Twisted Hochschild Homology of Quantum Hyperplanes

Derived Higher Hochschild Homology, Topological Chiral Homology and Factorization algebras

In this paper, we study the higher Hochschild functor and its relationship with factorization algebras and topological chiral homology. To this end, we emphasize that the higher Hochschild complex is a $(\infty,1)$-functor from the category $\hsset \times \hcdga$ to the category $\hcdga$ (where $\hsset$ and $\hcdga$ are the $(\infty,1)$-categories of simplicial sets and commutative differential graded algebras) and give an axiomatic characterization of this functor. From the axioms we deduce several properties and computational tools for this functor. We study the relationship between the high…

Hochschild homology of iterate skew polynomial rings

We study the Hochschild homology of the iterated skew polynomial rings introduced by D. Jordan in ``A simple localization of the quantized Weyl algebra''. First, we obtain a complex, smaller than the canonical one of Hochschild, given the homology of such an algebra, and then, we study this complex in order to compute the homology of some families of algebras. In particular we compute the homology of some quantum groups, in the generic case.

Hochschild and cyclic (co)homology of the Fomin-Kirillov algebra on 3 generators

The goal of this article is to explicitly compute the Hochschild (co)homology of the Fomin-Kirillov algebra on three generators over a field of characteristic different from 2 and 3. We also obtain the cyclic (co)homology of the Fomin-Kirillov algebra in case the characteristic of the field is zero. Moreover, we compute the algebra structure of the Hochschild cohomology.

Sutured Khovanov homology, Hochschild homology, and the Ozsvath-Szabo spectral sequence

In 2001, Khovanov and Seidel constructed a faithful action of the (m+1)-strand braid group on the derived category of left modules over a quiver algebra, A_m. We interpret the Hochschild homology of the Khovanov-Seidel braid invariant as a direct summand of the sutured Khovanov homology of the annular braid closure.

Hochschild (co)homology of the Dunkl operator quantization of $\Z_2$-singularity

We study Hochschild (co)homology groups of the Dunkl operator quantization of $\Z_2$-singularity constructed by Halbout and Tang. Further, we study traces on this algebra and prove a local algebraic index formula.

Hochschild and cyclic homology of central extensions of preprojective algebras of ADE quivers

Let A be the central extension of the preprojective algebra of an ADE quiver introduced by P. Etingof and E. Rains in math/0503393. The paper math/0606403 computes the structure of the zeroth Hochschild (co)homology of A. We generalize the results of math/0606403 by calculating the additive structure of all the Hochschild homology and cohomology groups of A and the cyclic homology of A, and to describe the universal deformation of A. Namely, we show that the (co)homology is periodic with period 4, and compute the first four (co)homology groups in each case.

量子拓扑Hochschild同调与环状Khovanov谱

Quantum topological Hochschild homology and annular Khovanov spectra

本文提出量子拓扑Hochschild同调（qTHH）并构建其稳定同调细化，用于量子环状Khovanov同调的构造，证明其与现有理论一致。

On the Hochschild (co)homology of Quantum Homogeneous Spaces

The recent result of Brown and Zhang establishing Poincare duality in the Hochschild (co)homology of a large class of Hopf algebras is extended to right coideal subalgebras over which the Hopf algebra is faithfully flat, and applied to the standard Podles quantum 2-sphere.

Maximal commutative subalgebras, Poisson geometry and Hochschild homology

A Poisson geometry arising from maximal commutative subalgebras is studied. A spectral sequence convergent to Hochschild homology with coefficients in a bimodule is presented. It depends on the choice of a maximal commutative subalgebra inducing appropriate filtrations. Its E^{2}_{p,q}-groups are computed in terms of canonical homology with values in a Poisson module defined by a given bimodule and a maximal commutative subalgebra.

Hochschild and cyclic homology of preprojective algebras of ADE quivers

We calculate the additive structure (together with the grading) of the Hochschild homology and cohomology and the cyclic homology of preprojective algebras of types ADE.

Twisted Hochschild Homology of Quantum Flag Manifolds and Kähler Forms

We study the twisted Hochschild homology of quantum flag manifolds, the twist being the modular automorphism of the Haar state. We prove that every quantum flag manifold admits a non-trivial class in degree two, with an explicit representative defined in terms of a certain projection. The corresponding classical two-form, via the Hochschild-Kostant-Rosenberg theorem, is identified with a Kähler form on the flag manifold.

Hochschild Homology and Cohomology of Klein Surfaces

Within the framework of deformation quantization, a first step towards the study of star-products is the calculation of Hochschild cohomology. The aim of this article is precisely to determine the Hochschild homology and cohomology in two cases of algebraic varieties. On the one hand, we consider singular curves of the plane; here we recover, in a different way, a result proved by Fronsdal and make it more precise. On the other hand, we are interested in Klein surfaces. The use of a complex suggested by Kontsevich and the help of Groebner bases allow us to solve the problem.

A remark on quantum Hochschild homology

Beliakova-Putyra-Wehrli studied various kinds of traces, in relation to annular Khovanov homology. In particular, to a graded algebra and a graded bimodule over it, they associate a quantum Hochschild homology of the algebra with coefficients in the bimodule, and use this to obtain a deformation of the annular Khovanov homology of a link. A spectral refinement of the resulting invariant was recently given by Akhmechet-Krushkal-Willis. In this short note we observe that quantum Hochschild homology is a composition of two familiar operations, and give a short proof that it gives an invariant of …

The calculus structure of the Hochschild homology/cohomology of preprojective algebras of Dynkin quivers

The Hochschild homology/cohomology an associative algebra, together with the Connes differential, the contraction map and the Lie derivative, forms the structure of calculus. In this paper we compute explicitely the calculus structure of preprojective algebras of Dynkin quivers over a field of characteristic zero. This also completes the work in math.AG/0502301, where the Batalin-Vilkovisky structure of the Hochschild cohomology of preprojective algebras of non-Dynkin quivers are computed and the calculus can be easily computed from that.

Hochschild homology of structured algebras

We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any PROP with $A_\infty$--multiplication---we think of such algebras as $A_\infty$--algebras "with extra structure". As applications, we obtain an integral version of the Costello-Kontsevich-Soibelman moduli space action on the Hochschild complex of open TCFTs, the Tradler-Zeinalian action of Sullivan diagrams on the Hochschild complex of strict Frobenius algebras, and give applications to string topology in characteristic zero. Our main tool is a generalization of the Hochschil…

Hochschild homology, lax codescent, and duplicial structure

We study the duplicial objects of Dwyer and Kan, which generalize the cyclic objects of Connes. We describe duplicial objects in terms of the decalage comonads, and we give a conceptual account of the construction of duplicial objects due to Bohm and Stefan. This is done in terms of a 2-categorical generalization of Hochschild homology. We also study duplicial structure on nerves of categories, bicategories, and monoidal categories.

On the Hochschild homology of quantum SL(N)

We show that the standard quantized coordinate ring A of quantum SL(N) satisfies van den Bergh's analogue of Poincare duality for Hochschild (co)homology with dualizing bimodule being A_sigma, the A-bimodule which is A as k-vector space with right multiplication twisted by the modular automorphism sigma of the Haar functional. This implies that H_{N^2-1} (A, A_sigma)=k, generalizing our previous results for quantum SL(2).

Invariance of the Goresky-Hingston algebra on reduced Hochschild homology

We prove that two quasi-isomorphic simply connected differential graded associative Frobenius algebras have isomorphic Goresky-Hingston algebras on their reduced Hochschild homology. Our proof is based on relating the Goresky-Hingston algebra on reduced Hochschild homology to the singular Hochschild cohomology algebra. For any simply connected oriented closed manifold $M$ of dimension $k$, the Goresky-Hingston algebra on reduced Hochschild homology induces an algebra structure of degree $k-1$ on $\bar{H}^*(LM;\mathbb{Q})$, the reduced rational cohomology of the free loop space of $M$. As a con…

Twisted Hochschild homology of quantum flag manifolds: 2-cycles from invariant projections

We study the twisted Hochschild homology of quantum full flag manifolds, with the twist being the modular automorphism of the Haar state. We show that non-trivial 2-cycles can be constructed from appropriate invariant projections. The main result is that $HH_2^θ(\mathbb{C}_q[G / T])$ is infinite-dimensional when $\mathrm{rank}(\mathfrak{g}) > 1$. We also discuss the case of generalized flag manifolds and present the example of quantum Grassmannians.

Factorization homology and calculus à la Kontsevich Soibelman

Using factorization homology techniques, we give a simple proof of the action of Kontsevich and Soibelman operad on the pair consisting of Hochschild cohomology and Hochschild homology. The proof obviously extends to higher Hochschild cohomology and factorization (also called chiral) homology. Note that our result works in categories of chain complexes but also in categories of modules over a commutative ring spectrum.

Coactions on Hochschild Homology of Hopf-Galois Extensions and Their Coinvariants

Let A be an H-Galois extension of B. If M is a Hopf bimodule then HH.(A,M), the Hochschild homology of A with coefficients in M, is a right comodule over the coalgebra C:=H/[H,H]. Given an injective left C-comodule V, we denote the cotensor product of M and V by N. Our aim is to investigate the relationship between the cotensor product of HH.(A,M) and V, on the one hand, and HH.(B,N) on the other hand. The roots of this problem can be found in Lorenz's work on the descent of Hochschild homology of centrally Galois extensions, where HH.(A)^G, the subspace of invariant cycles with respect to the…

实心闭环面的微分同胚群与Hochschild同调

The Diffeomorphism Group of the Solid Closed Torus and Hochschild Homology

本文证明了Boyarchenko-Drinfeld意义下自内射ribbon Grothendieck-Verdier范畴的Hochschild复形上的循环作用可扩张为实心闭环面微分同胚群的作用。

Hochschild homology, trace map and $ζ$-cycles

In this paper we consider two spectral realizations of the zeros of the Riemann zeta function. The first one involves all non-trivial (non-real) zeros and is expressed in terms of a Laplacian intimately related to the prolate wave operator. The second spectral realization affects only the critical zeros and it is cast in terms of sheaf cohomology. The novelty is that the base space is the Scaling Site playing the role of the parameter space for the $ζ$-cycles and encoding their stability by coverings.

奇 Khovanov 弧代数的 Hochschild（上）同调

Hochschild (co)homology for odd Khovanov arc algebras

本文将 Hochschild 同调与上同调推广至拟结合代数，并据此重新证明奇 Khovanov 同调的至多相差符号函子性，同时首次确立 Naisse–Putyra 缠结理论关于缠结配边的至多相差单位函子性。

Zeroth Hochschild homology of preprojective algebras over the integers

We determine the Z-module structure of the preprojective algebra and its zeroth Hochschild homology, for any non-Dynkin quiver (and hence the structure working over any base commutative ring, of any characteristic). This answers (and generalizes) a conjecture of Hesselholt and Rains, producing new $p$-torsion classes in degrees 2p^l, l >= 1, We relate these classes by p-th power maps and interpret them in terms of the kernel of Verschiebung maps from noncommutative Witt theory. An important tool is a generalization of the Diamond Lemma to modules over commutative rings, which we give in the ap…

On algebraic structures of the Hochschild complex

We first review various known algebraic structures on the Hochschild (co)homology of a differential graded algebras under weak Poincar{é} duality hypothesis, such as Calabi-Yau algebras, derived Poincar{é} duality algebras and closed Frobenius algebras. This includes a BV-algebra structure on $HH^*(A,A^\vee)$ or $HH^*(A,A)$, which in the latter case is an extension of the natural Gerstenhaber structure on $HH^*(A,A)$. As an example, after proving that the chain complex of the Moore loop space of a manifold $M$ is a CY-algebra and using Burghelea-Fiedorowicz-Goodwillie theorem we obtain a BV-st…