Connes-Moscovici characteristic map is a Lie algebra morphism
Luc Menichi
备注：
submitted version. Corollary 28 and Section 9 has been added. Section 9 computes the Batalin-Vilkovisky algebra on the rational cotor of an universal envelopping algebra of a lie algebra
Let $H$ be a Hopf algebra with a modular pair in involution $(\Character,1)$. Let $A$ be a (module) algebra over $H$ equipped with a non-degenerated $\Character$-invariant $1$-trace $τ$. We show that Connes-Moscovici characteristic map $φ_τ:HC^*_{(\Character,1)}(H)\rightarrow HC^*_λ(A)$ is a morphism of graded Lie algebras. We also have a morphism $Φ$ of Batalin-Vilkovisky algebras from the cotorsion product of $H$, $\text{Cotor}_H^*({\Bbbk},{\Bbbk})$, to the Hochschild cohomology of $A$, $HH^*(A,A)$. Let $K$ be both a Hopf algebra and a symmetric Frobenius algebra. Suppose that the square of its antipode is an inner automorphism by a group-like element. Then this morphism of Batalin-Vilkovisky algebras $Φ:\text{Cotor}_{K^\vee}^*(\mathbb{F},\mathbb{F})\cong \text{Ext}_{K}(\mathbb{F},\mathbb{F}) \hookrightarrow HH^*(K,K)$ is injective.
We prove that a certain bialgebroid introduced recently by Kadison is isomorphic to a bialgebroid introduced earlier by Connes and Moscovici. At the level of total algebras, the isomorphism is a consequence of the general fact that an L-R-smash product over a Hopf algebra is isomorphic to a diagonal crossed product.
We construct a universal deformation formula for Connes-Moscovici's Hopf algebra without any projective structure using Fedosov's quantization of symplectic diffeomorphisms.
We compute the second Hochschild cohomology space $HH^2(\mathcal{H}_1)$ of Connes-Moscovici's Hopf algebra $\mathcal{H}_1$, giving the infinitesimal deformations (up to equivalence) of the associative structure. $HH^2(\mathcal{H}_1)$ is shown to be one dimensional, and thus Connes-Moscovici's formal deformation of $\mathcal{H}_1$ using Rankin-Cohen brackets is unique up to equivalence.
In $1998$, Connes and Moscovici defined the cyclic cohomology of Hopf algebras. In $2010$, Khalkhali and Pourkia proposed a braided generalization: to any Hopf algebra $H$ in a braided category $\mathcal B$, they associate a paracocyclic object in $\mathcal B$. In this paper we explicitly compute the powers of the paracocyclic operator of this paracocyclic object. Also, we introduce twisted modular pairs in involution for $H$ and derive (co)cyclic modules from them. Finally, we relate the paracocyclic object associated with $H$ to that associated with an $H$-module coalgebra via a categorical …
We define a super version of the Connes-Moscovici Hopf algebra, $\mathcal{H}_1$. For that, we consider the supergroup $G^s=Diff^+(\mathbb{R}^{1,1})$ of orientation preserving diffeomorphisms of the superline $\mathbb{R}^{1,1}$ and define two (super) subgroups $G^s_1$ and $G^s_2$ of $G^s$ where $G^s_1$ is the supergroup of affine transformations. The super Hopf algebra $\mathcal{H} ^s_1$ is defined as a certain bicrossproduct super Hopf algebra of the super Hopf algebras attached to $G^s_1$ and $G^s_2$. We also give an explicit description of $\mathcal{H} ^s_1$ in terms of generators and relati…
We show that the (even and odd) index cocycles for theta-summable Fredholm modules are in the image of the Connes-Moscovici characteristic map. To show this, we first define a new range of asymptotic cohomologies, and then we extend the Connes-Moscovici characteristic map to our setting. The ordinary periodic cyclic cohomology and the entire cyclic cohomology appear as two instances of this setup. We then construct an asymptotic characteristic class, defined independently from the underlying Fredholm module. Paired with the $K$-theory, the image of this class under the characteristic map yield…
As the second part of the sequel, we investigate the variation of rearrangement operators (more precisely, the spectral functions behind) arising in the study of modular geometry on noncommutative (two) tori. We initiate a systematic approach by introducing transformations corresponding to basic operations in calculus, like differentiation and integration by parts. As for applications, we extend, in a uniform way, the Connes-Moscovici's functional relations on noncommutative two tori attached to the variation of second heat coefficients to noncommutative tori of arbitrary dimension. Moreover, …
We give a rigorous proof that the (codimension one) Connes-Moscovici Hopf algebra H_CM is isomorphic to a bicrossproduct Hopf algebra linked to a group factorisation of the group of positively-oriented diffeomorphisms of the real line. We construct a second bicrossproduct U_CM equipped with a nondegenerate dual pairing with H_CM. We give a natural quotient Hopf algebra of H_CM and Hopf subalgebra of U_CM which again are in duality. All these Hopf algebras arise as deformations of commutative or cocommutative Hopf algebras that we describe in each case. Finally we develop the noncommutative dif…
In this paper, we use the theory of deformation quantization to understand Connes' and Moscovici's results \cite{cm:deformation}. We use Fedosov's method of deformation quantization of symplectic manifolds to reconstruct Zagier's deformation \cite{z:deformation} of modular forms, and relate this deformation to the Weyl-Moyal product. We also show that the projective structure introduced by Connes and Moscovici is equivalent to the existence of certain geometric data in the case of foliation groupoids. Using the methods developed by the second author \cite{t1:def-gpd}, we reconstruct a universa…
The aim of this short note is to communicate an example of a finite-dimensional Hopf algebra that does not admit a modular pair in involution in the sense of Connes and Moscovici.
We show that the Connes-Moscovici cyclic cohomology of a Hopf algebra equipped with a character has a Lie bracket of degree -2. More generally, we show that a "cyclic operad with multiplication" is a cocyclic module whose cohomology is a Batalin-Vilkovisky algebra and whose cyclic cohomology is a graded Lie algebra of degree -2. This explain why the Hochschild cohomology algebra of a symmetric algebra is a Batalin-Vilkovisky algebra.
We give a construction of cyclic cocycles representing the equivariant characteristic classes of equivariant bundles. Our formulas generalize Connes' Godbillon-Vey cyclic cocycle. An essential tool of our construction is Connes-Moscovici's theory of cyclic cohomology of Hopf algebras.
We review recent progress in the study of cyclic cohomology of Hopf algebras, Hopf algebroids, and invariant cyclic homology starting with the pioneering work of Connes-Moscovici.
We construct cup products of two different kinds for Hopf-cyclic cohomology. When the Hopf algebra reduces to the ground field our first cup product reduces to Connes' cup product in ordinary cyclic cohomology. The second cup product generalizes Connes-Moscovici's characteristic map for actions of Hopf algebras on algebras.
We discuss the local index formula of Connes-Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum SU(2). We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula.
In this paper a general van Est type isomorphism is established. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and SAYD modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is found at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscov…
In this paper we propose still another approach to the Hopf-type cyclic homology of Hopf algebras, introduced by A. Connes and H. Moscovici. Our construction is based on the notion of "universal differential calculus" on an algebra. Few remarks, concerning the possible generalizations and applications of this approach are made. (This is only a preliminary version of the paper. Only outlines of some ideas are given.)
A cyclic cohomology theory adapted to Hopf algebras has been introduced recently by Connes and Moscovici. In this paper, we consider this object in the homological framework, in the spirit of Loday-Quillen and Karoubi's work on the cyclic homology of associative algebras. In the case of group algebras, we interpret the decomposition of the classical cyclic homology of a group algebra in terms of this homology. We also compute both cyclic homologies for truncated quiver algebras.
A pseudo-Galois extension is shown to be a depth two extension. Studying its left bialgebroid, we construct an enveloping Hopf algebroid for the semi-direct product of groups, or more generally involutive Hopf algebras, and their module algebras. It is a type of cofibered sum of two inclusions of the Hopf algebra into the semi-direct product and its derived right crossed product. Van Oystaeyen and Panaite observe that this Hopf algebroid is non-trivially isomorphic to a Connes-Moscovici Hopf algebroid, which raises interesting comparative questions.
The renormalization of quantum field theory twists the antipode of a noncocommutative Hopf algebra of rooted trees, decorated by an infinite set of primitive divergences. The Hopf algebra of undecorated rooted trees, ${\cal H}_R$, generated by a single primitive divergence, solves a universal problem in Hochschild cohomology. It has two nontrivial closed Hopf subalgebras: the cocommutative subalgebra ${\cal H}_{\rm ladder}$ of pure ladder diagrams and the Connes-Moscovici noncocommutative subalgebra ${\cal H}_{\rm CM}$ of noncommutative geometry. These three Hopf algebras admit a bigrading by …
Given a Hopf algebra H and an algebra A that is an H-module algebra we consider the category of left H-modules and A-bimodules, where morphisms are just right A-linear maps (not necessarily H-equivariant). Given a twist F of H we then quantize (deform) H to H^F, A to A_\star and correspondingly the category of left H-modules and A-bimodules to the category of left H^F-modules and A_\star-bimodules. If we consider a quasitriangular Hopf algebra H, a quasi-commutative algebra A and quasi-commutative A-bimodules, we can further construct and study tensor products over A of modules and of morphism…
We give an introductory survey to the use of Hopf algebras in several problems of noncommutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We show its relation to the algebra of transverse differential operators introduced by Connes and Moscovici in order to compute a local index formula in cyclic cohomology, and to the several Hopf algebras defined by Connes and Kreimer to simplify the combinatorics of perturbative renormalization. We explain how characteristic classes for a Hopf module algebra can …
We extend the formalism of Hopf cyclic cohomology to the context of braided categories. For a Hopf algebra in a braided monoidal abelian category we introduce the notion of stable anti-Yetter-Drinfeld module. We associate a para-cocyclic and a cocyclic object to a braided Hopf algebra endowed with a braided modular pair in involution in the sense of Connes and Moscovici. When the braiding is symmetric the full formalism of Hopf cyclic cohomology with coefficients can be extended to our categorical setting.
The local index formula of Connes--Moscovici for the isospectral noncommutative geometry recently constructed on quantum SU(2) is discussed. The cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula, are presented.
This note discusses the cyclic cohomology of a left Hopf algebroid ($\times_A$-Hopf algebra) with coefficients in a right module-left comodule, defined using a straightforward generalisation of the original operators given by Connes and Moscovici for Hopf algebras. Lie-Rinehart homology is a special case of this theory. A generalisation of cyclic duality that makes sense for arbitrary para-cyclic objects yields a dual homology theory. The twisted cyclic homology of an associative algebra provides an example of this dual theory that uses coefficients that are not necessarily stable anti Yetter-…
We show how to integrate a weak morphism of Lie algebra crossed-modules to a weak morphism of Lie 2-groups. To do so we develop a theory of butterflies for 2-term L_infty algebras. In particular, we obtain a new description of the bicategory of 2-term L_infty algebras. We use butterflies to give a functorial construction of connected covers of Lie 2-groups. We also discuss the notion of homotopy fiber of a morphism of 2-term L_infty algebras.
We define a noncommutative analogue of invariant de Rham cohomology. More precisely, for a triple $(A,\mathcal{H},M)$ consisting of a Hopf algebra $\mathcal{H}$, an $\mathcal{H}$-comodule algebra $A$, an $\mathcal{H}$-module $M$, and a compatible grouplike element $σ$ in $\mathcal{H}$, we define the cyclic module of invariant chains on $A$ with coefficients in $M$ and call its cyclic homology the invariant cyclic homology of $A$ with coefficients in $M$. We also develop a dual theory for coalgebras. Examples include cyclic cohomology of Hopf algebras defined by Connes-Moscovici and its dual th…
REVISED VERSION: We have re-organized the paper, and included some new results. Most important, we prove that the (truncated) Weil complexes compute the cyclic cohomology of the Hopf algebra (see the new Theorem 7.3). We also include a short discussion on the uni-modulare case, and the computation for $H= U_q(sl_2)$. THE OLD ABSTRACT: We give a construction of Connes-Moscovici's cyclic cohomology for any Hopf algebra equipped with a twisted antipode. Furthermore, we introduce a non-commutative Weil complex, which connects the work of Gelfand and Smirnov with cyclic cohomology. We show how the …
"Thick" or "microformal" morphisms of supermanifolds generalize ordinary maps. They were discovered as a tool for homotopy algebras. Namely, the corresponding pullbacks provide $L_{\infty}$-morphisms for $S_{\infty}$ or Batalin--Vilkovisky algebras. It was clear from the start that constructions used for thick morphisms closely resemble some fundamental notions in quantum mechanics and their classical limits (such as action, Schrödinger and Hamilton--Jacobi equations, etc.) There was also a natural question about any connection of thick morphisms with spinor representation. We answer both ques…
