Excision in Hopf Cyclic Homology

This chapter is devoted to the computation of Hochschild and cyclic homologies of some particular types of algebras: tensor algebras, symmetric algebras, universal enveloping algebras of Lie algebras and, finally, smooth algebras, on which we put some emphasis. One more important example, the case of group algebras, will be treated later, in Sect. 7.4. It is also shown in this chapter that Hochschild and cyclic homology are related to many other theories such as the homology of Lie algebras, André-Quillen homology of commutative algebras, and Deligne cohomology.KeywordsExact SequenceSpectral SequenceCommutative AlgebraPolynomial AlgebraFree ResolutionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this chapter, we introduce the reader to the homology theory for algebras known as cyclic homology. As we shall see in the next section, cyclic homology may be viewed as the “linearization” of K-theory, in the same sense in which the matrix ring M n (R) is the “linearization” of the general linear group GL(n, R). For motivation, it is useful to think of the case where the ring R is ℝ or ℂ. Then GL(n, R)is a Lie group, and the space BGL(n, R)+ giving rise to the higher K-groups is by its construction an H-space whose homology agrees with the homology of this Lie group (never mind for the moment that we are forgetting the topology!). On the other hand, one of the basic principles of Lie theory is that the best way to study Lie groups (which are “non-linear” objects) is often by way of their Lie algebras (which can be studied using linear algebra). For example, it is a famous result that for any compact Lie group G with Lie algebra g, the topological cohomology H●(G;ℝ) (here we are just thinking of G as a space and ignoring the group structure) is canonically isomorphic to the Lie algebra cohomology H●(g;ℝ), which can at least in principle be computed using only finite-dimensional linear algebra. This suggests that some construction with the Lie algebra M n (R) of GL(n, R)ought to yield a reasonable approximation to the K-theory of R, at least in the limit as n → ∞. While the actual construction of cyclic homology and of the Chern character which relates it to K-theory is a bit roundabout, this philosophy turns out to be basically correct.KeywordsExact SequenceCommutative RingHomology TheoryChern CharacterCyclic HomologyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper we study the cyclic cohomology of certain ×-Hopf algebras: universal enveloping algebras, quantum algebraic tori, the Connes-Moscovici ×-Hopf algebroids and the Kadison bialgebroids. Introducing their stable anti Yetter-Drinfeld modules and cocyclic modules, we compute their cyclic cohomology. Furthermore, we provide a pairing for the cyclic cohomology of ×-Hopf algebras which generalizes the Connes-Moscovici characteristic map to ×-Hopf algebras. This enables us to transfer the ×-Hopf algebra cyclic cocycles to algebra cyclic cocycles.

In their article [9] on cyclic homology, Feigin and Tsygan have given a spectral sequence for the cyclic homology of a crossed product algebra, generalizing Burghelea’s calculation [4] of the cyclic homology of a group algebra. For an analogous spectral sequence for the Hochschild homology of a crossed product algebra, see Brylinski [2], [3]. In this article, we give a new derivation of this spectral sequence, and generalize it to negative and periodic cyclic homology HC• (A) and HP•(A). The method of proof is itself of interest, since it involves a natural generalization of the notion of a cyclic module, in which the condition that the morphism τ ∈ Λ(n,n) is cyclic of order n + 1 is relaxed to the condition that it be invertible. We call this category the paracyclic category. Given a paracyclic module P , we can define a chain complex C(P ), with differentials b and B, which respectively lower and raise degree. The condition that the module P is paracyclic translates to the condition on C(P ) that 1− (bB +Bb) is invertible. Our main result is to show that there is an analogue of the Eilenberg-Zilber theorem for bi-paracyclic modules. It is then easy to obtain a new expression for the cyclic homology of a crossed product algebra which leads immediately to the spectral sequence of Feigin and Tsygan. If M is a module over a commutative ring k, we will denote by M (k) the iterated tensor product, defined by M (0) = k and M (k+1) =M (k) ⊗M . If M and N are graded modules, we will denote by M ⊗N their graded tensor product. We would like to thank J. Block, J.-L. Brylinski and C. Kassel for a number of interesting discussions on the results presented here.

Note on the cohomology of color Hopf and Lie algebras

We show that inner derivations act trivially on the cyclic cohomology of the normalized cyclic complex W(l)/3s().)where 2 is a differential graded algebra.This is then used to establish the fact that the map introduced in [GJ] defines a Chern character in K theory.J.-M.Bismut recently extended the classical theory of Chern characters to the equivariant theory of the free loop space LX ; he explicitly constructed an S ' -invariant equivariantly closed form a>(E, V) on LX whose restriction to X, the space of constant loops, is Ch(F) = tr(expV2).Closely related to the equivariant theory of the loop space of X is A. Connes' cyclic theory of the differential graded algebra Q. of differential forms on X.More precisely, Chen's theory of iterated integrals allows us to view cyclic cycles of forms on X as equivariantly closed forms on the loop space LX.This is a generalization of the point of view that a one-form co on X can be viewed as a function on LX by y -> fy oe.This suggests the possibility of interpreting the classical theory of Chern characters in the context of cyclic homology i.e., of extending Ch(E) off the fixed point set X of the circle action on LX by way of a cyclic cycle over Q.In [GJP], Getzler, Jones, and Petrack, gave an explicit formula for a cyclic cycle y/(E, e) which via iterated integrals maps to the Bismut Chern character il(E) over LX.A priori, their construction of y/(E, e) appears to depend on the choice of idempotent e describing the vector bundle E, but in fact, it is shown that y/(E, e) depends only on the isomorphism class of E in Kq(X) , and hence defines the desired Chern character.This is essentially proved by establishing the triviality of certain group actions on the cyclic homology of a differential graded algebra Q..These actions are natural generalizations of the usual adjoint action Ad(g), given by conjugation by an invertible element g, and its infinitesimal analogue ad, given by inner derivation, to Connes' cyclic bar complex of Q. Cyclic homologyIn this section we review the definition of the cyclic homology of a DGA.According to C. Kassel, by a mixed complex we mean a triplet (f* ; d; B) where g7* is a graded algebra and where d: W* - ff*+x, and B: W* - W*~x, are derivations of degrees +1 and -1 respectively, and subject to the rela-

In this paper, we introduce a paracyclic version of S -modules. These new objects are called para- S -modules . Paracyclic modules and parachain complexes give rise to para- S -modules much in the same way as cyclic modules and mixed complexes give rise to S -modules. More generally, para- S -modules provide us with a natural framework to get analogues for paracyclic modules and parachain complexes of various constructions and equivalence results for cyclic modules or mixed complexes. The datum of a para- S -module does not provide us with a chain complex, and so notions of homology and quasi-isomorphisms do not make sense. However, chain maps and chain homotopies continue to make sense, and so by equivalences we actually mean chain homotopy equivalences. We establish some generalizations for para- S -modules and parachain complexes of the basic perturbation lemma of differential homological algebra. These generalizations provide us with general recipes for converting deformation retracts of Hochschild chain complexes into deformation retracts of para- S -modules. By using ideas of Kassel this then allows us to get comparison results between the various para- S -modules associated with para-precyclic modules, and between them and Connes’ cyclic chain complex. These comparison results lead us to alternative descriptions of Connes’ periodicity operator. This has some applications in periodic cyclic homology. We also describe the counterparts of these results in cyclic cohomology. In particular, we obtain an explicit way to convert a periodic (b, B) -cocycle into a cohomologous periodic cyclic cocycle.

Geometric construction of Hopf cyclic characteristic classes

Hopf algebras have important connections to quantum theory, Lie algebras, knot and braid theory, operator algebras and other areas of physics and mathematics. They have been intensely studied in the past; in particular, the solution of a number of conjectures of Kaplansky from the 1970s has led to progress on the classification of semisimple Hopf algebras and on the structure of pointed Hopf algebras. Among the topics covered are results toward the classification of finite-dimensional Hopf algebras (semisimple and non-semisimple), as well as what is known about the extension theory of Hopf algebras. Some papers consider Hopf versions of classical topics, such as the Brauer group, while others are closer to work in quantum groups. The book also explores the connections and applications of Hopf algebras to other fields.

In this paper, we will study the theory of cyclic homology for regular multiplier Hopf algebras. We associate a cyclic module to a triple $(mathcal{R},mathcal{H},mathcal{X})$ consisting of a regular multiplier Hopf algebra $mathcal{H}$, a left $mathcal{H}$-comodule algebra $mathcal{R}$, and a unital left $mathcal{H}$-module $mathcal{X}$ which is also a unital algebra. First, we construct a paracyclic module to a triple $(mathcal{R},mathcal{H},mathcal{X})$ and then prove the existence of a cyclic structure associated to this triple.

The Hochschild and (cotriple) cyclic homologies of crossed modules of (not necessarily unital) associative algebras are investigated. Wodzicki’s excision theorem is extended for inclusion crossed modules in the category of crossed modules of algebras. The cyclic and cotriple cyclic homologies of crossed modules are compared in terms of a long exact homology sequence, generalising the relative cyclic homology exact sequence.

Quantum topology is the branch of mathematics that studies objects from the realm of low-dimensional topology, such as knots, surfaces or 3-manifolds, using algebraic structures that arose in the 1980s inspired by ideas from theoretical physics. This fascinating new area of topology combines various fields of mathematics, including monoidal category theory, Hopf algebras, representation theory, Lie algebras, topological quantum field theories, algebraic topology, etc., all revolving around the so-called quantum knot invariants.<br/><br/>This book focuses on two such invariants: the universal invariant subject to a ribbon Hopf algebra —that dominates the Reshetikhin-Turaev invariants coming from the representation theory of the Hopf algebra— and the Kontsevich invariant —which is universal among finte type invariants—. More precisely, Gaussian calculus techniques developed by Bar-Natan and van der Veen are exploited to study a family of knot polynomial invariants that determines the universal invariant subject to a specific algebra, and to investigate how the most elementary polynomial of this collection is closely related to the so-called 2-loop polynomial of the knot, an invariant that encodes a certain part of its Kontsevich invariant.<br/><br/>Although this book covers several topics of current research, it has been written entirely in a self-contained way, so that it can be accessible to a broad audience.

The Poincar-Birkhoff-Witt Theorem deals with the structure and universal property of the universal enveloping algebra U (L) of a Lie algebra L , e.g., over R or C .K. H. Hofmann and L. Kramer (HK) [On weakly complete group algebras of Compact Groups, J. Lie Theory 30 (2020) 407-426] recently introduced the weakly complete universal enveloping algebra U(g) of a profinite-dimensional topological Lie algebra g.Here it is shown that the classical universal enveloping algebra U (|g|) of the abstract Lie algebra underlying g is a dense subalgebra of U(g) , algebraically generated by g U(g) .It is further shown that, inspite of U being a left adjoint functor, it nevertheless preserves projective limits in the form U(lim i g/i) = lim i U(g/i) , for profinite-dimensional Lie algebras g represented as projective limits of their finite-dimensional quotients.The required theory is presented in an appendix which is of independent interest.In a natural way, a weakly complete enveloping algebra U(g) is a weakly complete symmetric Hopf algebra with a Lie subalgebra P(U(g)) of primitive elements containing g (indeed properly if g = {0}), and with a nontrivial multiplicative pro-Lie group G(U(g)) of grouplike units, having P(U(g)) as its Lie algebra -in contrast with the classical Poincar-Birhoff-Witt environment of U (L) , thus providing a new aspect of Lie's Third Fundamental Theorem: Indeed a canonical pro-Lie subgroup * (g) of G(U(g)) is identified whose Lie algebra is naturally isomorphic to g.The structure of U(g) is described in detail for dim g = 1 .The primitive and grouplike components and their mutual relationship are evaluated precisely.In (HK), cited above, and in the work of R. Dahmen and K. H. Hofmann [The pro-Lie group aspect of weakly complete algebras and weakly complete group Hopf algebras, J. Lie Theory 29 (2019) 413-455] the real weakly complete group Hopf algebra R[G] of a compact group G was described.In particular, the set P(R[G])) of primitive elements of R[G] was identified as the Lie algebra g of G .It is now shown that for any compact group G with Lie algebra g there is a natural morphism of weakly complete symmetric Hopf algebras g : U(g) R[G] , implementing the identity on g and inducing a morphism of pro-Lie groups * (G) G(R[G]) = G : yet another aspect of Sophus Lie's Third Fundamental Theorem !

Let [Formula: see text] be a groupoid and assume that [Formula: see text] is a family of algebras with identity. First, we introduce the notion of a weak Hopf (groupoid)[Formula: see text]-coalgebra by that if, for each pair [Formula: see text], there is given a unital homomorphism [Formula: see text] satisfying certain properties, generalizing the notion of Hopf group-coalgebras as introduced by Turaev from groups to groupoids and Hopf algebra structures to weak Hopf algebra structures. Then one considers now the direct sum [Formula: see text] of these algebras. It is an algebra, without identity, except when [Formula: see text] is a finite groupoid, but the product is non-degenerate. The maps [Formula: see text] can be used to define a coproduct [Formula: see text] on [Formula: see text] and the conditions imposed on these maps give that [Formula: see text] is a weak multiplier Hopf algebra. It is [Formula: see text]-cograded as explained in this paper. We study these so-called groupoid-cograded weak multiplier Hopf algebras. They are, as explained above, more general than the weak Hopf group-coalgebras (introduced by Van Daele and Wang), generalizing the Turaev’s Hopf group-coalgebras. Moreover, our point of view makes it possible to use results and techniques from the theory of weak multiplier Hopf algebras in the study of weak Hopf groupoid-coalgebras (and generalizations).

Algebraic homology and cohomology theories for quandles have been studied extensively in recent years. With a given quandle 2(3)-cocycle one can define a state-sum invariant for knotted curves(surfaces). In this paper we introduce another version of quandle (co)homology theory, say positive quandle (co)homology. Some properties of positive quandle (co)homology groups are given and some applications of positive quandle cohomology in knot theory are discussed.