology of groups. The abstract algebraic set-up is then realized in the context of semi-simple Lie groups. As an application continuous tensor products and factorizable representations of current groups are described. We end the survey with a concrete example from quantum mechanics.     

Cocycles Over Abelian TNS Actions

We study extensions of higher-rank Abelian TNS actions (i.e. hyperbolic and with a special structure of the stable distributions) by compact connected Lie groups. We show that up to a constant, there are only finitely many cohomology classes. We also show the existence of cocycles over higher-rank Abelian TNS actions that are not cohomologous to constant cocycles. This is in contrast to earlier results, showing that real valued cocycles, or small Lie group valued cocycles, over higher-rank Abelian actions are cohomologous to constants.     

On the second cohomology group in semi-abelian categories

We develop some new aspects of cohomology in the context of semi-abelian categories: we establish a Hochschild-Serre 5-term exact sequence extending the classical one for groups and Lie algebras; we prove that an object is perfect if and only if it admits a universal central extension; we show how the second Barr-Beck cohomology group classifies isomorphism classes of central extensions; we prove a universal coefficient theorem to explain the relationship with homology.     

Cohomology of discrete groups in harmonic cochains on buildings

Modules of harmonic cochains on the Bruhat-Tits building of the projective general linear group over ap-adic field were defined by one of the authors, and were shown to represent the cohomology of Drinfel’d’sp-adic symmetric domain. Here we define certain non-trivial natural extensions of these modules and study their properties. In particular, for a quotient of Drinfel’d’s space by a discrete cocompact group, we are able to define maps between consecutive graded pieces of its de Rham cohomology, which we show to be (essentially) isomorphisms. We believe that these maps are graded versions of the Hyodo-Kato monodromy operatorN.     

Equivariant homology and duality

This note is concerned with stable G-equivariant homology and cohomology theories (G a compact Lie group). In important cases, when H-equivariant theories are defined naturally for all closed subgroups H of G, we show that the G-(co)homology groups of G x_H X are isomorphic with H-(co)homology groups of X. We introduce the concept of orientability of G-vector bundles and manifolds with respect to an equivariant cohomology theory and prove a duality theorem which implies an equivariant analogue of Poincaré-Lefschetz duality.

     

On the Cohomology of the Invariant Euler–Lagrange Complex

Given a Lie group action G we show, using the method of equivariant moving frames, that the local cohomology of the invariant Euler–Lagrange complex is isomorphic to the Lie algebra cohomology of G.     

Cohomology structure for a Poisson algebra:Ⅱ

For a Poisson algebra, we prove that the Poisson cohomology theory introduced by Flato et al.(1995)is given by a certain derived functor. We show that the(generalized) deformation quantization is equivalent to the formal deformation for Poisson algebras under certain mild conditions. Finally we construct a long exact sequence, and use it to calculate the Poisson cohomology groups via the Yoneda-extension groups of certain quasi-Poisson modules and the Lie algebra cohomology groups.     

Central extensions and deformations of Hom-Lie triple systems

In this paper, we study the cohomology theory of Hom-Lie triple systems generalizing the Yamaguti cohomology theory of Lie triple systems. We introduce the central extension theory for Hom-Lie triple systems and show that there is a one-to-one correspondence between equivalent classes of central extensions of Hom-Lie triple systems and the third cohomology group. We develop the 1-parameter formal deformation theory of Hom-Lie triple systems and prove that it is governed by the cohomology group.     

Isomorphism classes of finite dimensional connected Hopf algebras in positive characteristic

We classify all finite-dimensional connected Hopf algebras with large abelian primitive spaces. We show that they are Hopf algebra extensions of restricted enveloping algebras of certain restricted Lie algebras. For any abelian matched pair associated with these extensions, we construct a cohomology group, which classifies all the extensions up to equivalence. Moreover, we present a 1–1 correspondence between the isomorphism classes and a group quotient of the cohomology group deleting some exceptional points, where the group respects the automorphisms of the abelian matched pair and the exceptional points represent those restricted Lie algebra extensions.     

A note on Sen��s theory in the imperfect residue field case

In Sen??s theory in the imperfect residue field case, Brinon defined a functor from the category of ${{\mathbb C}_p}$ -representations to the category of linear representations of a certain Lie algebra. We give a comparison theorem between the continuous Galois cohomology of ${{\mathbb C}_p}$ -representations and the Lie algebra cohomology of the associated representations. The key ingredients of the proof are Hyodo??s calculation of Galois cohomology and the effaceability of Lie algebra cohomology for solvable Lie algebras.     

The integral cohomology of toric manifolds

We prove that the integral cohomology of a smooth, not necessarily compact, toric variety X _Σ is determined by the Stanley-Reisner ring of Σ. This follows from a formality result for singular cochains on the Borel construction of X _Σ. As a consequence, we show that the cycle map from Chow groups to Borel-Moore homology is split injective.     

Compact symplectic nilmanifolds associated with graphs

In this paper we consider 2-step nilpotent Lie algebras, Lie groups and nilmanifolds associated with graphs. We present a combinatorial construction of the second cohomology group for these Lie algebras. This enables us to characterize those graphs giving rise to symplectic or contact nilmanifolds.     

Tight Homomorphisms and Hermitian Symmetric Spaces

We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetric spaces. We show that tight maps behave in a functorial way with respect to the Shilov boundary and use this to prove a general structure theorem for tight homomorphisms. Furthermore, we classify all tight embeddings of the Poincaré disk.