Characteristic classes for Hopf algebra actions and the Hopf-cyclic cohomology of quantum groups: the surviving core of the Connes-Moscovici program

In 'Cyclic Cohomology and Hopf Algebras' (Lett. Math. Phys. 48 (1999) 97-108), Connes and Moscovici associate to every Hopf algebra H equipped with a modular pair in involution $(\delta,\sigma)$ - a character $\delta$ and a group-like element $\sigma$ with $\delta(\sigma)=1$ such that the $\delta$-twisted antipode $\tilde{S}(h)=\delta(h_{(1)})S(h_{(2)})$ satisfies $\tilde{S}^{2}(h)=\sigma h\sigma^{-1}$ - a cocyclic module, and hence a twisted cyclic cohomology theory $HC^{*}_{(\delta,\sigma)}(\mathcal{H})$, presenting it as the key step toward a theory of characteristic classes for actions of Hopf algebras compatible not only with traces but with the modular theory of weights, applicable to ribbon and coribbon algebras as well as quantum groups and their duals. The paper computes essentially no new cohomology groups, so the announced program remained open. After twenty-five years of development, the surviving core remains open. The general characteristic-class machinery (characteristic maps, cup products, Weil complexes) was substantially built in 2000-2015, and the enveloping-algebra side of the computation problem was closed by Kaygun-Sütlü in 2016, who computed the Hopf-cyclic cohomology of $U_{q}(\mathfrak{g})$ for every semisimple Lie algebra g. What remains are: (i) the Hopf-cyclic (co)homology of the quantized function algebras $\mathcal{O}(G_{q})$ beyond $A(SL_{q}(2))$ at the full cyclic level - only Hochschild-level results are known for $GL_{q}(n)$, $SL_{q}(n)$, $M_{q}(n)$, and the orthogonal and symplectic families appear uncomputed - together with the sought general picture of these groups for polynomial functions on quantum groups; (ii) explicit evaluation of the characteristic map on K-theoretic and index-type invariants beyond the standard Podles sphere; and (iii) a structural characterization of which Hopf algebras admit modular pairs in involution - the exact domain of the whole construction, known since 2019 not to include all finite-dimensional Hopf algebras - including whether the passage through the ribbon double cover $\mathcal{H}[\theta]/(\theta^{2}-uS(u))$ needed to produce the canonical $\sigma$ in quasitriangular cases can be avoided or systematically controlled.

Background

Cyclic cohomology, introduced by Connes in the early 1980s, plays the role of de Rham cohomology for noncommutative algebras, and its pairing with K-theory yields noncommutative analogues of the Atiyah-Singer index theorem and of Chern-Weil characteristic classes. Connes and Moscovici's proof of the transverse index theorem for foliations (Commun. Math. Phys. 198 (1998) 199-246) revealed that the correct symmetry object for transverse geometry is not a Lie algebra but a Hopf algebra $\mathcal{H}(n)$ of transverse vector fields, and that the relevant cocycles live in a cyclic cohomology theory adapted to Hopf algebras. Their 1999 paper constructed this theory: to a Hopf algebra H carrying a modular pair in involution $(\delta,\sigma)$ - a character $\delta$ and a group-like element $\sigma$ with $\delta(\sigma)=1$ such that the $\delta$-twisted antipode $\tilde{S}(h)=\delta(h_{(1)})S(h_{(2)})$ satisfies $\tilde{S}^{2}(h)=\sigma h\sigma^{-1}$ for all h, a condition built in to make the cyclic operator square to the identity - they associated a cocyclic module whose cohomology is $HC^{*}_{(\delta,\sigma)}(\mathcal{H})$. They verified the axioms, quoted from the 1998 paper that for enveloping algebras the theory recovers Lie algebra (co)homology with coefficients twisted by $\delta$, and analyzed where such pairs come from: compact and locally compact quantum groups with their modular Haar functionals supply intrinsic pairs; coribbon function algebras such as $\mathcal{O}(GL_{q}(N))$, $\mathcal{O}(SL_{q}(N))$, $\mathcal{O}(SO_{q}(N))$, $\mathcal{O}(O_{q}(N))$, $\mathcal{O}(Sp_{q}(N))$ supply pairs $(\delta,1)$; and in quasitriangular (ribbon) cases the canonical group-like element is $\sigma=\theta^{-1}u$, where $u=\sum S(t^{i})s^{i}$ is built from the universal R-matrix and $\theta$ is a central square root of $uS(u)$, which generally requires passage to the double cover $\mathcal{H}[\theta]/(\theta^{2}-uS(u))$. The published abstract frames the construction as the key to developing a theory of characteristic classes for Hopf algebra actions compatible with the modular theory of weights, and to computing the new cohomology for quantum groups and their duals. Subsequent work of Hajac-Khalkhali-Rangipour-Sommerhäuser (2004) identified the correct general coefficient systems, stable anti-Yetter-Drinfeld modules, of which one-dimensional instances correspond exactly to modular pairs in involution; Hopf-cyclic theory is now the standard cohomological machine for quantum group symmetries acting on noncommutative spaces, with the Connes-Moscovici characteristic map playing the role of the Chern character homomorphism from the Hopf algebra's cohomology to the cyclic cohomology of the acted-on algebra.

Problem Statement

Within the Connes-Moscovici framework of Hopf-cyclic cohomology (Lett. Math. Phys. 48 (1999) 97-108), determine the surviving open core of the program announced there. It has three linked parts. (1) Computations: compute the periodic and non-periodic Hopf-cyclic (co)homology groups $HC^{*}_{(\delta,\sigma)}(\mathcal{H})$ with their canonical modular-pair-in-involution coefficients - equivalently, with coefficients in the associated one-dimensional stable anti-Yetter-Drinfeld modules - for the important quantum groups and their duals beyond the settled cases ($U_{q}(\mathfrak{g})$ for semisimple g, $U_{q}(\mathfrak{sl}_{2})$, $A(SL_{q}(2))$, and the Hochschild level of $M_{q}(n)$, $GL_{q}(n)$, $SL_{q}(n)$): in particular the full cyclic-level groups for the quantized function algebras $\mathcal{O}(G_{q})$ for rank at least 2 (e.g. $SL_{q}(n)$ for n ≥ 3, the orthogonal and symplectic families $O_{q}(N)$, $SO_{q}(N)$, $Sp_{q}(N)$, and multiparameter deformations), and formulate the general picture ('conjecture') governing these groups for algebras of polynomial functions on quantum groups, as explicitly requested in the 2003 Khalkhali-Rangipour survey. (2) Characteristic classes and pairings: complete the theory of characteristic classes for actions of Hopf algebras compatible with the modular theory of weights (that is, built from the $\delta$-invariant $\sigma$-trace or the general modular weight carried by the pair $(\delta,\sigma)$, not from an ordinary trace alone) at the level of explicit invariants - given an H-module algebra A carrying a $\delta$-invariant $\sigma$-trace or a general modular weight, evaluate the Connes-Moscovici characteristic map $HC^{*}_{(\delta,\sigma)}(\mathcal{H})\to HC^{*}(A)$ concretely, identifying which classes pair nontrivially with K-theory classes or represent index cocycles, for a specified quantum homogeneous space beyond the standard Podles sphere or a specified class of foliations of codimension greater than 2; a single worked instance of such a space or foliation settles this part. (3) Modular pairs in involution: characterize the Hopf algebras that admit a modular pair in involution - the exact class on which the entire cocyclic construction is defined - and clarify the quasitriangular caveat of the source paper: in ribbon cases the canonical group-like $\sigma=\theta^{-1}u$ requires the central square root $\theta$ of $uS(u)$, obtained by passing to the double cover $\mathcal{H}[\theta]/(\theta^{2}-uS(u))$; determine in which cases this passage can be avoided or systematically controlled. The three parts are independent, each a self-contained open problem.

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