Twisted Hochschild Homology of Quantum Hyperplanes*

The twisted Hochschild homology groups of generic quantum hyperplanes are calculated using the Koszul resolution. For the example of the two-dimensional quantum plane also the twisted cyclic homology groups are determined. *Partially supported by Polish State Committee for Scientific Research (KBN) under grant 2 P03B 022 25     

Hochschild and cyclic homology of finite type algebras

We study Hochschild and cyclic homology of finite type algebras using abelian stratifications of their primitive ideal spectrum. Hochschild homology turns out to have a quite complicated behavior, but cyclic homology can be related directly to the singular cohomology of the strata. We also briefly discuss some connections with the representation theory of reductive p-adic groups.     

Khovanov homology of torus links

In this paper we show that there is a cut-off in the Khovanov homology of (2k,2kn)-torus links, namely that the maximal homological degree of non-zero homology groups of (2k,2kn)-torus links is 2k²n. Furthermore, we calculate explicitly the homology group in homological degree 2k²n and prove that it coincides with the center of the ring Hk of crossingless matchings, introduced by M. Khovanov in [M. Khovanov, A functor-valued invariant for tangles, Algebr. Geom. Topol. 2 (2002) 665-741, arXiv:math.QA/0103190]. This gives the proof of part of a conjecture by M. Khovanov and L. Rozansky in [M. Khovanov, L. Rozansky, A homology theory for links in S²×S¹, in preparation]. Also we give an explicit formula for the ranks of the homology groups of (3,n)-torus knots for every n∈N.     

Noncommutative orlicz-hardy spaces

Let (Φ, Ψ) be a pair of complementary N-functions and HΦ(A)$H^{\Phi }\left(A\right)$ and HΨ(A)$H^{\Psi }\left(A\right)$ be the associated noncommutative Orlicz-Hardy spaces. We extend the Riesz, Szegö and inner-outer type factorization theorems of Hp(A)$H^p\left(A\right)$ to this case.     

Analytic Hardy spaces on the quantum torus

Analytic Hardy and BMO spaces on the quantum torus are introduced. Some basic properties of these spaces are presented. In particular, the associated H 1-BMO duality theorem is proved. Finally, we discuss some possible extensions of the obtained results.     

Some Remarks on Absolute Mathematics

We calculate Hochschild cohomology groups of the integers treated as an algebra over so-called field with one element. We compare our results with calculation of the topological Hochschild cohomology groups of the integers—this is the case when one considers integers as an algebra over the sphere spectrum.     

Cyclic homology of Dedekind domains

We compute the cyclic homology of the algebraR[t]/(P(t)) relative to an arbitrary integral domainR. As an application, we compute the cyclic homology of number rings.Supported by N.S.F. Grant No. DMS-8807203.     

量子外代数的Galois覆盖代数的Hochschild上同调

本文研究一类量子代数$\Lambda^n_q$的Hochschild上同调.量子代数$\Lambda^n_q$的极小投射双模分解被构造, $\Lambda^n_q$的各阶Hochschild上同调群的维数被清晰的给出.此外,对一些特殊的情况, $\Lambda^n_q$的上同调环也被清晰的刻画.     

Homologies cyclique et de Hochschild de certains espaces homogènes quantiques

Nous quantifions certaines inclusions d'algèbres de Lie semi-simpleshg. Nous calculons les homologies associées aux quantifications, surC((h)), d'une part des algèbres de fonctions formelles surG/H, pourHG une inclusion de groupes de Lie semi-simples associée, et d'autre part des fonctions algébriques sur SL(2,C)/T.We quantize certain inclusions of semisimple Lie algebrashg. We compute the cyclic and Hochschild homologies for theC((h))-quantizations of

Batalin-Vilkovisky Algebras and Cyclic Cohomology of Hopf Algebras

We show that the Connes–Moscovici negative cyclic cohomology of a Hopf algebra equipped with a character has a Lie bracket of degree -2. More generally, we show that a cyclic operad with multiplication is a cocyclic module whose simplicial cohomology is a Batalin–Vilkovisky algebra and whose negative cyclic cohomology is a graded Lie algebra of degree -2. This generalizes the fact that the Hochschild cohomology algebra of a symmetric algebra is a Batalin–Vilkovisky algebra.     

非交换陈特征的一个注记

此注记利用Moscovici的一个想法和热核渐近展开技术,给出了Clifford模 上Dirac算子的整循环陈特征的一个计算公式.     

Hochschild cohomologies and deformations of the pointwise superproduct

We consider the associative superalgebra of smooth compactly supported functions on ℝ ⁿ taking values in a Grassmann algebra. We find the lower Hochschild cohomologies and also the deformations of this superalgebra under certain additional conditions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 323–346, March, 2009.     

Twisted Homology of Quantum SL(2)

We calculate the twisted Hochschild and cyclic homology (in the sense of Kustermans, Murphy and Tuset) of the coordinate algebra of the quantum SL(2) group relative to twisting automorphisms acting by rescaling the standard generators a,b,c,d. We discover a family of automorphisms for which the “twisted” Hochschild dimension coincides with the classical dimension of , thus avoiding the “dimension drop” in Hochschild homology seen for many quantum deformations. Strikingly, the simplest such automorphism is the canonical modular automorphism arising from the Haar functional. In addition, we identify the twisted cyclic cohomology classes corresponding to the three covariant differential calculi over quantum SU(2) discovered by Woronowicz.     

Computing the Hochschild Cohomology Groups of Some Families of Incidence Algebras

The purpose of this article is to present some computations of Hochschild cohomology groups of particular classes of incidence algebras using one-point extensions and one-point coextensions.     

Gerbes and twisted orbifold quantum cohomology

In this paper,we construct an orbifold quantum cohomology twisted by a flat gerbe. Then we compute these invariants in the case of a smooth manifold and a discrete torsion on a global quotient orbifold.     

Gromov-Witten invariants and quantum cohomology

This article is an elaboration of a talk given at an international conference on Operator Theory, Quantum Probability, and Noncommutative Geometry held during December 20–23, 2004, at the Indian Statistical Institute, Kolkata. The lecture was meant for a general audience, and also prospective research students, the idea of the quantum cohomology based on the Gromov-Witten invariants. Of course there are many important aspects that are not discussed here. Dedicated to Professor K B Sinha on the occasion of his 60th birthday     

The Hochschild cohomology ring of a cyclic block

Suppose is a block of a group algebra with cyclic defect group. We calculate the Hochschild cohomology ring of , giving a complete set of generators and relations. We then show that if is the principal block, the canonical map from to the Hochschild cohomology ring of induces an isomorphism modulo radicals.