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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]

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]

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 cocommutative Hopf algebra H over a commutative ring K and a symmetric partial action of H on a K -algebra A , we obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the smash product A # H , involving the Hochschild homology of A and the partial homology of H. An analogous third quadrant cohomological spectral sequence is also obtained. The definition of the partial (co)homology of H under consideration is based on the category of the partial representations of H. A specific partial representation of H on a subalgebra B of the partial "Hopf" algebra H p a r is involved in the definition and we construct a projective resolution of B. [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]

A typical example of a Cartan calculus consists of the Lie derivative and the contraction with vector fields of a manifold on the derivation ring of the de Rham complex. In this manuscript, a second stage of the Cartan calculus is investigated. In a general setting, the stage is formulated with operators obtained by the André–Quillen cohomology of a commutative differential graded algebra A on the Hochschild homology of A in terms of the homotopy Cartan calculus in the sense of Fiorenza and Kowalzig. Moreover, the Cartan calculus is interpreted geometrically with maps from the rational homotopy group of the monoid of self-homotopy equivalences on a space M to the derivation ring on the loop cohomology of M. We also give a geometric description to Sullivan's isomorphism, which relates the geometric Cartan calculus to the algebraic one, via the Γ 1 map due to Félix and Thomas. [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]

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]

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]

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]

We define twisted Hochschild homology for Green functors. This construction is the algebraic analogue of the relative topological Hochschild homology THH C n (−) , and under flatness conditions it describes the E 2 term of the Künneth spectral sequence for relative THH. Applied to ordinary rings, we obtain new algebraic invariants. Extending Hesselholt's construction of the Witt vectors of noncommutative rings, we interpret our construction as providing Witt vectors for Green functors. [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]

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 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]

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]

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]

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]

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]

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]

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]

It is easy to find algebras T ∈ C in a finite tensor category C that naturally come with a lift to a braided commutative algebra T ∈ Z (C) in the Drinfeld center of C. In fact, any finite tensor category has at least two such algebras, namely the monoidal unit I and the canonical end ∫ X ∈ C X ⊗ X ∨. Using the theory of braided operads, we prove that for any such algebra T the homotopy invariants, i.e. the derived morphism space from I to T , naturally come with the structure of a differential graded E 2 -algebra. This way, we obtain a rich source of differential graded E 2 -algebras in the homological algebra of finite tensor categories. We use this result to prove that Deligne's E 2 -structure on the Hochschild cochain complex of a finite tensor category is induced by the canonical end, its multiplication and its non-crossing half braiding. With this new and more explicit description of Deligne's E 2 -structure, we can lift the Farinati-Solotar bracket on the Ext algebra of a finite tensor category to an E 2 -structure at cochain level. Moreover, we prove that, for a unimodular pivotal finite tensor category, the inclusion of the Ext algebra into the Hochschild cochains is a monomorphism of framed E 2 -algebras, thereby refining a result of Menichi. [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]

For a commutative ring R and an ADE Dynkin quiver Q , we prove that the multiplicative preprojective algebra of Crawley-Boevey and Shaw, with parameter q = 1 , is isomorphic to the (additive) preprojective algebra as R -algebras if and only if the bad primes for Q – 2 in type D, 2 and 3 for Q = E 6 , E 7 and 2, 3 and 5 for Q = E 8 – are invertible in R. We construct an explicit isomorphism over Z [ 1 / 2 ] in type D, over Z [ 1 / 2 , 1 / 3 ] for Q = E 6 , E 7 and over Z [ 1 / 2 , 1 / 3 , 1 / 5 ] for Q = E 8. Conversely, if some bad prime is not invertible in R , we show that the additive and multiplicative preprojective algebras differ in zeroth Hochschild homology, and hence are not isomorphic. In fact, one only needs the vanishing of certain classes in zeroth Hochschild homology of the multiplicative preprojective algebra, utilizing a rigidification argument for isomorphisms that may be of independent interest. In the setting of Ginzburg dg-algebras, our obstructions are new in type E and give a more elementary proof of the negative result of Etgü–Lekili [5, Theorem 13] in type D. Moreover, the zeroth Hochschild homology of the multiplicative preprojective algebra, computed in Section 4, can be interpreted as the space of unobstructed deformations of the multiplicative Ginzburg dg-algebra by Van den Bergh duality. Finally, we observe that the multiplicative preprojective algebra is not symmetric Frobenius if Q ≠ A 1 , a departure from the additive preprojective algebra in characteristic 2 for Q = D 2 n , n ≥ 2 and Q = E 7 , E 8. [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 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 2 p 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 appendix. [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 Γ ⊂ SL ( 2 , Z ) be a finite subgroup acting on the irrational rotational algebra A θ via the restriction of the canonical action of SL ( 2 , Z ) . Consider the crossed product algebra A θ alg ⋊ Γ obtained by the restriction of the Γ action on the algebraic irrational rotational algebra. In this paper we prove many results on the homology group of the crossed product algebra A θ alg ⋊ Γ . We also analyse the case of the smooth crossed product algebra, A θ ⋊ Γ and calculate some of its homology groups. [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]

We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any prop with A ∞ -multiplication—we think of such algebras as A ∞ -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 and Kaufmann actions 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 Hochschild complex. [ABSTRACT FROM AUTHOR]