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Consider a reductive p-adic group G, its (complex-valued) Hecke algebra H(G), and the Harish-Chandra-Schwartz algebra S(G). We compute the Hochschild homology groups of H(G) and of S(G), and we describe the outcomes in several ways. Our main tools are algebraic families of smooth G-representations. With those we construct maps from HHn(H(G)) and HHn(S(G)) to modules of differential n-forms on affine varieties. For n=0, this provides a description of the cocentres of these algebras in terms of nice linear functions on the Grothendieck group of finite length (tempered) G-representations. It is known from [J. Algebra 606 (2022), 371-470] that every Bernstein ideal H(G)s of H(G) is closely related to a crossed product algebra of the form O(T)⋊W. Here O(T) denotes the regular functions on the variety T of unramified characters of a Levi subgroup L of G, and W is a finite group acting on T. We make this relation even stronger by establishing an isomorphism between HH∗(H(G)s) and HH∗(O(T)⋊W), although we have to say that in some cases it is necessary to twist C[W] by a 2-cocycle. Similarly, we prove that the Hochschild homology of the two-sided ideal S(G)s of S(G) is isomorphic to HH(C∞(Tu)⋊W), where Tu denotes the Lie group of unitary unramified characters of L. In these pictures of HH*(H(G)) and HH∗(S(G)), we also show how the Bernstein centre of H(G) acts. Finally, we derive similar expressions for the (periodic) cyclic homology groups of H(G) and of S(G) and we relate that to topological K-theory. [ABSTRACT FROM AUTHOR]

In this paper, we show that higher Hochschild homology groups do not vanish for a class of Hochschild extension algebras of truncated quiver algebras by the standard duality module. Consequently, this result extends the result of the trivial extension algebra by the standard duality module for truncated quiver algebras. Moreover, as an application, we show that for any Hochschild extension algebra of selfinjective Nakayama algebras by the standard duality module, its Hochschild homology dimension is infinite. [ABSTRACT FROM AUTHOR]

Given a group G and a partial factor set σ of G, we introduce the twisted partial group algebra κ par σ G , which governs the partial projective σ -representations of G into algebras over a field κ. Using the relation between partial projective representations and twisted partial actions we endow κ par σ G with the structure of a crossed product by a twisted partial action of G on a commutative subalgebra of κ par σ G. Then, we use twisted partial group algebras to obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the crossed product A ∗ Θ G , involving the Hochschild homology of A and the partial homology of G, where Θ is a unital twisted partial action of G on a κ -algebra A with a κ -based twist. An analogous third quadrant cohomological spectral sequence is also obtained. [ABSTRACT FROM AUTHOR]

Using Chern character, we construct a natural transformation from the local Hilbert functor to a functor of Artin rings defined from Hochschild homology. This enables us to realize (after slight modification) the infinitesimal Abel-Jacobi map as a morphism between tangent spaces of two functors of Artin rings and also enables us to reconstruct the semi-regularity map together with giving a different proof of a theorem of Bloch stating that the semi-regularity map annihilates certain obstructions to embedded deformations of a closed subvariety which is a locally complete intersection. [ABSTRACT FROM AUTHOR]

This work is part of a research program to compute the Hochschild homology groups HH ∗ (C [ x 1 , ... , x d ] / (x 1 , ... , x d) 3 ; C) in the case d = 2 through a lesser-known invariant called Coxeter cohomology, motivated by the isomorphism HH i (C [ x 1 , ... , x d ] / (x 1 , ... , x d) 3 ; C) ≅ ∑ 0 ≤ j ≤ i H C j S i + j , V ⊗ (i + j) provided by Larsen and Lindenstrauss. Here, H C ∗ denotes Coxeter cohomology, S i + j denotes the symmetric group on i + j letters, and V is the standard representation of GL d (C) on C d . We compute the Euler characteristic of the Coxeter cohomology (the alternating sum of the ranks of the Coxeter cohomology groups) of several representations of S n . In particular, the aforementioned tensor representation, and also several classes of irreducible representations of S n . Although the problem and its motivation are algebraic and topological in nature, the techniques used are largely combinatorial. [ABSTRACT FROM AUTHOR]

We compute the Hochschild homology and cohomology of A (1) , the subalgebra of the 2-primary Steenrod algebra generated by the first two Steenrod squares, Sq 1 , Sq 2 . The computation is accomplished using several May-type spectral sequences. [ABSTRACT FROM AUTHOR]

In this paper, we discuss Hochschild chain models for some of the string topology operations. We use these models to simplify the proofs and computations of some of the results in string topology. Along the way we also make some new observations. We further discuss how nonnilpotent local level homology classes with respect to the Chas–Sullivan and the Goresky–Hingston product detect closed geodesics with optimal index growth rates. [ABSTRACT FROM AUTHOR]

The cohomology algebra of the space H∗ (X) defines neither cohomology modules of the loop space H∗ (ΩX) nor cohomologies of the free loop space H∗ (ΛX). But by the author's minimality theorem, there exists a structure of A(∞)-algebra (H∗ (X), {mi}) on H∗ (X), which determines H∗ (ΩX). Here will be shown that the same A(∞)-algebra (H∗(X), {mi}) determines also cohomology modules H∗ (ΛX). [ABSTRACT FROM AUTHOR]

We describe two major string topology operations, the Chas–Sullivan product and the Goresky–Hingston coproduct, from geometric and algebraic perspectives. The geometric construction uses Thom–Pontrjagin intersection theory while the algebraic construction is phrased in terms of Hochschild homology. We give computations of products and coproducts on lens spaces via geometric intersection, and deduce that the coproduct distinguishes 3-dimensional lens spaces. Algebraically, we describe the structure these operations define together on the Tate–Hochschild complex. We use rational homotopy theory methods to sketch the equivalence between the geometric and algebraic definitions for simply-connected manifolds and real coefficients, emphasizing the role of configuration spaces. Finally, we study invariance properties of the operations, both algebraically and geometrically. [ABSTRACT FROM AUTHOR]

In this paper, we study a generalization of Kähler differentials, which correspond to the secondary Hochschild homology associated to a triple. We establish computations in low dimension, while also showing how this connects with the kernel of a multiplication map. [ABSTRACT FROM AUTHOR]

We study the Hochschild homology of the convolution algebra of a proper Lie groupoid by introducing a convolution sheaf over the space of orbits. We develop a localization result for the associated Hochschild homology sheaf, and we prove that the Hochschild homology sheaf at each stalk is quasi-isomorphic to the stalk at the origin of the Hochschild homology of the convolution algebra of its linearization, which is the transformation groupoid of a linear action of a compact isotropy group on a vector space. We then explain Brylinski's ansatz to compute the Hochschild homology of the transformation groupoid of a compact group action on a manifold. We verify Brylinski's conjecture for the case of smooth circle actions that the Hochschild homology is given by basic relative forms on the associated inertia space. [ABSTRACT FROM AUTHOR]

The Hochschild homology of the ring k [ x 1 , x 2 , ... , x d ] / (x 1 , x 2 , ... , x d) 2 has been known and calculated several ways. This paper uses those calculations to calculate cyclic, negative cyclic, and periodic cyclic homology of k [ x 1 , x 2 , ... , x d ] / (x 1 , x 2 , ... , x d) 2 over k for k = Z. These calculations have been known for k = Q or fields containing it, but this paper studies the torsion which one gets for k = Z and through it, via the universal coefficient theorem, for general k. The computations are related to homology, cohomology, and Tate cohomology of cyclic groups. [ABSTRACT FROM AUTHOR]

We prove a conjecture of Shklyarov concerning the relationship between K. Saito's higher residue pairing and a certain pairing on the periodic cyclic homology of matrix factorization categories. Along the way, we give new proofs of a result of Shklyarov (Corollary 2 of [Adv. Math. 292 (2016), 181--209]) and Polishchuk--Vaintrob's Hirzebruch--Riemann--Roch formula for matrix factorizations (Theorem 4.1.4 (i) of [Duke Math. J. 161 (2012), 1863--1926]). [ABSTRACT FROM AUTHOR]

These are slightly expanded notes of lectures given in April 2019 at the Isfahan school and conference on representations of algebras. We recall the formalism of derived categories and functors and survey invariance results for the Hochschild (co)homology of differential graded algebras and categories. [ABSTRACT FROM AUTHOR]

In this paper, we prove the commutativity of the square in the chain level of both Chevalley Eilenberg complex of Lie homology and Hochschild complex, where the antisymmetrization map is used between the complexes. Originally, Loday proved this isomorphism by constructing a certain map satisfying relations of a presimplicial homotopy to prove the commutativity of the square mentioned above. Here, we present a different approach for the proof of the commutativity without constructing that certain map satisfying the relations of the presimplicial homotopy. [ABSTRACT FROM AUTHOR]

For every n ≥ 1, we calculate the Hochschild homology of the quantum monoids Mq(n), and the quantum groups GLq(n) and SLq(n) with coefficients in a 1-dimensional module coming from a modular pair in involution. [ABSTRACT FROM AUTHOR]

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 nontrivial 2 -cycles can be constructed from appropriate invariant projections. Moreover, we show that HH 2 𝜃 (ℂ q [ G / T ]) has dimension at least rank (𝔤). We also discuss the case of generalized flag manifolds and present the example of the quantum Grassmannians. [ABSTRACT FROM AUTHOR]

This paper interprets Hesselholt and Madsen's real topological Hochschild homology functor THR in terms of the multiplicative norm construction. We show that THR satisfies cofinality and Morita invariance, and that it is suitably multiplicative. We then calculate its geometric fixed points and its Mackey functor of components, and show a decomposition result for group algebras. Using these structural results we determine the homotopy type of THR(Fp) and show that its bigraded homotopy groups are polynomial on one generator over the bigraded homotopy groups of HFp. We then calculate the homotopy type of THR.Z/away from the prime 2, and the homotopy ring of the geometric fixed points spectrum 8Z=2 THR(Z). [ABSTRACT FROM AUTHOR]

We present a version of higher Hochschild homology for spaces equipped with principal bundles for a structure group G. As coefficients, we allow E∞-algebras with G-action. For this homology theory, we establish an equivariant version of excision and prove that it extends to an equivariant topological field theory with values in the (∞,1)-category of cospans of E∞-algebras. [ABSTRACT FROM AUTHOR]

In this paper, we develop the theory of modules over (A , δ) , where A is an algebra and δ : A ⟶ A is a derivation. Our approach is heavily influenced by Lie derivative operators in noncommutative geometry, which make the Hochschild homologies H H ∙ (A) of A into a module over (A , δ) . We also consider modules over (A , Δ) , where Δ = { Δ n } n ≥ 0 is a higher derivation on A. Further, we obtain a Cartan homotopy formula for an arbitrary higher derivation on A. [ABSTRACT FROM AUTHOR]

We consider the derived category DbG(V) of coherent sheaves on a complex vector space V equivariant with respect to an action of a finite reflection group G. In some cases, including Weyl groups of type A, B, G2, F4, as well as the groups G(m, 1, n) = (μm)n × Sn, we construct a semiorthogonal decomposition of this category, indexed by the conjugacy classes of G. The pieces of this decompositions are equivalent to the derived categories of coherent sheaves on the quotient-spaces Vg/C(g), where C(g) is the centralizer subgroup of g ε G. In the case of the Weyl groups the construction uses some key results about the Springer correspondence, due to Lusztig, along with some formality statement generalizing a result of Deligne [23]. We also construct global analogs of some of these semiorthogonal decompositions involving derived categories of equivariant coherent sheaves on Cn, where C is a smooth curve. [ABSTRACT FROM AUTHOR]

In this note, we use the pairing induced by the interchange map in conjunction with the strongly homotopy commutative algebra structure to define products on the Eilenberg–Moore differential Tor and give a simplified proof of an improved outcome of Jones's result due to Ndombol and Thomas. As a result, we establish an isomorphism of graded algebras between the Hochschild homology and the free loop space cohomology of a simply connected topological space. [ABSTRACT FROM AUTHOR]

We show that faithfully flat smooth extensions of associative unital algebras are reduced flat, and therefore, fit into the Jacobi-Zariski exact sequence in Hochschild homology and cyclic (co)homology even when the algebras are noncommutative or infinite dimensional. We observe that such extensions correspond to étale maps of affine schemes, and we propose a definition for generic noncommutative fibrations using distributive laws and homological properties of the induction and restriction functors. Then we show that Galois fibrations do produce the right exact sequence in homology. We then demonstrate the versatility of our model on a geometro-combinatorial example. For a connected unramified covering of a connected graph , we construct a smooth Galois fibration and calculate the homology of the corresponding local coefficient system. [ABSTRACT FROM AUTHOR]

Higher Hochschild homology is the analog of the homology of spaces, where the context for the coefficients—which usually is that of abelian groups—is that of commutative algebras. Two spaces that are equivalent after a suspension have the same homology. We show that this is not the case for higher Hochschild homology, providing a counterexample to a behavior so far observed in stable homotopy theory. [ABSTRACT FROM AUTHOR]

The notion of non-abelian Hom-Leibniz tensor product is introduced and some properties are established. This tensor product is used in the description of the universal (-)central extensions of Hom-Leibniz algebras. We also give its application to the Hochschild homology of Hom-associative algebras. [ABSTRACT FROM AUTHOR]

In this note, we use the notion of semi-free resolution to give a new proof of J. D. S. Jones’ theorem (Invent Math 87:403-423, 1987) which establishes an isomorphism of graded vector spaces between the Hochschild homology and the cohomology of the free loop space of a topological space. [ABSTRACT FROM AUTHOR]

In this paper we study properties of the secondary Hochschild homology of the triple ( A, B, ε) with coefficients in M. We establish a type of Morita equivalence between two triples and show that H•(( A, B, ε); M) is invariant under this equivalence. We also prove the existence of an exact sequence which connects the usual and the secondary Hochschild homologies in low dimension, allowing one to perform easy computations. The functoriality of H•(( A, B, ε); M) is also discussed. [ABSTRACT FROM AUTHOR]

In a previous paper, the author computed the dimension of Hochschild cohomology groups of Jacobian algebras from (unpunctured) triangulated surfaces, and gave a geometric interpretation of those numbers in terms of the number of internal triangles, the number of vertices and the existence of certain kind of boundaries. The aim of this note is to compute the cyclic (co)homology and the Hochschild homology of the same family of algebras and to give an interpretation of those dimensions through elements of the triangulated surface. [ABSTRACT FROM AUTHOR]

Let be a commutative algebra with and let be a cleft extension of . We obtain a new mixed complex, simpler than the canonical one, giving the Hochschild and cyclic homologies of relative to . This complex resembles the canonical reduced mixed complex of an augmented algebra. We begin the study of our complex showing that it has a harmonic decomposition like the one considered by Cuntz and Quillen for the normalized mixed complex of an algebra. [ABSTRACT FROM AUTHOR]

We construct and study the map from Leibniz homologyHL∗(𝔥) of an abelian extension𝔥of a simple real Lie algebra𝔤to the Hochschild homologyHH∗−1(U(𝔥)) of the universal envelopping algebraU(𝔥). To calculate some homology groups, we use the Hochschild-Serre spectral sequences and Pirashvili spectral sequences. The result shows what part of the non-commutative Leibniz theory is detected by classical Hochschild homology, which is of interest today in string theory. [ABSTRACT FROM PUBLISHER]

We give a classification of open Klein topological conformal field theories in terms of Calabi-Yau $$A_\infty $$ -categories endowed with an involution. Given an open Klein topological conformal field theory, there is a universal open-closed extension whose closed part is the involutive variant of the Hochschild chains of the open part. [ABSTRACT FROM AUTHOR]

We prove that if an algebra is either selfinjective, local or graded, then the Hochschild homology dimension of its trivial extension is infinite. [ABSTRACT FROM AUTHOR]

In our paper “Jacobi–Zariski exact sequence for Hochschild homology and cyclic (co)homology” [Homology Homotopy Appl., 14(1):65–78, 2012], in Theorem 2.3 and in all remaining results based on this theorem, it is stated that the Jacobi–Zariski long exact sequence works for the range p⩾0. The range should be corrected to p⩾1. [ABSTRACT FROM AUTHOR]

In this work we prove the so-called dimension property for the cohomological field theory associated with a homogeneous polynomial $W$ with an isolated singularity, in the algebraic framework of [A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016), 1–122]. This amounts to showing that some cohomology classes on the Deligne–Mumford moduli spaces of stable curves, constructed using Fourier–Mukai-type functors associated with matrix factorizations, live in prescribed dimension. The proof is based on a homogeneity result established in [A. Polishchuk and A. Vaintrob, Algebraic construction of Witten’s top Chern class, in Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (American Mathematical Society, Providence, RI, 2001), 229–249] for certain characteristic classes of Koszul matrix factorizations of $0$. To reduce to this result, we use the theory of Fourier–Mukai-type functors involving matrix factorizations and the natural rational lattices in the relevant Hochschild homology spaces, as well as a version of Hodge–Riemann bilinear relations for Hochschild homology of matrix factorizations. Our approach also gives a proof of the dimension property for the cohomological field theories associated with some quasihomogeneous polynomials with an isolated singularity. [ABSTRACT FROM PUBLISHER]

We introduce BV-algebra structures on the homology of the space of framed long knots in $${\mathbb {R}}^n$$ in two ways. The first one is given in a similar fashion to Chas-Sullivan's string topology. The second one is defined on the Hochschild homology associated with a cyclic, multiplicative operad of graded modules. The latter can be applied to Bousfield-Salvatore spectral sequence converging to the homology of the space of framed long knots. Conjecturally these two structures coincide with each other. [ABSTRACT FROM AUTHOR]

We develop a theory of 'ad hoc' Chern characters for twisted matrix factorizations associated to a scheme X, a line bundle $$\mathcal {L}$$ , and a regular global section $$W \in \Gamma (X, \mathcal {L})$$ . As an application, we establish the vanishing, in certain cases, of $$h_c^R(M,N)$$ , the higher Herbrand difference, and, $$\eta _c^R(M,N)$$ , the higher codimensional analogue of Hochster's theta pairing, where R is a complete intersection of codimension c with isolated singularities and M and N are finitely generated R-modules. Specifically, we prove such vanishing if $$R = Q/(f_1, \dots , f_c)$$ has only isolated singularities, Q is a smooth k-algebra, k is a field of characteristic 0, the $$f_i$$ 's form a regular sequence, and $$c \ge 2$$ . Such vanishing was previously established in the general characteristic, but graded, setting in Moore et al. (Math Z 273(3-4):907-920, 2013). [ABSTRACT FROM AUTHOR]

This article is the sequel to (Marcolli and Tabuada in Sel Math 20(1):315-358, ). We start by developing a theory of noncommutative (=NC) mixed motives with coefficients in any commutative ring. In particular, we construct a symmetric monoidal triangulated category of NC mixed motives, over a base field k, and a full subcategory of NC mixed Artin motives. Making use of Hochschild homology, we then apply Ayoub's weak Tannakian formalism to these motivic categories. In the case of NC mixed motives, we obtain a motivic Hopf dg algebra, which we describe explicitly in terms of Hochschild homology and complexes of exact cubes. In the case of NC mixed Artin motives, we compute the associated Hopf dg algebra using solely the classical category of mixed Artin-Tate motives. Finally, we establish a short exact sequence relating the Hopf algebra of continuous functions on the absolute Galois group with the motivic Hopf dg algebras of the base field k and of its algebraic closure. Along the way, we describe the behavior of Ayoub's weak Tannakian formalism with respect to orbit categories and relate the category of NC mixed motives with Voevodsky's category of mixed motives. [ABSTRACT FROM AUTHOR]

Let A be a dg algebra over 픽2 and let M be a dg A-bimodule. We show that under certain technical hypotheses on A, a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product M ⊗AL M and converges to the Hochschild homology of M. We apply this result to bordered Heegaard Floer theory, giving spectral sequences associated to Heegaard Floer homology groups of certain branched and unbranched double covers. [ABSTRACT FROM AUTHOR]

In this article, we compute the Hochschild homology group ofA = KΓ/(f(Xs)), whereKΓ is the path algebra of the cyclic quiver Γ withsvertices andsarrows over a commutative ringK,f(x) is a monic polynomial overK, andXis the sum of all arrows inKΓ. Moreover, we compute the cyclic homology group ofAin the casef(x) = (x − a)m, wherea ∈ K, so that we can determine the cyclic homology ofAin general whenKis an algebraically closed field. [ABSTRACT FROM PUBLISHER]

We prove that the bounded derived category of the surface S constructed by Barlow admits a length 11 exceptional sequence consisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of S in the moduli space of determinantal Barlow surfaces, the generic surface has a semiorthogonal decomposition of its derived category into a length 11 exceptional sequence of line bundles and a category with trivial Grothendieck group and Hochschild homology, called a phantom category. This is done using a deformation argument and the fact that the derived endomorphism algebra of the sequence is constant. Applying Kuznetsov's results on heights of exceptional sequences, we also show that the sequence on S itself is not full and its (left or right) orthogonal complement is also a phantom category. [ABSTRACT FROM AUTHOR]

We consider the polynomial algebra $${\mathbb{C}[{\bf z}] := \mathbb{C}[z_1, z_2, z_3]}$$ and the polynomial $${f := z^{3}_{1} + z^3_2 + z^3_3 + 3qz_1z_2z_3}$$ , where $${q \in \mathbb{C}}$$ . Our aim is to compute the Hochschild homology and cohomology of the cubic surface $${\mathcal{X}_f := \{{\bf z} \in \mathbb{C}^3/f({\bf z}) = 0\}}$$ . For explicit computations, we shall make use of a method suggested by M. Kontsevich. Then, we shall develop it in order to determine the Hochschild homology and cohomology by means of multivariate division and Groebner bases. Some formal computations with Maple are also used. [ABSTRACT FROM AUTHOR]

LetAbe a (not necessarily commutative) monoid object in an abelian symmetric monoidal category (C, ⊗,1) satisfying certain conditions. In this paper, we continue our study of the localizationMSof anyA-moduleMwith respect to a subsetS ⊆ HomA−Bimod(A,A) that is closed under composition. In particular, we prove the following theorem: ifPis anA-bimodule such thatPis symmetric as a bimodule over the centerZ(A) ofA, we have isomorphismsHH*(A,P)S ≅ HH*(A,PS) ≅ HH*(AS,PS) of Hochschild homology groups. [ABSTRACT FROM AUTHOR]

In this paper we prove a version of curved Koszul duality for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{Z}/2\mathbb{Z}$-graded curved coalgebras and their cobar differential graded algebras. A curved version of the homological perturbation lemma is also obtained as a useful technical tool for studying curved (co)algebras and precomplexes. The results of Koszul duality can be applied to study the category of matrix factorizations $\mathsf{MF}(R,W)$. We show how Dyckerhoff’s generating results fit into the framework of curved Koszul duality theory. This enables us to clarify the relationship between the Borel–Moore Hochschild homology of curved (co)algebras and the ordinary Hochschild homology of the category $\mathsf{MF}(R,W)$. Similar results are also obtained in the orbifold case and in the graded case. [ABSTRACT FROM AUTHOR]

We prove an analog of the holomorphic Lefschetz formula for endofunctors of smooth compact dg-categories. We deduce from it a generalization of the Lefschetz formula of Lunts (J Algebra 356:230-256, ) that takes the form of a reciprocity law for a pair of commuting endofunctors. As an application, we prove a version of Lefschetz formula proposed by Frenkel and Ngô (Bull Math Sci 1(1):129-199, ). Also, we compute explicitly the ingredients of the holomorphic Lefschetz formula for the dg-category of matrix factorizations of an isolated singularity $${\varvec{w}}$$ . We apply this formula to get some restrictions on the Betti numbers of a $${\mathbb Z}/2$$ -equivariant module over $$k[[x_1,\ldots ,x_n]]/({\varvec{w}})$$ in the case when $${\varvec{w}}(-x)={\varvec{w}}(x)$$ . [ABSTRACT FROM AUTHOR]

From any algebra A defined by a single non-degenerate homogeneous quadratic relation f, we prove that the quadratic algebra B defined by the potential w = fz is 3-Calabi-Yau. The algebra B can be viewed as a 3-Calabi-Yau completion of Keller. The algebras A and B are both Koszul. The classification of the algebras B in three generators, i.e., when A has two generators, leads to three types of algebras. The second type (the most interesting one) is viewed as a deformation of a Poisson algebra S whose Poisson bracket is non-diagonalizable quadratic. Although the potential of S has non-isolated singularities, the homology of S is computed. Next the Hochschild homology of B is obtained. [ABSTRACT FROM AUTHOR]

We show that if $A$ and $H$ are Hopf algebras that have equivalent tensor categories of comodules, then one can transport what we call a free Yetter–Drinfeld resolution of the counit of $A$ to the same kind of resolution for the counit of $H$, exhibiting in this way strong links between the Hochschild homologies of $A$ and $H$. This enables us to obtain a finite free resolution of the counit of $\mathcal {B}(E)$, the Hopf algebra of the bilinear form associated with an invertible matrix $E$, generalizing an earlier construction of Collins, Härtel and Thom in the orthogonal case $E=I_n$. It follows that $\mathcal {B}(E)$ is smooth of dimension 3 and satisfies Poincaré duality. Combining this with results of Vergnioux, it also follows that when $E$ is an antisymmetric matrix, the $L^2$-Betti numbers of the associated discrete quantum group all vanish. We also use our resolution to compute the bialgebra cohomology of $\mathcal {B}(E)$in the cosemisimple case. [ABSTRACT FROM AUTHOR]

We adapt to the case of deformation quantization modules a formula of Lunts (Lefschetz fixed point theorems for Fourier-Mukai functors and DG algebras. arXiv:1102.2884. ArXiv e-prints, ) who calculates the trace of a kernel acting on the Hochschild homology of a DQ-algebroid. [ABSTRACT FROM AUTHOR]

We study the Hochschild (co)homology of curved A∞ alge- bras that arise in the study of Landau-Ginzburg (LG) models in physics. We show that the ordinary Hochschild homology and cohomology of these algebras vanish. To correct this we introduce modified versions of these theories, Borel-Moore Hochschild homology and compactly sup- ported Hochschild cohomology. For LG models the new invariants yield the answer predicted by physics, shifts of the Jacobian ring. We also study the relationship between graded LG models and the geometry of hypersurfaces. We prove that Orlov's derived equivalence descends from an equivalence at the differential graded level, so in particular the CY/LG correspondence is a dg equivalence. This leads us to study the equivariant Hochschild homology of orbifold LG models. The results we get can be seen as noncommutative analogues of the Lefschetz hyperplane and Griffiths transversality theorems. [ABSTRACT FROM AUTHOR]

Let X be a smooth projective complex variety. The Hochschild homology HH( X) of X is an important invariant of X, which is isomorphic to the Hodge cohomology of X via the Hochschild-Kostant-Rosenberg isomorphism. On HH( X), one has the Mukai pairing constructed by Caldararu. An explicit formula for the Mukai pairing at the level of Hodge cohomology was proven by the author in an earlier work (following ideas of Markarian). This formula implies a similar explicit formula for a closely related variant of the Mukai pairing on HH( X). The latter pairing on HH( X) is intimately linked to the study of Fourier-Mukai transforms of complex projective varieties. We give a new method to prove a formula computing the aforementioned variant of Caldararu's Mukai pairing. Our method is based on some important results in the area of deformation quantization. In particular, we use part of the work of Kashiwara and Schapira on Deformation Quantization modules together with an algebraic index theorem of Bressler, Nest and Tsygan. Our new method explicitly shows that the 'Noncommutative Riemann-Roch' implies the classical Riemann-Roch. Further, it is hoped that our method would be useful for generalization to settings involving certain singular varieties. [ABSTRACT FROM AUTHOR]