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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]