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     

关于例外序列自同态代数的Hochschild同调

LetAbeafinitedimensionalassociatealgebrawithidentityoverafieldkandmodAbethecategoryoffinitedimensionalA_modules.LetAe =A k Aop betheenvelopingalgebraofA ,denotebyAithei_foldtensorproductoverkofAwithitself,andby (a1 ,a2 ,… ,ai)theelementa1 a2 … aiinAi,thentheHochschildcomplex (A + 1 ,b)ofAisdefinedviab :An+ 1 →An withb(a1 ,a2 ,… ,an) =∑ni=1( - 1 ) i-1 (a1 ,… ,aiai+ 1 ,… ,an+ 1 ) + ( - 1 ) n(an+ 1 a1 ,… ,an)andH (A) =H(A + 1 ,b)iscalled ththeHochschildhomologygroupofA .Then…     

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.     

一个Cluster-Tilted代数的Hochschild同调与循环同调

基于Furuya构造的一个Cluster-Tilted代数的极小投射双模分解,用组合的方法计算了Cluster-Tilted代数的Hochschild同调空间的维数与基.当基础域的特征为零时,也计算了代数的循环同调群的维数.     

Hochschild and Cyclic Homology of Ore Extensions and Some Examples of Quantum Algebras

For every Ore extension we construct a chain complex giving its Hochschild homology. As an application we compute the Hochschild and cyclic homology of an arbitrary multiparametric affine space and the Hochschild homology of the algebra of differential operators over this space, in the generic case.     

一类自入射Koszul特殊双列代数的Hochschild同调群

基于Snashall与Taillefer构造的极小投射双模分解,用组合的方法,清晰地计算出一类自入射Koszul特殊双列代数∧_N的各阶Hochschild同调群的维数,从而以计算的方式直观地表明了韩阳的猜想对这类代数∧_N成立.     

Finiteness Conditions for Hochschild Homology Algebra and Free Loop Space Cohomology Algebra

We prove that over a characteristic zero field, in most cases, neither the Hochschild homology algebra of a commutative algebra, nor the free loop space cohomology algebra of a topological space, is finitely generated.     

Homology of Centralizers

Given a group Π, we study the group homology of centralizers Π _g , g ? Π, and of their central quotients Π _g /〈 g〉. This study is motivated by the structure of the Hochschild and the cyclic homology of group algebras, and is based on Quillen's approach to the cyclic homology of algebras via algebra extensions. A method of computing the de Rham complex of a group algebra by means of a Gruenberg resolution is also developed.     

Hochschild (Co)Homology of Hopf Crossed Products*

For a general crossed product E = A#_f H, of an algebra A by a Hopf algebra H, we obtain complexes smaller than the canonical ones, giving the Hochschild homology and cohomology of E. These complexes are equipped with natural filtrations. The spectral sequences associated to them coincide with the ones obtained using a natural generalization of the direct method introduced in Trans. Amer. Math. Soc. 74 (1953) 110–134. We also get that if the 2-cocycle f takes its values in a separable subalgebra of A, then the Hochschild (co)homology of E with coefficients in M is the (co)homology of H with coefficients in a (co)chain complex.     

Hochschild Cohomology of the Algebra of Smooth Functions on the Torus

We calculate the Hochschild cohomology of the algebra of smooth functions on a finite-dimensional real torus with coefficients in the adjoint representation, generalizing the previously developed technique to the discrete case for this.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 3, pp. 435–452, September, 2005.     

Symmetric Spectra and Topological Hochschild Homology

A functor is defined which detects stable equivalences of symmetric spectra. As an application, the definition of topological Hochschild homology on symmetric ring spectra using the Hochschild complex is shown to agree with Bökstedts original ad hoc definition. In particular, this shows that Bökstedts definition is correct even for nonconnective, nonconvergent symmetric ring spectra.     

Leibniz Homology of Extended Lie Algebras

Leibniz homology is a noncommutative homology theory for Lie algebras. In this paper, we compute low-dimensional Leibniz homology of extended Lie algebras.     

Hochschild Homology of Twisted Tensor Products

We compute the Hochschild homology of some twisted tensor products of algebras, which are a natural generalization of the Ore extensions. We apply our result to the ring D _Q,P(X,/X) of differential operators of the multiparametric affine space, the ring of coordinates of the quantum symplectic 2v-dimensional space and the ring of coordinates of the quantum 2v-dimensional Euclidean space.     

Hochschild (Co)homology of Exterior Algebras

The minimal projective bimodule resolutions of the exterior algebras are explicitly constructed. They are applied to calculate the Hochschild (co)homology of the exterior algebras. Thus the cyclic homology of the exterior algebras can be calculated in case the underlying field is of characteristic zero. Moreover, the Hochschild cohomology rings of the exterior algebras are determined by generators and relations.     

More on the Hochschild and Cyclic Homologies of Crossed Modules of Algebras

We investigate the Hochschild and cyclic homologies of crossed modules of algebras in some special cases. We prove that the cotriple cyclic homology of a crossed module of algebras (I, A, ρ) is isomorphic to HC _*(ρ): HC _*(I) → HC _*(A), provided I is H-unital and the ground ring is a field with characteristic zero. We also calculate the Hochschild and cyclic homologies of a crossed module of algebras (R, 0, 0) for each algebra R with trivial multiplication. At the end, we give some applications proving a new five term exact sequence.     

From Leibniz Homology to Cyclic Homology

For an algebra R over a commutative ring k, a natural homomorphism _*: HL_*+1(R) HH_* (R) from Leibniz to Hochschild homology is constructed that is induced by an antisymmetrization map on the chain level. The map _* is surjective when R = gl(A), A an algebra over a characteristic zero field. If f: A B is an algebra homomorophism, the relative groups HL_* (gl(f)) are studied, where gl(f): gl(A) gl(B) is the induced map on matrices. Letting HC_* denote cyclic homology, if f is surjective with nilpotent kernel, there is a natural surjection HL_*+1(gl(f)) HC_* (f) in the characteristic zero setting.     

(Co)Homology Theories for Oriented Algebras

We study three different (co)homology theories for a family of pullbacks of algebras that we call oriented. We obtain a Mayer Vietoris long exact sequence of Hochschild and cyclic homology and cohomology groups for these algebras. We give examples showing that our sequence for Hochschild cohomology groups is different from the known ones. In case the algebras are given by quiver and relations, and that the simplicial homology and cohomology groups are defined, we obtain a similar result in a slightly wider context. Finally, we also study the fundamental groups of the bound quivers involved in the pullbacks.     

Z_n分次代数的Hochschild上同调群

本文给出了Z_n分次代数A的Hochschild上同调群的定义,对低阶Hochschild上同调群进行了刻画.利用第一阶Hochschild上同调群给出了Z_n分次代数为分次可分代数的充要条件,证明了第二阶Hochschild上同调群的零次分支与A的Hochschild扩张之间的一一对应关系.