We study some aspects of the theory of non-commutative differential calculi over complex algebras, especially over the Hopf algebras associated to compact quantum groups in the sense of S.L. Woronowicz. Our principal emphasis is on the theory of twisted graded traces and their associated twisted cyclic cocycles. One of our principal results is a new method of constructing differential calculi, using twisted graded traces.
For two covariant differential *-calculi, the twisted cyclic cocycle associated with the volume form is represented in terms of commutators [F,ρ(x)] for some self-adjoint operator F and some *-representation $ρ$ of the underlying *-algebra.
In this report we give an intrinsic treatment of the results we developed in a previous work connecting the differential calculi on Hopf algebras to the Drinfeld double. In the first place we recover that bicovariant bimodules are in one to one correspondence with the Drinfeld double representations; we then introduce a Hochschild cohomology of the algebra of functions and discuss the main result stating that each differential calculus is associated to a 1-cocycle satisfying an additional invariance condition with respect to a natural action. Defining a Hochschild cohomology of the double, the…
We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasi-isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof. We give a new proof of Nest-Tsygan's algebraic higher index theorem by computing the pairing between such cyclic cocycles and the $K$-theory of the formal deformation quantization. Furthermore, we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold…
A Dirac operator D on the standard Podles sphere is defined and investigated. It yields a spectral triple such that |D|^{-z} is of trace class for Re z>0. Commutators with the Dirac operator give the distinguished 2-dimensional covariant differential calculus on the standard Podles sphere. The twisted cyclic cocycle associated with the volume form of the differential calculus is expressed by means of the Dirac operator.
We show that bicovariant bimodules as defined by Woronowicz are in one to one correspondence with the Drinfeld quantum double representations. We then prove that a differential calculus associated to a bicovariant bimodule of dimension n is connected to the existence of a particular (n+1)--dimensional representation of the double. An example of bicovariant differential calculus on the non quasitriangular quantum group E_q(2) is developed. The construction is studied in terms of Hochschild cohomology and a correspondence between differential calculi and 1-cocycles is proved. Some differences of…
We continue the investigation of twisted homology theories in the context of dimension drop phenomena. This work unifies previous equivariant index calculations in twisted cyclic cohomology. We do this by proving the existence of the resolvent cocycle, a finitely summable analogue of the JLO cocycle, under weaker smoothness hypotheses and in the more general setting of `modular' spectral triples. As an application we show that using our twisted resolvent cocycle, we can obtain a local index formula for the Podles sphere. The resulting twisted cyclic cocycle has non-vanishing Hochschild class w…
We integrate the Lifting cocycles $Ψ_{2n+1},Ψ_{2n+3},Ψ_{2n+5},...$ ([Sh1], [Sh2]) on the Lie algebra $\Dif_n$ of holomorphic differential operators on an $n$-dimensional complex vector space to the cocycles on the Lie algebra of holomorphic differential operators on a holomorphic line bundle $λ$ on an $n$-dimensional complex manifold $M$ in the sense of Gelfand-Fuks cohomology [GF] (more precisely, we integrate the cocycles on the sheaves of the Lie algebras of finite matrices over the corresponding associative algebras). The main result is the following explicit form of the Feigin-Tsygan theo…
We use monoidal category methods to study the noncommutative geometry of nonassociative algebras obtained by a Drinfeld-type cochain twist. These are the so-called quasialgebras and include the octonions as braided-commutative but nonassociative coordinate rings, as well as quasialgebra versions $\CC_{q}(G)$ of the standard q-deformation quantum groups. We introduce the notion of ribbon algebras in the category, which are algebras equipped with a suitable generalised automorphism $σ$, and obtain the required generalisation of cyclic cohomology. We show that this \emph{braided cyclic cocohomolo…
We give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. For proper étale groupoids, Tu and Xu provide a map between the periodic cyclic cohomology of a gerbe-twisted convolution algebra and twisted cohomology groups which is similar to a construction of Mathai and Stevenson. When the groupoid is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial techniques to construct a simplicial curvature 3-form representing the class of the gerbe. Then by using …
We show that the crossed modules and bicovariant different calculi on two Hopf algebras related by a cocycle twist are in 1-1 correspondence. In particular, for quantum groups which are cocycle deformation-quantisations of classical groups the calculi are obtained as deformation-quantisation of the classical ones. As an application, we classify all bicovariant differential calculi on the Planck scale Hopf algebra $\C[x]\bicross_{\hbar,\grav}\C[p]$. This is a quantum group which has an $\hbar\to 0$ limit as the functions on a classical but non-Abelian group and a $\grav\to 0$ limit as flat spac…
We have previously shown that the truncated Weil algebra of any Lie algebra is a Hopf-cyclic type complex with nontrivial coefficients. In this paper we apply this result to transfer the characteristic classes of transversely orientable foliations into the cyclic cohomology of the groupoid action algebra. Our result in codimension 1 matches with the only existing explicit computation done by Connes-Moscovici. In codimension 2 case, we carry out a constructive and explicit computation, by which we present the transverse fundamental class, the Godbillon-Vey class, and the other four residual cla…
In this thesis we give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. In his seminal book, Connes constructs a map from the equivariant cohomology of a manifold carrying the action of a discrete group into the periodic cyclic cohomology of the associated convolution algebra. Furthermore, for proper étale groupoids, J.-L. Tu and P. Xu provide a map between the periodic cyclic cohomology of a gerbe twisted convolution algebra and twisted cohomology groups. Our focus will be the convolution algebra with a product defined by a gerbe…
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 present an intelligible review of recent results concerning cyclic cocycles in the spectral action and one-loop quantization. We show that the spectral action, when perturbed by a gauge potential, can be written as a series of Chern-Simons actions and Yang-Mills actions of all orders. In the odd orders, generalized Chern-Simons forms are integrated against an odd $(b,B)$-cocycle, whereas, in the even orders, powers of the curvature are integrated against $(b,B)$-cocycles that are Hochschild cocycles as well. In both cases, the Hochschild cochains are derived from the Taylor series expansion…
We show that the spectral action, when perturbed by a gauge potential, can be written as a series of Chern--Simons actions and Yang--Mills actions of all orders. In the odd orders, generalized Chern--Simons forms are integrated against an odd $(b,B)$-cocycle, whereas, in the even orders, powers of the curvature are integrated against $(b,B)$-cocycles that are Hochschild cocycles as well. In both cases, the Hochschild cochains are derived from the Taylor series expansion of the spectral action Tr$(f(D+V))$ in powers of $V=π_D(A)$, but unlike the Taylor expansion we expand in increasing order of…
We study the "quantized calculus" corresponding to the algebraic ideas related to "twisted cyclic cohomology" introduced in [KMT]. With very similar definitions and techniques as those used in [jlo], we define and study "twisted entire cyclic cohomology" and the "twisted Chern character" associated with an appropriate operator-theoretic data called "twisted spectral data", which consists of a spectral triple in the conventional sense of noncommutative geometry ([Con]) and an additional positive operator having specified properties. Furthermore, it is shown that given a spectral triple (in the …
The algebras obtained as fixed points of the action of the cyclic group $Z_N$ on the coordinate algebra of the quantum disc are studied. These can be understood as coordinate algebras of quantum or non-commutative cones. The following observations are made. First, contrary to the classical situation, the actions of $Z_N$ are free and the resulting algebras are homologically smooth. Second, the quantum cone algebras admit differential calculi that have all the characteristics of calculi on smooth complex curves. Third, the corresponding volume forms are exact, indicating that the constructed al…
We present an analysis of the cocycle appearing in the vertex operator representation of simply-laced, affine, Kac-Moody algebras. We prove that it can be described in the context of $R$-commutative geometry, where $R$ is a Yang-Baxter operator, as a strong $R$-commutative algebra. We comment on the Hochschild, cyclic and dihedral homology theories that appear in non-commutative geometry and their potential relation to string theory.
We consider the 2-cocycle deformation of algebras graded by discrete groups. The action of the Maurer-Cartan form on cyclic cohomology is shown to be cohomologous to the cup product action of the group cocycle. This allows us to compute the monodromy of the Gauss-Manin connection in the strict deformation setting.
We introduce and begin to analyse a class of algebras, associated to congruence subgroups, that extend both the algebra of modular forms of all levels and the ring of classical Hecke operators. At the intuitive level, these are algebras of `polynomial coordinates' for the `transverse space' of lattices modulo the action of the Hecke correspondences. Their underlying symmetry is shown to be encoded by the same Hopf algebra that controls the transverse geometry of codimension 1 foliations. Its action is shown to span the `holomorphic tangent space' of the noncommutative space, and each of its th…
Connes and Cuntz showed in [Comm. Math. Phys. 114 (1988), 515-526] that suitable cyclic cocycles can be represented as Chern characters of finitely summable semifinite Fredholm modules. We show an analogous result in twisted cyclic cohomology using Chern characters of modular Fredholm modules. We present examples of modular Fredholm modules arising from Podleś spheres and from $SU_q(2)$.
The purpose of the paper is twofold: First, known results of the noncommutative spin geometry of the standard Podles sphere are extended by discussing Poincare duality and orientability. In the discussion of orientability, Hochschild homology is replaced by a twisted version which avoids the dimension drop. The twisted Hochschild cycle representing an orientation is related to the volume form of the distinguished covariant differential calculus. Integration over the volume form defines a twisted cyclic 2-cocycle which computes the q-winding numbers of quantum line bundles. Second, a "twisted" …
The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to group extension theories for group cocycles. Explicit formulas for cocycles corresponding to extensions are given, and used to prove non-triviality of cohomology groups for some quandles. The corresponding generalization of the quandle cocycle knot invariants is given, by using the…
We give a description of cyclic cohomology and its pairing with K-groups for 2-cocycle deformation of algebras graded over discrete groups. The proof relies on a realization of monodromy for the Gauss-Manin connection on periodic cyclic cohomology in terms of the cup product action of group cohomology.
We present the main ingredients of twistor theory leading up to and including the Penrose-Ward transform in a coordinate algebra form which we can then `quantise' by means of a functorial cocycle twist. The quantum algebras for the conformal group, twistor space CP^3, compactified Minkowski space CMh and the twistor correspondence space are obtained along with their canonical quantum differential calculi, both in a local form and in a global *-algebra formulation which even in the classical commutative case provides a useful alternative to the formulation in terms of projective varieties. We o…
We consider Hopf crossed products of the the type $A#_σ\mathcal{H}$, where $\mathcal{H}$ is a cocommutative Hopf algebra, $A$ is an $\mathcal{H}$-module algebra and $σ$ is a "numerical" convolution invertible 2-cocycle on $\mathcal{H}$. we give an spectral sequence that converges to the cyclic homology of $A#_σ\mathcal{H}$ and identify the $E^1$ and $E^2$ terms of the spectral sequence.
We show that if $G$ is a compact Lie group and $\mathfrak{g}$ is its Lie algebra, then there is a map from the Hopf-cyclic cohomology of the quantum enveloping algebra $U_q(\mathfrak{g})$ to the twisted cyclic cohomology of quantum group algebra $\mathcal{O}(G_q)$. We also show that the Schmüdgen-Wagner index cocycle associated with the volume form of the differential calculus on the standard Podleś sphere $\mathcal{O}(S^2_q)$ is in the image of this map.
We construct a Hopf action, with an invariant trace, of a bicrossed product Hopf algebra $\cH=\big( \cU(\Fg_1) \acr \cR(G_2) \big)^{\cop}$ constructed from a matched pair of Lie groups $G_1$ and $G_2$, on a convolution algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^δ$. We give an explicit way to construct Hopf cyclic cohomology classes of our Hopf algebra $\cH$ and then realize these classes in terms of explicit representative cocycles in the cyclic cohomology of the convolution algebra $\cA$.
We calculate the twisted Hochschild and cyclic homology (in the sense of Kustermans, Murphy and Tuset) of all Podles quantum spheres relative to arbitary automorphisms. Our calculations are based on a free resolution due to Masuda, Nakagami and Watanabe. The dimension drop in Hochschild homology can be overcome by twisting by automorphisms induced from the canonical modular automorphism associated to the Haar state on quantum SU(2). We specialize our results to the standard quantum sphere, and identify the class in twisted cyclic cohomology of the 2-cocycle discovered by Schmuedgen and Wagner …
The cup and cap product in twisted Hochschild (co)homology is computed for the standard quantum 2-sphere and used to construct a cyclic 2-cocycle that represents the fundamental Hochschild class.
We show that much of the structure of the 2-sphere as a complex curve survives the q-deformation and has natural generalizations to the quantum 2-sphere - which, with additional structures, we identify with the quantum projective line. Notably among these is the identification of a quantum homogeneous coordinate ring with the coordinate ring of the quantum plane. In parallel with the fact that positive Hochschild cocycles on the algebra of smooth functions on a compact oriented 2-dimensional manifold encode the information for complex structures on the surface, we formulate a notion of twisted…
Let $H$ be a Hopf algebra and let $\mathcal D_H$ be a Hopf-module category. We describe the cocycles and coboundaries for the Hopf cyclic cohomology of $\mathcal D_H$, which correspond respectively to categorified cycles and vanishing cycles over $\mathcal D_H$. An important role in our work is played by semicategories, which are categories that may not contain identity maps. In particular, a cycle over $\mathcal D_H$ consists of a differential graded $H$-module semicategory equipped with a trace on endomorphism groups satisfying some conditions. Using a pairing on cycles, we obtain a pairing …
We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan-Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the image of the canonical homomorphism from the Gelfand-Fuks complex to the Bott complex for equivariant cohomology. This provides a convenient new model for the Hopf cyclic cohomology of the geometric Hopf algebras, which allows for an efficient transport of the Hopf cyclic classes v…
We complete the calculation of the twisted cyclic homology of the quantised coordinate ring of SL(2) that we began in math.QA/0405249. In particular, a nontrivial cyclic 3-cocycle is constructed which also has a nontrivial class in Hochschild cohomology and thus should be viewed as a noncommutative geometry analogue of a volume form.
We re-examine all the contractions related with the ${\cal U}_q(su(2))$ deformed algebra and study the consequences that the contraction process has for their structure. We also show using ${\cal U}_q(su(2))\times{\cal U}(u(1))$ as an example that, as in the undeformed case, the contraction may generate Hopf algebra cohomology. We shall show that most of the different Hopf algebra deformations obtained have a bicrossproduct or a cocycle bicrossproduct structure, for which we shall also give their dual `group' versions. The bicovariant differential calculi on the deformed spaces associated with…
We calculate the twisted Hochschild and cyclic homology (in the sense of Kustermans, Murphy and Tuset) of the coordinate algebra of the quantum SL(2) group relative to twisting automorphisms acting by rescaling the standard generators a,b,c,d. We discover a family of automorphisms for which the "twisted" Hochschild dimension coincides with the classical dimension of SL(2, C), thus avoiding the "dimension drop" in Hochschild homology seen for many quantum deformations. Strikingly, the simplest such automorphism is the canonical modular automorphism arising from the Haar functional. In addition,…
We construct a variant $\mathcal{K}_n$ of the Hopf algebra $\mathcal{H}_n$, which acts directly on the noncommutative model for the generic space of leaves rather than on its frame bundle. We prove that the Hopf cyclic cohomology of $\mathcal{K}_n$ is isomorphic to that of the pair $(\mathcal{H}_n, {\mathop{\rm GL}_n})$ and thus consists of the universal Hopf cyclic classes. We then realize these classes in terms of geometric cocycles.
In this paper we use the description of free group factors as the von Neumann algebras of Berezin's deformation of the upper half-plane, modulo PSL$(2,{\Bbb Z})$. The derivative, in the deformation parameter, of the product in the corresponding algebras, is a positive Hochschild 2-cocycle, defined on a dense subalgebra. By analyzing the structure of the cocycle we prove that there is a generator $\cal L$ for a quantum dynamical semigroup that implements the cocycle on a strongly dense subalgebra. For $x$ in the dense subalgebra, ${\cal L}(x)$ is the (diffusion) operator $$ {\cal L}(x)=Λ(x)-(1/…
Let $k$ be a field, $k^*=k\setminus\{0\}$ and $C_2$ the cyclic group of order 2. In this note we compute all the braided monoidal structures on the category of $k$-vector spaces graded by the Klein group $C_2\times C_2$. Actually, for the monoidal structures we will compute the explicit form of the 3-cocycles on $C_2\times C_2$ with coefficients in $k^*$, while for the braided monoidal structures we will compute the explicit form of the abelian 3-cocycles on $C_2\times C_2$ with coefficients in $k^*$. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided…
We analyse the symmetries underlying nonassociative deformations of geometry in non-geometric R-flux compactifications which arise via T-duality from closed strings with constant geometric fluxes. Starting from the non-abelian Lie algebra of translations and Bopp shifts in phase space, together with a suitable cochain twist, we construct the quasi-Hopf algebra of symmetries that deforms the algebra of functions and the exterior differential calculus in the phase space description of nonassociative R-space. In this setting nonassociativity is characterised by the associator 3-cocycle which cont…
On the configuration space of projections in a noncommutative algebra, and for an automorphism of the algebra, we use a twisted Hochschild cocycle for an action functional, and a twisted cyclic cocycle for a topological term. The latter is Hochschild-cohomologous to the former and positivity in twisted Hochschild cohomology results into a lower bound for the action functional. While the equations for the critical points are rather complicate, the use of the positivity and the bound by the topological term leads to self-duality equations (thus yielding twisted noncommutative sigma-model soliton…
We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a `quantum group' counterpart of the infinite-dimensional primitive Lie algebra of the pseudogroup. It is first constructed via its action on the étale groupoid associated to the pseudogroup, and then realized as a bicrossed product of a universal enveloping algebra by a Hopf algebra of regular functions on a formal group. The bicrossed product structure …
The Kontsevich-Soibelman solution of the cyclic version of Deligne's conjecture and the formality of the operad of little discs on a cylinder provide us with a natural homotopy calculus structure on the pair (C^*(A), C_*(A)) ``Hochschild cochains + Hochschild chains'' of an associative algebra A. We show that for an arbitrary smooth algebraic variety X with the structure sheaf O_X the sheaf (C^*(O_X), C_*(O_X)) of homotopy calculi is formal. This result was announced in paper [29] by the second and the third author.
In this article, we show under what additional ingredients a comp (or opposite) module over an operad with multiplication can be given the structure of a cyclic k-module and how the underlying simplicial homology gives rise to a Batalin-Vilkovisky module over the cohomology of the operad. In particular, one obtains a generalised Lie derivative and a generalised (cyclic) cap product that obey a Cartan-Rinehart homotopy formula, and hence yield the structure of a noncommutative differential calculus in the sense of Nest, Tamarkin, Tsygan, and others. Examples include the calculi known for the Ho…
We analyse the noncommutative space underlying the quantum group SUq(2) from the spectral point of view which is the basis of noncommutative geometry, and show how the general theory developped in our joint work with H. Moscovici applies to the specific spectral triple defined by Chakraborty and Pal. This provides the pseudo-differential calculus, the Wodzciki-type residue, and the local cyclic cocycle giving the index formula. The cochain whose coboundary is the difference between the original Chern character and the local one is given by the remainders in the rational approximation of the lo…
This is a contribution to the project of quiver approaches to quasi-quantum groups initiated in arXiv:0902.1620. We classify Majid bimodules over groups with 3-cocycles by virtue of projective representations. This leads to a theoretic classification of graded pointed Majid algebras over path coalgebras, or equivalently cofree pointed coalgebras, and helps to provide a projective representation-theoretic description of the gauge equivalence of graded pointed Majid algebras. We apply this machinery to construct some concrete examples and obtain a classification of finite-dimensional graded poin…
It is known that irreducible noncommutative differential structures over $\Bbb F_p[x]$ are classified by irreducible monics $m$. We show that the cohomology $H_{\rm dR}^0(\Bbb F_p[x]; m)=\Bbb F_p[g_d]$ if and only if ${\rm Tr}(m)\ne 0$, where $g_d=x^{p^d}-x$ and $d$ is the degree of $m$. This implies that there are ${p-1\over pd}\sum_{k|d, p\nmid k}μ_M(k)p^{d\over k}$ such noncommutative differential structures ($μ_M$ the Möbius function). Motivated by killing this zero'th cohomology, we consider the directed system of finite-dimensional Hopf algebras $A_d=\Bbb F_p[x]/(g_d)$ as well as their i…
We report on the following highlights from among the many discoveries made in Noncommutative Geometry since year 2000: 1) The interplay of the geometry with the modular theory for noncommutative tori, 2) Advances on the Baum-Connes conjecture, on coarse geometry and on higher index theory, 3) The geometrization of the pseudo-differential calculi using smooth groupoids, 4) The development of Hopf cyclic cohomology, 5) The increasing role of topological cyclic homology in number theory, and of the lambda operations in archimedean cohomology, 6) The understanding of the renormalization group as a…
We introduce the notion of a {\vartheta}-summable Fredholm module over a locally convex dg algebra Ω and construct its Chern character as a cocycle on the entire cyclic complex of Ω, extending the construction of Jaffe, Lesniewski and Osterwalder to a differential graded setting. Using this Chern character, we prove an index theorem involving an abstract version of a Bismut-Chern character constructed by Getzler, Jones and Petrack in the context of loop spaces. Our theory leads to a rigorous construction of the path integral for N=1/2 supersymmetry which satisfies a Duistermaat-Heckman type lo…
Given a Hopf algebra $A$ graded by a discrete group together with an action of the same group preserving the grading, we define a new Hopf algebra, which we call the graded twisting of $A$. If the action is adjoint, this new Hopf algebra is a twist of $A$ by a pseudo-$2$-cocycle. Analogous construction can be carried out for monoidal categories. As examples we consider graded twistings of the Hopf algebras of nondegenerate bilinear forms, their free products, hyperoctahedral quantum groups and $q$-deformations of compact semisimple Lie groups. As applications, we show that the analogues of the…
