On a conjecture of Lynch

Abstract

We comment on a conjecture of Lynch on annihilators of local cohomology.

Communicated by Lawrence Ein     

On Hom-Lie antialgebra

Abstract

In this paper, we introduced the notion of Hom-Lie antialgebras. The representations and cohomology theory of Hom-Lie antialgebras are investigated. We prove that the equivalent classes of abelian extensions of Hom-Lie antialgebras are in one-to-one correspondence to elements of the second cohomology group. We also prove that 1-parameter infinitesimal deformation of a Hom-Lie antialgebra are characterized by 2-cocycles of this Hom-Lie antialgebra with adjoint representation in itself. The notion of Nijenhuis operators of Hom-Lie antialgebra is introduced to describe trivial deformations.

Communicated by Dr. Pavel Kolesnikov     

Cohomology operations and the deligne conjecture

The aim of this note, which raises more questions than it answers, is to study natural operations acting on the cohomology of various types of algebras. It contains a lot of very surprising partial results and examples.     

Bounded differential forms,generalized Milnor–Wood inequality and an application to deformation rigidity

We establish sufficient conditions for a cohomology class of a discrete subgroup Γ of a connected semisimple Lie group with finite center to be representable by a bounded differential form on the quotient by Γ of the associated symmetric space; furthermore if \(\rho : \Gamma\to\mathrm{PU}(1,q)\) is any representation of any discrete subgroup Γ of SU (1, p), we give an explicit closed bounded differential form on the quotient by Γ of complex hyperbolic space which is a representative for the pullback via ρ of the Kähler class of PU(1,q). If G,G′ are Lie groups of Hermitian type, we generalize to representations \(\rho : \Gamma\to G'\) of lattices Γ < G the invariant defined in [Burger, M., Iozzi, A.: Bounded cohomology and representation variates in PU (1,n). Preprint announcement, April 2000] for which we establish a Milnor–Wood type inequality. As an application we study maximal representations into PU(1, q) of lattices in SU(1,1).     

Dirac cohomology of unitary representations of equal rank exceptional groups

In this paper, we consider the unitary representations of equal rank exceptional groups of type E with a regular lambda-lowest K-type and classify those unitary representations with the nonzero Dirac cohomology.     

经典N=2李共形超代数的导子和第二上同调群

研究了经典N=2李共形超代数的导子和第二上同调群的结构,并应用第二上同调群的结果确定了该李共形超代数的泛中心扩张.     

Complete Cohomology for Extriangulated Categories

Let (C,E,s) be an extriangulated category with a proper class ξ of E-triangles.We study complete cohomology of objects in (C,E,s) by applying ξ-projective resolutions and ξ-injective coresolutions constructed in (C,E,s).Vanishing of complete cohomology detects objects with finite ξ-projective dimension and finite ξ-injective dimension.As a consequence,we obtain some criteria for the validity of the Wakamatsu tilting conjecture and give a necessary and sufficient condition for a virtually Gorenstein algebra to be Gorenstein.Moreover,we give a general technique for computing complete cohomology of objects with finite ξ-Gprojective dimension.As an application,the relations between ξ-projective dimension and ξ-Gprojective dimension for objects in (C,E,s) are given.     

Track Extensions of Categories and Cohomology

We introduce the notion of n-fold track extensions of a category C by a natural system D and prove that such extensions represent classes in the cohomology of C with coefficients in D introduced by Baues–Wirsching. This generalizes a result of Huebschmann on the cohomology of groups.