The cyclic and simplicial cohomology of

Let be the unital semigroup algebra of . We show that the cyclic cohomology groups vanish when is odd and are one dimensional when is even (). Using Connes' exact sequence, these results are used to show that the simplicial cohomology groups vanish for . The results obtained are extended to unital algebras for some other semigroups of .

     

On the simplicial cohomology of Banach operator algebras

We investigate the simplicial cohomology of certain Banach operator algebras. The two main examples considered are the Banach algebra of all bounded operators on a Banach space and its ideal of approximable operators. Sufficient conditions will be given forcing Banach algebras of this kind to be simplicially trivial.     

Cyclic cohomology of Banach algebras of quivers and their inverse semigroups

In this paper, we consider the path semigroup ?¹${?}^1$-algebra for a quiver and the inverse semigroup ?¹${?}^1$-algebra of a quiver, the latter of which can be used in the construction of Cuntz–Krieger algebras. The main objectives of the paper are to determine the simplicial and cyclic cohomology groups of these algebras. First, we determine the simplicial and cyclic cohomology of the path algebra of the quiver, showing the simplicial cohomology groups of dimension n   vanish for n>1$n>1$. We then determine the simplicial and cyclic cohomology of the inverse semigroup algebra. The work uses the Connes–Tzygan long exact sequence.     

Vanishing of cohomology groups of random simplicial complexes

We consider k‐dimensional random simplicial complexes generated from the binomial random (k + 1)‐uniform hypergraph by taking the downward‐closure. For 1 ≤ j ≤ k ? 1, we determine when all cohomology groups with coefficients in from dimension one up to j vanish and the zero‐th cohomology group is isomorphic to . This property is not deterministically monotone for this model, but nevertheless we show that it has a single sharp threshold. Moreover we prove a hitting time result, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction. We also study the asymptotic distribution of the dimension of the j‐th cohomology group inside the critical window. As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced by Linial and Meshulam, previously only known for dimension two.     

Embeddings of

We show that there is only one embedding of in at the prime , up to self-maps of . We also describe the effect of the group of self-equivalences of at the prime on this embedding and then show that the Friedlander exceptional isogeny composed with a suitable Adams map is an involution of whose homotopy fixed point set coincide with

     

Some characterizations of the automorphisms of

We present some nonlinear characterizations of the automorphisms of the operator algebra and the function algebra by means of their spectrum preserving properties.

     

Dirac cohomology of the Dunkl-Opdam subalgebra via inherited Drinfeld properties

Abstract

In this paper, we define a new presentation for the Dunkl-Opdam subalgebra of the rational Cherednik algebra. This presentation uncovers the Dunkl-Opdam subalgebra as a Drinfeld algebra. We use this fact to define Dirac cohomology for the DO subalgebra. We also formalize generalized graded Hecke algebras and extend a Langlands classification to generalized graded Hecke algebras.     

Higher-dimensional virtual diagonals and ideal cohomology for triangular algebras

We investigate the cohomology of non-self-adjoint algebras using virtual diagonals and their higher-dimensional generalizations. We show that infinite dimensional nest algebras always have non-zero second cohomology by showing that they cannot possess 2-virtual diagonals. In the case of the upper triangular atomic nest algebra we exhibit concrete modules for non-vanishing cohomology.

     

Exotic cohomology for

We show that for the mod group cohomology of is not detected on diagonal matrices.

     

The cohomology of the Morava stabilizer group at the prime 3

We compute the cohomology of the Morava stabilizer group at the prime by resolving it by a free product and analyzing the ``relation module.'

     

action on the cohomology of a rank two abelian group with arbitrary coefficient domain

A rank two abelian group is in a natural way an -module. This induces an action of on its group cohomology for any trivial coefficient domain . In the present note we determine this module, including the question of when the universal coefficient theorem sequence splits.

     

Nilpotency degree of cohomology rings in characteristic

Let be an odd prime number. The purpose of this paper is to provide a -group whose mod- cohomology ring has a nilpotent element satisfying .

     

Generalized Burnside rings and group cohomology

In this paper we give by a unified formula the classification of exceptional compact simple Kantor triple systems defined on tensor products of composition algebras corresponding to realifications of exceptional simple Lie algebras.     

Group homomorphisms inducing cohomology monomorphisms

Let be a homomorphism of -groups such that
is injective, for . We prove that the non-bijectivity of implies the existence of a quotient of containing as a proper direct factor. This gives a refined proof of a result of Evens, which asserts that is bijective if is.

     

have the Dunford-Pettis property

Denote by either the disc algebra , or the space of bounded analytic functions on the disc, or any of their even duals. Then has the Dunford-Pettis property.

     

The cohomology of homotopy categories and the general linear group

A homotopy categoryC (of co-H-groups resp.H-groups) represents an element C in the third cohomology ofC. This element determines all Toda brackets and secondary homotopy operations inC. Moreover, in caseC =VS ⁿ consists of all one-point unions ofn-spheres, the bracket is actually a /2-generator which restricts to Igusa's class(1) in casen3; an explicit new cocycle for(1) is obtained by automorphisms of free nil(2)-groups.