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Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik–Zamolodchikov equations and Letzter–Kolb coideals

Published online by Cambridge University Press:  02 May 2023

Kenny De Commer
Affiliation:
Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium; E-mail: kenny.de.commer@vub.be
Sergey Neshveyev
Affiliation:
Universitetet i Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway; E-mail: sergeyn@math.uio.no
Lars Tuset
Affiliation:
OsloMet - storbyuniversitetet, P.O Box 4, St. Olavs plass, 0130 Oslo, Norway; E-mail: larst@oslomet.no
Makoto Yamashita*
Affiliation:
Universitetet i Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway;

Abstract

We establish an equivalence between two approaches to quantization of irreducible symmetric spaces of compact type within the framework of quasi-coactions, one based on the Enriquez–Etingof cyclotomic Knizhnik–Zamolodchikov (KZ) equations and the other on the Letzter–Kolb coideals. This equivalence can be upgraded to that of ribbon braided quasi-coactions, and then the associated reflection operators (K-matrices) become a tangible invariant of the quantization. As an application we obtain a Kohno–Drinfeld type theorem on type $\mathrm {B}$ braid group representations defined by the monodromy of KZ-equations and by the Balagović–Kolb universal K-matrices. The cases of Hermitian and non-Hermitian symmetric spaces are significantly different. In particular, in the latter case a quasi-coaction is essentially unique, while in the former we show that there is a one-parameter family of mutually nonequivalent quasi-coactions.

Type
Analysis
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

Introduction

This paper is about quantization of symmetric spaces of compact type. It will be sufficient to concentrate on the irreducible simply connected symmetric spaces of type I, that is, the spaces of the form $U/U^{\sigma }$ for a compact simply connected simple Lie group U with an involutive automorphism $\sigma $ . Our approach is motivated by the groundbreaking work of Drinfeld [Reference DrinfeldDri89b], in which he gave a new algebraic proof of Kohno’s theorem [Reference KohnoKoh87] on equivalence of the braid group representations that appear as deformations of representations of the symmetric group on tensor powers of some representation of $\mathfrak {g}=\mathfrak {u}^{\mathbb {C}}$ . The representations in question are defined by the monodromy of the Knizhnik–Zamolodchikov (KZ) equations, on the one hand, and by the universal R-matrix of the Hopf algebraic deformation $U_h(\mathfrak {g})$ of the universal enveloping algebra $U(\mathfrak {g})$ on the other.

Drinfeld developed a framework of quasi-triangular quasi-bialgebras, which captures both types of representations. He showed that a deformation of $U(\mathfrak {g})$ among such quasi-bialgebras is controlled by the co-Hochschild cohomology of the coalgebra $U(\mathfrak {g})$ , up to a natural notion of equivalence derived from tensor categorical considerations. This cohomology is the exterior algebra $\bigwedge \mathfrak {g}$ , and the part giving the deformation parameter is the one-dimensional space $(\bigwedge ^3 \mathfrak {g})^{\mathfrak {g}}$ . Moreover, this parameter is detected by the eigenvalues of the square of the braiding.

In the course of developing the theory, Drinfeld also clarified the geometric structures behind such deformations. Namely, the first order terms of the deformations correspond to Poisson–Lie group structures on U, or structures of a Lie bialgebra on $\mathfrak {u}$ . The two types of representations of the braid groups arise from different models of quantizations of Poisson–Lie groups, and Drinfeld’s result says that such quantizations are essentially unique. In hindsight, his result can be interpreted as an instance of the formality principle, which roughly says that deformations of algebraic structures are controlled by first order terms through a quasi-isomorphism of differential graded Lie algebras.

Having understood quantizations of Poisson–Lie groups, one natural next direction is to look at quantizations of the Poisson homogeneous spaces. The first important step towards a classification of such spaces was again made by Drinfeld [Reference DrinfeldDri93]: For the standard Poisson–Lie group structure on U, they correspond to the real Lagrangian subalgebras of $\mathfrak {g}$ . A complete classification of these spaces (with connected stabilizers) was then given by Karolinsky [Reference KarolinskiĭKar96].

The first classification result for quantizations of Poisson homogeneous spaces was obtained by Podleś [Reference PodleśPod87]. He classified the actions of Woronowicz’s compact quantum group $\mathrm {SU}_q(2)$ [Reference WoronowiczWor87] with the same spectral pattern as that of $\mathrm {SU}(2)$ acting on (the functions on) the $2$ -sphere $S^2$ . In other words, he considered coactions of the C $^*$ -bialgebra $C(\mathrm {SU}_q(2))$ , which is a deformation of the algebra of continuous functions on $\mathrm {SU}(2)$ and is dual to (an analytic version of) $U_h(\mathfrak {sl}_2)$ . Podleś showed that there is a one-parameter family of isomorphism classes of such coactions. From the geometric point of view, this is explained by the fact that the covariant Poisson structures on $S^2$ form a Poisson pencil [Reference Sheu, Lu and WeinsteinShe91].

Tensor categorical counterparts of Hopf algebra coactions are module categories. Although the precise correspondence, through a Tannaka–Krein type duality, came later [Reference OstrikOst03Reference De Commer and YamashitaDCY13Reference NeshveyevNes14], in the context of quantization of Poisson homogeneous spaces there is already a rich accumulation of results obtained from various angles, all related to the reflection equation.

This equation was introduced by Cherednik [Reference CherednikChe84] to study quantum integrable systems on the half-line. While braiding (Yang–Baxter operator) represents scattering of two particles colliding in a one-dimensional system, a solution of the reflection equation (reflection operator) represents the interaction of a particle with a boundary. Adding this operator to a braided tensor category (where the Yang–Baxter operators live) gives rise to a new category with a larger space of morphisms, which admits a monoidal product of the braided tensor category from one side, thus yielding a module category [Reference tom Dieck and Häring-OldenburgtDHO98], or more precisely, a braided module category [Reference BrochierBro13].

Matrix solutions of the reflection equation for the universal R-matrix of quasi-triangular Hopf algebras lead to coideal subalgebras, as originally pointed out by Noumi [Reference NoumiNou96] and further clarified by Kolb–Stokman [Reference Kolb and StokmanKS09]. In this direction, the best understood class is that of quantum symmetric pairs, that is, the coideals which are deformations of $U(\mathfrak {g}^{\theta })$ for a conjugate $\theta $ of $\sigma $ such that $\mathfrak {g}^{\theta }$ is maximally noncompact relative to the Cartan subalgebra defining the deformation $U_h(\mathfrak {g})$ . Following Koornwinder’s work [Reference KoornwinderKoo93] on the dual coideals of the Podleś spheres, Letzter [Reference LetzterLet99] developed a systematic way of constructing such coideal subalgebras $U_h^{\mathbf {t}}(\mathfrak {g}^{\theta }) < U_h(\mathfrak {g})$ for finite type Lie algebras, which was refined and extended by Kolb to Kac–Moody Lie algebras [Reference KolbKol14]. Next, a universal K-matrix for $U_h^{\mathbf {t}}(\mathfrak {g}^{\theta })$ , which gives reflection operators in the representations of $U_h^{\mathbf {t}}(\mathfrak {g}^{\theta })$ , was defined by Kolb and Balagović [Reference KolbKol08Reference Balagović and KolbBK19] expanding on the earlier work of Bao and Wang [Reference Bao and WangBW18] on the (quasi-split) type $\mathrm {AIII}$ and $\mathrm {AIV}$ cases. The construction relied on a coideal analogue of Lusztig’s bar involution [Reference Bao and WangBW18Reference Balagović and KolbBK15]. Kolb [Reference KolbKol20] further showed, developing on the ideas from [Reference tom Dieck and Häring-OldenburgtDHO98Reference BrochierBro13], that these structures give rise to ribbon twist-braided module categories.

On the dual side, a deformation quantization of $U/U^{\sigma }$ from the reflection equation was developed by Gurevich, Donin, Mudrov and others [Reference Gurevich and SaponovGS99Reference Donin, Gurevich and ShniderDGS99Reference Donin and MudrovDM03bReference Donin and MudrovDM03a]. Here, one sees a close connection to the theory of dynamical r-matrices [Reference FelderFel95Reference Etingof and VarchenkoEV98].

There is a parallel theory of module categories over the Drinfeld category, that is, the tensor category of finite-dimensional $\mathfrak {g}$ -modules with the associator defined by the monodromy of the KZ-equations. The basic idea is to add an extra pole in these equations, then the reflection operator appears as a suitably normalized monodromy around it. Conceptually, the usual KZ-equations give flat connections on the configuration space of points in the complement of type $\mathrm A$ hyperplane configurations, and the modified equations are obtained by looking at the complement of type $\mathrm B$ hyperplane configurations. Following early works of Leibman [Reference LeibmanLei94] and Golubeva–Leksin [Reference Golubeva and LeksinGL00] on monodromy of such equations, Enriquez [Reference EnriquezEnr07] introduced cyclotomic KZ-equations. He also defined quasi-reflection algebras, a particular class of quasi-coactions of quasi-bialgebras, which can be considered as type $\mathrm B$ analogues of quasi-triangular quasi-bialgebras. This formalism turned out to have powerful applications to quantization of Poisson homogeneous spaces, where the associator of a quasi-coaction gives rise to a quantization of a dynamical r-matrix [Reference Enriquez and EtingofEE05].

Based on these developments and guided by the categorical duality between module categories and Hopf algebraic coactions, we proposed a conjecture on equivalence between the following structures [Reference De Commer, Neshveyev, Tuset and YamashitaDCNTY19]:

  • a category of finite-dimensional representations of $\mathfrak {g}^{\sigma }$ , considered as a ribbon twist-braided module category over the Drinfeld category, with the associator and ribbon twist-braid defined by the cyclotomic KZ-equations;

  • a category of finite-dimensional modules over a Letzter–Kolb coideal $U_h^{\mathbf {t}}(\mathfrak {g}^{\theta })$ , considered as a ribbon twist-braided module category over the category of $U_h(\mathfrak {g})$ -modules, with the ribbon twist-braid defined by the Balagović–Kolb universal K-matrix.

To be precise, the conjecture was formulated in the analytic setting, that is, $q=e^h$ was assumed to be a real number and the categories carried unitary structures. In this paper, we give a proof of the corresponding conjecture in the formal setting using the framework of quasi-coactions.

It should be mentioned that Brochier [Reference AdrienBro12] has already proved an interesting equivalence between two quasi-coactions on $U_h(\mathfrak {h})$ , where $\mathfrak {h}<\mathfrak {g}$ is the Cartan subalgebra and one of the quasi-coactions comes from the cyclotomic KZ-equations associated with a finite order automorphism $\sigma $ such that $\mathfrak {g}^{\sigma } = \mathfrak {h}$ . In his setting, the extra deformation parameter space is the formal group generated by the Cartan algebra. The construction of the equivalence follows the strategy of [Reference DrinfeldDri89b], this time relying on the co-Hochschild cohomology studied by Calaque [Reference CalaqueCal06].

Now, let us sketch what we concretely carry out:

  • Show that the quasi-coactions of Drinfeld’s quasi-bialgebra induced by the cyclotomic KZ-equations are generically universal among the quasi-coactions deforming $\Delta $ on $U(\mathfrak {g}^{\sigma })$ .

  • Give a complete classification of the corresponding ribbon twist-braids and show that the corresponding K-matrices give a complete invariant of the quasi-coactions.

  • In the Hermitian case (see below), when there is a one-parameter family of nonequivalent quasi-coactions, establish a correspondence with Poisson structures on $U / U^{\sigma }$ by studying coisotropic subgroups which are conjugates (‘Cayley transforms’) of $U^{\sigma }$ .

  • Make a concrete comparison with the Letzter–Kolb coideals and the Balagović–Kolb braided module categorical structures.

In the first step, the main idea is to reduce the problem to vanishing of obstructions in a suitable version of the co-Hochschild cohomology. This strategy is quite standard; see [Reference DrinfeldDri89bReference AdrienBro12], but while these papers relied on the braiding/ribbon braids to have a good control of the cohomology, we work with the cohomology classes directly, analogously to Donin–Shnider’s approach [Reference Donin and ShniderDS97] to Lie bialgebra quantization, and the identification of the ribbon twist-braids comes only towards the end. The relevant co-Hochschild cohomology turns out to be isomorphic to $\bigwedge \mathfrak {m}^{\mathbb {C}}$ for $\mathfrak {m}^{\mathbb {C}} = \mathfrak {g} \ominus \mathfrak {g}^{\sigma }$ , and the deformation of a quasi-coaction is controlled by the invariant part of the second cohomology, that is, $(\bigwedge ^2 \mathfrak {m}^{\mathbb {C}})^{\mathfrak {g}^{\sigma }}$ . Up to complexification, this space can be interpreted as the space of U-invariant bivectors on $U/U^{\sigma }$ , hence there is a direct connection to equivariant deformation quantization. This is where one sees the formality principle in action.

At this point, we encounter an important dichotomy between the Hermitian and the non-Hermitian cases. Although we already discussed it in [Reference De Commer, Neshveyev, Tuset and YamashitaDCNTY19] based on the parameters $\mathbf {t}$ for the coideals $U_h^{\mathbf {t}}(\mathfrak {g}^{\theta })$ , the following observation is perhaps more illuminating: The dimension of $(\bigwedge ^2 \mathfrak {m}^{\mathbb {C}})^{\mathfrak {g}^{\sigma }}$ is either zero or one and is equal to that of the center of $\mathfrak {g}^{\sigma }$ . In the Hermitian case, and only in this case, this dimension is one and the corresponding homogeneous space $U/U^{\sigma }$ has an invariant Hermitian structure, induced by an element of the center of $\mathfrak {g}^{\sigma }$ (hence the name).

In the non-Hermitian case, the triviality of the center eliminates cohomological obstructions, quickly leading to rigidity of the algebra structure and coaction homomorphisms on $U(\mathfrak {g}^{\sigma })$ . Our results in this case can be summarized as follows.

Theorem A (Section 2.2 and Theorem 2.18)

Let $\mathfrak {u}^{\sigma } < \mathfrak {u}$ be a non-Hermitian irreducible symmetric pair. Suppose that and define a quasi-coaction of Drinfeld’s quasi-bialgebra that deforms $\Delta \colon U(\mathfrak {g}^{\sigma }) \to U(\mathfrak {g}^{\sigma }) \otimes U(\mathfrak {g})$ , and let $(\alpha ', \Psi ')$ be another such pair. Then $(\alpha , \Psi )$ and $(\alpha ', \Psi ')$ are obtained from each other by twisting. Moreover, the quasi-coaction admits a unique ribbon $\sigma $ -braid $\mathcal {E}$ with prescribed constant term $\mathcal {E}^{(0)}\in 1\otimes Z(U)$ .

In the above formulation, the ribbon twist-braid is allowed to live in a certain completion of . Namely, consider the multiplier algebra of the algebra of finitely supported functions on the dual of $U^{\sigma }$ [Reference Van DaeleVD96], which is the direct product of full matrix algebras

$$\begin{align*}\mathcal{U}(G^{\sigma}) = \prod_{\pi} \operatorname{\mathrm{End}}(V_{\pi}), \end{align*}$$

where $\pi $ runs over the irreducible finite-dimensional representations of $\mathfrak {g}^{\sigma }$ which appear in finite-dimensional representations of $\mathfrak {g}$ . We can further define

$$\begin{align*}\mathcal{U}(G^{\sigma} \times G^n) = \prod_{\pi, \pi_1, \dots, \pi_n} \operatorname{\mathrm{End}}(V_{\pi}) \otimes \operatorname{\mathrm{End}}(V_{\pi_1}) \otimes \dots \otimes \operatorname{\mathrm{End}}(V_{\pi_n}), \end{align*}$$

where $\pi _1, \dots \pi _n$ run over the irreducible finite-dimensional representations of $\mathfrak {g}$ . Then we take $\mathcal {E}$ as an element of .

The situation is more interesting in the Hermitian case. Even up to equivalence defined by twisting, the quasi-coactions are no longer unique. In this case, we show that generic quasi-coactions are equivalent to the ones arising from the cyclotomic KZ-equations with prescribed coefficients [Reference Enriquez and EtingofEE05Reference De Commer, Neshveyev, Tuset and YamashitaDCNTY19]: The associator $\Psi _{\mathrm {KZ},s; \mu }$ , for parameters $s \in \mathbb {C}\setminus i\mathbb {Q}^{\times }$ and , is given as the normalized monodromy from $w = 0$ to $w = 1$ of the differential equation

$$\begin{align*}H'(w) = \left( \frac{\hbar(t^{\mathfrak{k}}_{12} - t^{\mathfrak{m}}_{12})}{w + 1} + \frac{\hbar t^{\mathfrak{u}}_{12}}{w-1} + \frac{\hbar(2 t^{\mathfrak{k}}_{01} + C^{\mathfrak{k}}_1) + (s+\mu) Z_1}{w} \right) H(w). \end{align*}$$

Here, we put $\hbar = \frac {h}{\pi i}$ , and the coefficients are given as follows: $t^{\mathfrak {u}}$ , $t^{\mathfrak {k}}$ , $t^{\mathfrak {m}} $ are the canonical $2$ -tensors of $\mathfrak {u}$ , $\mathfrak {k} = \mathfrak {u}^{\sigma }$ , and $\mathfrak {m} = \mathfrak {u} \ominus \mathfrak {k}$ respectively, $C^{\mathfrak {k}}$ is the Casimir element of $\mathfrak {k}$ associated to $t^{\mathfrak {k}}$ , and Z is a normalized element of $\mathfrak {z}(\mathfrak {k})$ .

If $s=0$ , then $\Psi _{\mathrm {KZ},s;\mu }$ makes sense in , but otherwise we can only say that $\Psi _{\mathrm {KZ},s;\mu }$ is in . It is therefore convenient to start working with the multiplier algebras throughout instead of the universal enveloping algebras. Fortunately, the concepts of quasi-bialgebras and quasi-coactions have straightforward formulations in this setting, and from the categorical point of view this formalism is actually even more natural when dealing with semisimple module categories. Then is a quasi-coaction of , and our results can be summarized as follows.

Theorem B (Theorems 2.16 and 2.19)

Let $\mathfrak {u}^{\sigma } < \mathfrak {u}$ be an irreducible Hermitian symmetric pair, and let $\omega $ be an invariant symplectic form on $U / U^{\sigma }$ . There is a countable subset $A \subset \mathbb {C}$ with the following property: If and define a quasi-coaction of that deforms $\Delta \colon \mathcal {U}(G^{\sigma }) \to \mathcal {U}(G^{\sigma }\times G)$ , and the first order term $\Psi ^{(1)}$ of $\Psi $ satisfies $\langle \omega , \Psi ^{(1)} \rangle \in \mathbb {C} \setminus A$ , then there is a pair $(s, \mu )$ , unique up to translation by $(2i\mathbb {Z},0)$ , such that is equivalent to . Moreover, admits a unique ribbon $\sigma $ -braid $\mathcal {E}$ with prescribed constant term $\mathcal {E}^{(0)}\in 1\otimes \exp (-\pi i s Z)Z(U)$ .

We resolve the cohomological obstruction to equivalence by looking at the expansion of $\Psi _{\mathrm {KZ},s; \mu }$ , where we follow Enriquez and Etingof’s work [Reference Enriquez and EtingofEE05] on quantization of dynamical r-matrices. Up to a coboundary, $\Psi _{\mathrm {KZ},s; \mu }$ has the expansion

$$ \begin{align*} \Psi_{\mathrm{KZ}, s; \mu} \sim 1 - \frac{h}{2}\tanh \left(\frac{\pi (s + \mu)}{2}\right) \sum_{\alpha \in \Phi_{\mathrm{nc}}^+} \frac{(\alpha,\alpha)}{2} 1 \otimes \left( X_{\alpha} \otimes X_{-\alpha} - X_{-\alpha} \otimes X_{\alpha} \right) + \cdots, \end{align*} $$

where $\Phi _{\mathrm {nc}}^+$ is the set of positive roots in $\mathfrak {m}^{\mathbb {C}}$ with respect to a choice of Cartan subalgebra in $\mathfrak {g}^{\sigma }$ , and $X_{\pm \alpha }$ is a normalized root vector for $\pm \alpha $ ; see Sections 2.3 and 2.5 for details. This shows that, under a perturbation of $\mu $ , the associator changes in the term one order higher than the perturbation, with a precise control of the cohomology class (formal Poisson structure) of the difference in that term. This leads to the universality of quasi-coactions with the associators $\Psi _{\mathrm {KZ},s; \mu }$ and can be interpreted as ‘poor man’s formality’ for equivariant deformation quantization.

We next apply these results to the Letzter–Kolb coideals. Since our classification is formulated in the framework of multiplier algebras, we show that the coideals indeed give rise to such structures, essentially by taking a completion. It should be stressed that the formalism of multiplier algebras is important not only for making sense of $\Psi _{\mathrm {KZ}, s; \mu }$ . The second and even more important reason is that it allows us to check that the coactions defined by the Letzter–Kolb coideals are twistings of $\Delta $ . The point is that, since $\mathfrak {g}^{\sigma }$ is not semisimple in the Hermitian case, the standard arguments based on Whitehead’s first lemma are not applicable. By working with the multiplier algebras, which are built out of semisimple algebras, we can circumvent the nonvanishing of Lie algebraic cohomological obstructions. We still need to use Letzter’s result [Reference LetzterLet00] on existence of spherical vectors for this, which means that we have to consider $*$ -coideals $U^{\mathbf {t}}_h(\mathfrak {g}^{\theta })$ .

Next, in the Hermitian case, we have to verify the condition on the first order term $\Psi ^{(1)}$ . For this, we study Poisson homogeneous structures on $U/U^{\sigma }$ . More precisely, we have to compare two Poisson structures, corresponding to two ways we obtain the quasi-coactions. On the one hand, from the cyclotomic KZ-equations we obtain a Poisson pencil [Reference Donin and GurevichDG95], where one takes the sum of the left action of the standard r-matrix r on $U / U^{\sigma }$ and a scalar multiple of the Kostant–Kirillov–Souriau bracket, which agrees with the bracket defined by the right action of r. On the other hand, from the coideals we obtain the reduction of the Sklyanin bracket to quotients by coisotropic subgroups.

Starting from the model $\sigma = \theta $ in the maximally noncompact position, where the subgroup is coisotropic [Reference Foth and LuFL04], we take a distinguished one-parameter family of subgroups $U^{\theta _{\phi }}$ that are conjugate to $U^{\theta }$ by interpolated Cayley transforms and show that the associated fixed point subgroups $U^{\theta _{\phi }}$ remain coisotropic. At the level of Lie algebras, this construction interpolates between the maximally noncompact subalgebra $\mathfrak {g}^{\theta }$ and the maximally compact one $\mathfrak {g}^{\nu }$ (which contains $\mathfrak {h}$ ). Moreover, the Lie algebras $\mathfrak {g}^{\theta _{\phi }}$ turn out to be the classical limits of the Letzter–Kolb $*$ -coideals $U^{\mathbf {t}}_h(\mathfrak {g}^{\theta })$ . By a detailed analysis of the Cayley transforms, we are able to find the relation between the parameters $\phi $ and $\mathbf {t}$ , as well as to compute the cohomology classes of $\Psi ^{(1)}$ for the associators we get. In a bit imprecise form, these results are summarized as follows.

Theorem C (Theorems 5.4, 5.5, 5.8 and 5.10)

There is a parameter set $\mathcal {T}^*$ (consisting of one point $\mathbf {t}=0$ in the non-Hermitian case) defining $*$ -coideals $U^{\mathbf {t}}_h(\mathfrak {g}^{\theta })$ and satisfying the following properties. For every $\mathbf {t}\in \mathcal {T}^*$ , the coideal $U^{\mathbf {t}}_h(\mathfrak {g}^{\theta })$ gives rise to a coaction of a multiplier bialgebra which is equivalent to the quasi-coaction of , where $G^{\theta }_{\mathbf {t}}<G$ is a subgroup conjugate to $G^{\theta }$ , while $s\in \mathbb {R}$ and are uniquely determined parameters (equal to $0$ in the non-Hermitian case), with s given by an explicit formula. Under this equivalence, the Balagović–Kolb ribbon twist-braids correspond to the ones coming from the cyclotomic KZ-equations.

This implies a Kohno–Drinfeld type result (Theorems 5.12 and 5.13) for quantum symmetric pairs, stating that representations of type $\mathrm B$ braid groups arising from the coideals and the cyclotomic KZ-equations are equivalent.

A formula for the parameter $\mu $ in Theorem C can in principle be obtained by comparing the eigenvalues of the reflection operators in the two pictures. In the general case, this step might be somewhat involved, but at least for the $\mathrm {AIII}$ case (which corresponds to the symmetric pairs $\mathfrak {s}(\mathfrak {u}_p \oplus \mathfrak {u}_{N-p}) < \mathfrak {su}_{N}$ ) this can be done thanks to the classification of reflection operators by Mudrov [Reference MudrovMud02].

So far we have discussed the case of irreducible symmetric spaces of type I, that is, $U/U^{\sigma }$ with U simple. However, the type II case, corresponding to U itself as a symmetric space, or the quotient of $U \times U$ by the diagonal subgroup, can be handled in essentially the same way as the non-Hermitian type I cases. In particular, Theorems A and C can be adapted to this case. This implies that an analogue of Theorem C holds in general for Letzter–Kolb $*$ -coideals of $U_h(\mathfrak {g})$ with $\mathfrak {g}$ semisimple.

Let us now briefly summarize the contents of the paper. In Section 1, we recall basic definitions and introduce conventions which are used throughout the paper.

In Section 2, we prove our main conceptual results on classification of quasi-coactions and ribbon twist-braids. As explained above, the non-Hermitian case is done by a more or less standard cohomological argument, while in the Hermitian case we look into the structure of the associators arising from the cyclotomic KZ-equations.

In Section 3, we focus on the Hermitian case and look at conjugates of $\mathfrak {u}^{\sigma } < \mathfrak {u}$ in the maximally compact position by interpolated Cayley transforms. We show that these conjugates generate coisotropic subgroups and relate them to models arising from the cyclotomic KZ-equations, with an explicit formula for the first order term.

In Section 4, we explain how the quantized universal enveloping algebra and the Letzter–Kolb coideals fit into our setting of multiplier quasi-bialgebras and their quasi-coactions.

Finally, in Section 5, we combine the results of the previous sections and prove our main comparison theorems. We finish the section with a detailed analysis of the $\mathrm {AIII}$ case.

There are three appendices, in which we collect some technical but not fundamentally new results used in the paper.

Let us close the introduction with some further problems. First of all, a general formula for $\mu $ in Theorem C would be nice to find, especially if this can be done in a unified way rather than via a case-by-case analysis. Second, the analytic version of the conjecture, as originally proposed in [Reference De Commer, Neshveyev, Tuset and YamashitaDCNTY19], remains to be settled, together with a comparison with the ‘Vogan picture’ introduced there. On the geometric side, one would like to extend the above results in the Hermitian case to all coadjoint orbits of U.

1 Preliminaries

1.1 Conventions

We treat h as a formal variable and put $h^* = h$ when we consider $*$ -algebraic structures. We put

$$\begin{align*}q=e^h\quad\text{and}\quad\hbar = \frac{h}{\pi i},\\[-10pt] \end{align*}$$

the latter is mostly reserved for the KZ-equations. We denote the space of formal power series with coefficients in A by

and the space of Laurent series by

For

, we denote the smallest n such that $a^{(n)} \neq 0$ by $\operatorname {\mathrm {ord}}(a)$ .

We denote the h-adically completed tensor product of -modules by $\mathbin {\hat \otimes }$ . In particular, we have .

When $A = \mathbb {C}$ and has constant term $a^{(0)}>0$ , we take its nth root $b = a^{\frac {1}{n}}$ to be the unique solution of $b^n = a$ such that $b^{(0)}$ is positive. A similar convention is used for $\log $ .

1.2 Simple Lie groups

Throughout the entire paper, $\mathfrak {u}$ denotes a compact simple Lie algebra and $\mathfrak {g}$ denotes its complexification. The connected and simply connected Lie groups corresponding to $\mathfrak {g}$ and $\mathfrak {u}$ are denoted by G and U.

We denote by $(\cdot ,\cdot )_{\mathfrak {g}}$ the unique invariant symmetric bilinear form on $\mathfrak {g}$ such that, for any Cartan subalgebra $\mathfrak {h} < \mathfrak {g}$ , its dual form on $\mathfrak {h}^*$ has the property that $(\alpha ,\alpha )=2$ for every short root $\alpha $ . Let $t^{\mathfrak {u}}\in \mathfrak {u}^{\otimes 2}$ be the corresponding invariant tensor:

(1.1) $$ \begin{align} t^{\mathfrak{u}}=\sum_i X_i\otimes X^i,\\[-17pt]\nonumber \end{align} $$

where $(X_i)_i$ is a basis in $\mathfrak {g}$ and $(X^i)_i$ is the dual basis.

Recall that $(\cdot ,\cdot )_{\mathfrak {g}}$ is negative definite on $\mathfrak {u}$ . Therefore, if we define an antilinear involution $*$ on $\mathfrak {g}$ by letting $X^*=-X$ for $X\in \mathfrak {u}$ , then $(X,Y^*)_{\mathfrak {g}}$ becomes an $(\operatorname {\mathrm {Ad}} U)$ -invariant Hermitian scalar product on $\mathfrak {g}$ .

We denote the category of finite-dimensional algebraic representations of the linear algebraic group G (equivalently, finite-dimensional representations of $\mathfrak {g}$ ) by $\operatorname {\mathrm {Rep}} G$ . It is equivalent to the category of finite-dimensional unitary representations of U. We write $\pi \in \operatorname {\mathrm {Rep}} G$ to say that $\pi $ is a finite-dimensional representation of G, its underlying space is denoted by $V_{\pi }$ . We also fix a set $\operatorname {\mathrm {Irr}} G$ of representatives of the isomorphism classes of irreducible representations.

We will often have to extend the scalars to . Denote the category we get by . Thus, the objects of are the G-modules over that are isomorphic to the modules of the form for $\pi \in \operatorname {\mathrm {Rep}} G$ .

1.3 Multiplier algebras

For $n = 1, 2, \dots $ , we put

$$\begin{align*}\mathcal{U}(G^n) = \prod_{\substack{\pi_i \in \operatorname{\mathrm{Irr}} G,\\ i = 1, \dots, n}} \operatorname{\mathrm{End}}(V_{\pi_1}) \otimes \dots \otimes \operatorname{\mathrm{End}}(V_{\pi_n}).\\[-17pt] \end{align*}$$

We view G and $\mathfrak {g}$ as subsets of $\mathcal {U}(G) = \mathcal {U}(G^1)$ .

Since for every irreducible $\pi \in \operatorname {\mathrm {Rep}} G$ , there is a unique up to a scalar factor U-invariant Hermitian scalar product on $V_{\pi }$ , we have a canonical involution $*$ on $\mathcal {U}(G^n)$ . There is also a unique homomorphism

$$\begin{align*}\Delta\colon \mathcal{U}(G) \to \mathcal{U}(G^2)\\[-17pt] \end{align*}$$

characterized by the identities $(\pi _1 \otimes \pi _2)(\Delta (T))S=S\pi (T)$ for all intertwiners $S\colon V_{\pi }\to V_{\pi _1}\otimes V_{\pi _2}$ . Then $\Delta (g) = g \otimes g$ for $g \in G$ . This characterizes the elements of G among the nonzero elements of $\mathcal {U}(G)$ . Similarly, the identity $\Delta (X) = X \otimes 1 + 1 \otimes X$ for $X \in \mathfrak {g}$ characterizes $\mathfrak {g}$ inside $\mathcal {U}(G)$ .

Denote by $\mathcal {O}(G)$ the Hopf algebra of regular functions (matrix coefficients of finite-dimensional representations) on G. We occasionally write $\mathcal {O}(U)$ instead of $\mathcal {O}(G)$ when we think of it as a function algebra on U.

There is a nondegenerate pairing between $\mathcal {U}(G)$ and $\mathcal {O}(G)$ that allows us to identify $\mathcal {U}(G)$ with the dual space of $\mathcal {O}(G)$ . Concretely, if $\pi $ is irreducible, $T\in \operatorname {\mathrm {End}}(V_{\pi })$ , $v\in V_{\pi }$ , $\ell \in V_{\pi }^*$ , then for the matrix coefficient $a_{v,\ell }\in \mathcal {O}(G)$ , $a_{v,\ell }(g)=\ell (\pi (g)v)$ , we have

$$\begin{align*}\langle a_{v,\ell},T\rangle=\ell(T v),\\[-17pt] \end{align*}$$

and $\langle f,T\rangle =0$ for the matrix coefficients f of the irreducible representations $\pi '$ inequivalent to $\pi $ . Similarly, $\mathcal {U}(G^n)$ is the linear dual of $\mathcal {O}(G)^{\otimes n}$ . With respect to this duality, the bialgebra structures are related by

$$ \begin{align*} \langle f_1 \otimes f_2, \Delta(T) \rangle &= \langle f_1 f_2, T \rangle,& \langle \Delta(f), T_1 \otimes T_2 \rangle &= \langle f, T_1 T_2 \rangle\\[-17pt] \end{align*} $$

for $f_i \in \mathcal {O}(G)$ and $T_i \in \mathcal {U}(G)$ .

We can do the same constructions for any reductive linear algebraic group H over $\mathbb {C}$ . We then also define

for $0 \le n < \infty $ . In a more invariant form, $\mathcal {U}(H \times G^n)$ is the linear dual of $\mathcal {O}(H\times G^n)$ .

Assume in addition that H is an algebraic subgroup of G. Then the embedding $H\to G$ extends to an embedding of $\mathcal {U}(H^{n+1})$ into $\mathcal {U}(H\times G^n)$ . In particular, the comultiplication $\Delta \colon \mathcal {U}(H)\to \mathcal {U}(H^2)$ can be viewed as a homomorphism $\mathcal {U}(H)\to \mathcal {U}(H\times G)$ .

Note that in general H is not simply connected. In Lie algebraic terms, the category $\operatorname {\mathrm {Rep}} H$ consists of the finite-dimensional representations of $\mathfrak {h}$ that are subrepresentations of the finite-dimensional representations of $\mathfrak {g}$ restricted to $\mathfrak {h}$ .

1.4 Quasi-coactions and ribbon twist-braids

The notion of a quasi-bialgebra [Reference DrinfeldDri89b] has a straightforward adaptation to the setting of multiplier algebras, cf. [Reference Neshveyev and TusetNT11, Section 2]. We will be interested in multiplier quasi-bialgebras of the form . Thus, $\Delta _h$ is a nondegenerate homomorphism , meaning that the images of the idempotents $\Delta _h(\mathrm {id}_{V_{\pi }})$ ( $\pi \in \operatorname {\mathrm {Irr}} G$ ) in add up to $1$ , is a nondegenerate homomorphism, and is an invertible element (with $\Phi ^{(0)}=1$ ) satisfying the same identities as in [Reference DrinfeldDri89b, Section 1].

The assumption of nondegeneracy for the counit $\epsilon _h$ implies that it is determined by its restrictions to the blocks of . Since there are no nonzero ( -linear) homomorphisms for $\dim V>1$ and there is a unique such homomorphism for $\dim V=1$ , we conclude that $\epsilon _h$ coincides with the standard counit $\epsilon $ on . From now on, we will therefore omit $\epsilon _h$ from the notation for a multiplier quasi-bialgebra.

Given a reductive algebraic subgroup H of G, a quasi-coaction of on is given by a nondegenerate homomorphism and an associator satisfying $\Psi ^{(0)}=1$ ,

$$\begin{align*}(\mathrm{id}\otimes\epsilon)\alpha=\mathrm{id}, \end{align*}$$
(1.2)

the mixed pentagon equation

(1.3) $$ \begin{align} \Phi_{1,2,3} \Psi_{0, 12, 3} \Psi_{0, 1, 2} = \Psi_{0, 1, 23} \Psi_{01, 2, 3}, \end{align} $$

with $\Psi _{01, 2, 3} = (\alpha \otimes \mathrm {id})(\Psi )$ , $\Psi _{0, 12, 3} = (\mathrm {id}_{\mathcal {U}(H)} \otimes \Delta _h \otimes \mathrm {id})(\Psi )$ , etc., and the normalization conditions

$$\begin{align*}(\mathrm{id}\otimes\epsilon\otimes\mathrm{id})(\Psi) = (\mathrm{id}\otimes\mathrm{id}\otimes\epsilon)(\Psi)=1. \end{align*}$$

A multiplier quasi-bialgebra defines a tensor category , where the tensor product $\otimes _h$ on is defined using $\Delta _h$ and the associativity isomorphism is given by the action of $\Phi $ . A quasi-coaction as above defines then the structure of a right -module category on . Namely, the functor defining the module category structure is induced by $\alpha $ , while the associativity morphisms are defined by the action of $\Psi $ . See [Reference De Commer, Neshveyev, Tuset and YamashitaDCNTY19, Section 1] for more details, but note that in [Reference De Commer, Neshveyev, Tuset and YamashitaDCNTY19] we worked in the analytic setting, meaning that $q=e^h$ was a real number and $\Phi \in \mathcal {U}(G^3)$ , $\Psi \in \mathcal {U}(H\times G^2)$ .

Next, let be an R-matrix (with $\mathcal {R}^{(0)}=1$ ) for , that is, $\mathcal {R}\Delta _h(\cdot )=\Delta _h^{\mathrm {op}}(\cdot )\mathcal {R}$ and $\mathcal {R}$ satisfies the hexagon relations. Let $\beta $ be an automorphism of the quasi-triangular multiplier quasi-bialgebra . A ribbon $\beta $ -braid is given by an invertible element satisfying

(1.4)
(1.5) $$ \begin{align} (\alpha \otimes \mathrm{id})(\mathcal{E}) = \Psi^{-1} \mathcal{R}_{21} \Psi_{021} \mathcal{E}_{02} (\mathrm{id} \otimes\mathrm{id}\otimes \beta)(\Psi_{021}^{-1} \mathcal{R}_{12} \Psi), \end{align} $$
(1.6) $$ \begin{align} (\mathrm{id} \otimes \Delta_h)(\mathcal{E}) = \mathcal{R}_{21} \Psi_{021} \mathcal{E}_{02} (\mathrm{id} \otimes\mathrm{id}\otimes \beta)(\Psi_{021}^{-1} \mathcal{R}_{12} \Psi) \mathcal{E}_{01} (\mathrm{id} \otimes \beta \otimes \beta)(\Psi^{-1}). \end{align} $$

When $\beta $ is the identity map, we just say ‘ribbon braid’ instead of ‘ribbon $\mathrm {id}$ -braid’. We want to stress that, as opposed to $\Phi $ , $\Psi $ and $\mathcal {R}$ , we do not require $\mathcal {E}^{(0)}=1$ . A quadruple satisfying equations (1.4) and (1.5) is a version of a quasi-reflection algebra [Reference EnriquezEnr07]. In categorical terms, the action of $\mathcal {E}$ on $M\odot _{\alpha } N$ defines the structure of a ribbon $\beta $ -braided module category on . See again [Reference De Commer, Neshveyev, Tuset and YamashitaDCNTY19, Section 1] for more details.

1.5 Twisting

We can transform a quasi-coaction of into a new one as follows. Suppose that we are given elements and such that $\mathcal {F}^{(0)}=1$ , $\mathcal {G}^{(0)}=1$ and

$$ \begin{align*} (\epsilon\otimes\mathrm{id})(\mathcal{F}) &= (\mathrm{id}\otimes\epsilon)(\mathcal{F})=1,& (\mathrm{id}\otimes\epsilon)(\mathcal{G}) &= 1. \end{align*} $$

Then the twisting of the quasi-coaction by $(\mathcal {F},\mathcal {G})$ is the quasi-coaction of the multiplier quasi-bialgebra , where

$$ \begin{align*} \Delta_{h,\mathcal{F}}&=\mathcal{F}\Delta_h(\cdot)\mathcal{F}^{-1}, & \Phi_{\mathcal{F}}&=(1\otimes \mathcal{F})(\mathrm{id}\otimes \Delta_h)(\mathcal{F})\Phi(\Delta_h\otimes\mathrm{id})(\mathcal{F}^{-1})(\mathcal{F}^{-1}\otimes 1),\\ \alpha_{\mathcal{G}}&=\mathcal{G}\alpha(\cdot)\mathcal{G}^{-1}, & \Psi_{\mathcal{F},\mathcal{G}}&=(1\otimes\mathcal{F})(\mathrm{id}\otimes \Delta_h)(\mathcal{G})\Psi (\alpha\otimes \mathrm{id})(\mathcal{G}^{-1})(\mathcal{G}^{-1}\otimes 1). \end{align*} $$

Twisting defines an equivalence relation on the quasi-coactions. In categorical terms, it means that we pass from

to the equivalent tensor category

, with the tensor product defined by $\Delta _{h,\mathcal {F}}$ , and, up to this equivalence, the

-module category

is equivalent to the

-module category

As the following result shows, twisting often allows one to push all the information on a quasi-coaction into the associators.

Lemma 1.1. Assume H is a reductive algebraic subgroup of G and is a quasi-coaction of such that both $\alpha $ and $\Delta _h$ equal $\Delta $ modulo h. Then this quasi-coaction is a twisting of a quasi-coaction of for some $\Psi '$ and $\Phi '$ .

Proof. Take irreducible representations $\pi _1$ and $\pi _2$ of G. Consider the homomorphisms $f=(\pi _1\otimes \pi _2)\Delta $ and $f_h=(\pi _1\otimes \pi _2)\Delta _h$ from into . The assumption of nondegeneracy for $\Delta _h$ implies that there exists a finite set $F\subset \operatorname {\mathrm {Irr}} G$ such that $f_h$ factors through . By taking F large enough, we may assume that the same is true for f. Since the algebra $\bigoplus _{\pi \in F}\operatorname {\mathrm {End}}(V_{\pi })$ is semisimple, there are no nontrivial deformations of any given homomorphism $\bigoplus _{\pi \in F}\operatorname {\mathrm {End}}(V_{\pi })\to \operatorname {\mathrm {End}}(V_{\pi _1}\otimes V_{\pi _2})$ . Hence, there exists such that $\mathcal {F}_{\pi _1,\pi _2}^{(0)}=1$ and $f_h=(\operatorname {\mathrm {Ad}} \mathcal {F}_{\pi _1,\pi _2})f$ . Then satisfies $\mathcal {F}^{(0)}=1$ and $\Delta _h=\mathcal {F}\Delta (\cdot )\mathcal {F}^{-1}$ . Furthermore, since the counit of is $\epsilon $ , we could take $\mathcal {F}_{\pi _1,\pi _2}=1$ if either $\pi _1$ or $\pi _2$ were trivial representations. In this case, $\mathcal {F}$ would additionally satisfy $(\epsilon \otimes \mathrm {id})(\mathcal {F})= (\mathrm {id}\otimes \epsilon )(\mathcal {F})=1$ .

In a similar way, we can find such that $\mathcal {G}^{(0)}=1$ , $(\iota \otimes \epsilon )(\mathcal {G})=1$ and $\alpha =\mathcal {G}\Delta (\cdot )\mathcal {G}^{-1}$ . Then the twisting by $(\mathcal {F}^{-1},\mathcal {G}^{-1})$ gives the required quasi-coaction.

Next, given a quasi-coaction of , assume in addition we have an automorphism $\beta $ of . If $\mathcal {F}$ satisfies $(\beta \otimes \beta )(\mathcal {F})=\mathcal {F}$ , then $\beta $ remains an automorphism of . Assume also that is an R-matrix for that is fixed under $\beta $ . Then $\mathcal {R}_{\mathcal {F}}=\mathcal {F}_{21}\mathcal {R}\mathcal {F}^{-1}$ is an R-matrix for , again fixed by $\beta $ . Given a ribbon $\beta $ -braid $\mathcal {E}$ for the original quasi-coaction we get a ribbon $\beta $ -braid $\mathcal {E}_{\mathcal {G}}$ for the twisted quasi-coaction of defined by

(1.7) $$ \begin{align} \mathcal{E}_{\mathcal{G}}=\mathcal{G}\mathcal{E}(\mathrm{id}\otimes\beta)(\mathcal{G})^{-1}. \end{align} $$

The condition $(\beta \otimes \beta )(\mathcal {F})=\mathcal {F}$ can be relaxed; we will return to this in Section 5.1.

1.6 Symmetric pairs

Let $\mathfrak {k}$ be a proper Lie subalgebra of $\mathfrak {u}$ . We say that $\mathfrak {k}<\mathfrak {u}$ is a symmetric pair, or more precisely, an irreducible symmetric pair of type I, if there is a (necessarily unique) involutive automorphism $\sigma $ of $\mathfrak {u}$ such that $\mathfrak {k}=\mathfrak {u}^{\sigma }$ . Whenever convenient, we extend $\sigma $ to $\mathcal {U}(G)$ , in particular, to $\mathfrak {g}$ . Let $K=U^{\sigma }$ . The compact group K is connected by [Reference HelgasonHel01, Theorem VII.8.2]. Using the Cartan decomposition of G, we can also conclude that $G^{\sigma }$ is connected.

Given such a symmetric pair, put

$$\begin{align*}\mathfrak{m}=\{X\in\mathfrak{u}\mid \sigma(X)=-X\}, \end{align*}$$

which is the orthogonal complement of $\mathfrak {k}$ in $\mathfrak {u}$ with respect to the invariant inner product. We also write $\mathfrak {m}^{\mathbb {C}} = \mathfrak {m} \otimes _{\mathbb {R}} \mathbb {C}$ for its complexification.

We say that a symmetric pair $\mathfrak {k}<\mathfrak {u}$ is Hermitian, if $U/K$ is a Hermitian symmetric space. Such symmetric pairs are equivalently characterized by either of the following conditions; see [Reference BorelBor98, Proposition VI.1.3]:

  • The center $\mathfrak {z}(\mathfrak {k})$ is nontrivial (and $1$ -dimensional);

  • The space $\mathfrak {m}$ has a (unique up to a sign) $\mathfrak {k}$ -invariant complex structure.

The following closely related characterization will be crucial for us.

Lemma 1.2. For any symmetric pair $\mathfrak {k}<\mathfrak {u}$ , we have $\dim (\bigwedge ^2 \mathfrak {m})^{\mathfrak {k}} = 1$ if $\mathfrak {k} < \mathfrak {u}$ is Hermitian, and $\dim (\bigwedge ^2 \mathfrak {m})^{\mathfrak {k}} = 0$ otherwise. We always have $\mathfrak {m}^{\mathfrak {k}} = 0$ .

Proof. Since U is simple by assumption, $U/K$ is an irreducible symmetric space, so K acts irreducibly on $\mathfrak {m} = T_{[e]}(U/K)$ . As $U/K$ is not one-dimensional, this cannot be the trivial action, and we get $\mathfrak {m}^{\mathfrak {k}} = 0$ .

Next, since $\mathfrak {m}$ has a $\mathfrak {k}$ -invariant inner product, the space $(\bigwedge ^2 \mathfrak {m})^{\mathfrak {k}}$ is isomorphic to the space of $\mathfrak {k}$ -invariant skew-adjoint operators on $\mathfrak {m}$ . Assume we are given such a nonzero operator A. Then $A^2=-A^*A$ is self-adjoint, with negative eigenvalues. Hence, $A^2$ is diagonalizable, and by irreducibility of the action of $\mathfrak {k}$ on $\mathfrak {m}$ we conclude that $A^2$ must be a strictly negative scalar. Therefore, by rescaling A we get a $\mathfrak {k}$ -invariant complex structure on $\mathfrak {m}$ . Since there is a unique such structure up to a sign in the Hermitian case and no such structure in the non-Hermitian case, we get the result.

Remark 1.3. In the non-Hermitian case, the centralizer $Z_U(K)$ of K in U is a finite group that either agrees with the center $Z(U)$ of U or contains it as a subgroup of index $2$ . Indeed, we have $\mathfrak {z}_{\mathfrak {u}}(\mathfrak {k})=0$ from the above lemma, which implies the finiteness of $Z_U(K)$ . To see that the index of $Z(U)$ in $Z_U(K)$ is at most $2$ , observe that any K-intertwiner on $\mathfrak {m}$ has to be a (real) scalar by the vanishing of $(\bigwedge ^2 \mathfrak {m})^{\mathfrak {k}}$ . Then, given $g \in Z_U(K)$ , the restriction of the finite-order K-intertwiner $\operatorname {\mathrm {Ad}} g$ to $\mathfrak {m}$ should be $\pm 1$ , which implies that either $g \in Z(U)$ or $\operatorname {\mathrm {Ad}} g = \sigma $ . If the inclusion $\mathfrak {k} < \mathfrak {u}$ is of equal rank, there are elements g satisfying $\operatorname {\mathrm {Ad}} g = \sigma $ , hence we obtain $[Z_U(K) : Z(U)] = 2$ . Otherwise there is no such g, hence we obtain $Z_U(K) = Z(U)$ .

An irreducible symmetric pair of type II is an inclusion that is isomorphic to the diagonal inclusion of $\mathfrak {u}$ into $\mathfrak {u} \oplus \mathfrak {u}$ (with a simple compact Lie algebra $\mathfrak {u}$ ). This corresponds to the involution $\sigma (X,Y) = (Y,X)$ on $\mathfrak {u} \oplus \mathfrak {u}$ . For such a pair, we can put $\mathfrak {m} = \{(X, -X) \mid X \in \mathfrak {u}\}$ . Since both $\mathfrak {m}^{\mathfrak {u}}$ and $({\textstyle \bigwedge }^2 \mathfrak {m})^{\mathfrak {u}}$ are trivial, such pairs behave in many respects similarly to the non-Hermitian type I pairs. We will therefore mostly focus on the type I case and only make a few remarks on the type II case.

Back to type I symmetric pairs, in the Hermitian case, it is known that an invariant complex structure on $\mathfrak {m}$ is defined by an element of $\mathfrak {z}(\mathfrak {k})$ . The correct normalization is given by the following.

Lemma 1.4. Assuming that $\mathfrak {k}<\mathfrak {u}$ is a Hermitian symmetric pair, let $Z \in \mathfrak {z}(\mathfrak {k})$ be a vector such that $(Z,Z)_{\mathfrak {g}}=-1$ . Then on $\mathfrak {m}$ we have $(\operatorname {\mathrm {ad}} Z)^2=-a_{\sigma }^2\mathrm {id}$ , where

$$ \begin{align*} a_{\sigma} = \sqrt{\frac{2 h^{\vee} c}{\dim \mathfrak{m}}},\\[-15pt] \end{align*} $$

$c\in \{1, 2, 3\}$ is the ratio of the square lengths of long and short roots of $\mathfrak {g}$ and $h^{\vee }$ is the dual Coxeter number of $\mathfrak {g}$ .

Proof. As $(\operatorname {\mathrm {ad}} Z)|_{\mathfrak {m}}$ is $\mathfrak {k}$ -invariant and skew-adjoint, by the proof of the previous lemma, we have $(\operatorname {\mathrm {ad}} Z)^2=-a^2$ on $\mathfrak {m}$ for some scalar $a\ge 0$ . Hence, for the Killing form $B_{\mathrm {Kill}}$ on $\mathfrak {g}$ we have $B_{\mathrm {Kill}}(Z,Z)=\operatorname {\mathrm {Tr}}((\operatorname {\mathrm {ad}} Z)^2) = - a^2 \dim \mathfrak {m}$ . The Killing form and the normalized bilinear form $(\cdot ,\cdot )_{\mathfrak {g}}$ are related by $B_{\mathrm {Kill}}=2 h^{\vee } c (\cdot ,\cdot )_{\mathfrak {g}}$ ; see [Reference KacKac90, Chapter 6, Exercise 2]. Combining this with $(Z, Z)_{\mathfrak {g}} = -1$ , we get that $a=a_{\sigma }$ .

Corollary 1.5. The $\mathfrak {k}$ -invariant complex structures on $\mathfrak {m}$ are given by $\pm \frac {1}{a_{\sigma }} (\operatorname {\mathrm {ad}} Z)|_{\mathfrak {m}}$ . For the involutive automorphism $\sigma $ such that $\mathfrak {k}=\mathfrak {u}^{\sigma }$ , we have

$$ \begin{align*} \sigma = \exp\left(\frac{\pi}{a_{\sigma}} \operatorname{\mathrm{ad}} Z\right).\\[-15pt] \end{align*} $$

In particular, we see that K is the stabilizer of Z in U with respect to the adjoint action. As the adjoint and coadjoint representations are equivalent, this leads to yet another known characterization of the Hermitian symmetric pairs: A symmetric pair $\mathfrak {k}<\mathfrak {u}$ is Hermitian if and only if the homogeneous U-space $U/K$ is isomorphic to a coadjoint orbit of U.

2 Classification of quasi-coactions and ribbon braids

Throughout this section, $\mathfrak {k}=\mathfrak {u}^{\sigma } < \mathfrak {u}$ denotes a symmetric pair. Our goal is to classify using the co-Hochschild cohomology a class of quasi-coactions of on .

2.1 Co-Hochschild cohomology for multiplier algebras

The co-Hochschild cochains will play a central role in this paper. Let H be a reductive algebraic subgroup of G. Put $\tilde {B}_{G,H}^n = \mathcal {U}(H \times G^n)$ for $0 \le n < \infty $ , and define a differential $\tilde {B}_{G,H}^n\to \tilde {B}_{G,H}^{n+1}$ by

(2.1) $$ \begin{align} d_{\mathrm{cH}}(T) = T_{01,2,\dots,n+1} - T_{0,1 2, \dots, n+1} + \dots + (-1)^n T_{0,1,\dots,n(n+1)} + (-1)^{n+1} T_{0,1,\dots,n},\\[-15pt]\nonumber \end{align} $$

where $T_{0, \dots , j j + 1, \dots , n+1} = (\mathrm {id}_{\mathcal {U}(H \times G^{j-1})} \otimes \Delta \otimes \mathrm {id}_{\mathcal {U}(G^{n-j})})(T)$ and $T_{0,1,\dots ,n}=T\otimes 1$ . The group H acts diagonally by conjugation on $\mathcal {U}(H \times G^n)$ , the differential $d_{\mathrm {cH}}$ is equivariant with respect to this action. We put

$$\begin{align*}B_{G,H}^n=(\tilde{B}_{G,H}^n)^{H}. \end{align*}$$

Proposition 2.1. The cohomology of $\tilde {B}_{G,H}$ is isomorphic to the exterior algebra $\bigwedge \mathfrak {g}/\mathfrak {h}$ as a graded H-module.

Proof. The complex $\tilde {B}_{G,H}$ is the algebraic linear dual of $\tilde {B}_{G,H}' = (\mathcal {O}(H) \otimes \mathcal {O}(G)^{\otimes n})_{n=0}^{\infty }$ with the differential $d\colon \mathcal {O}(H) \otimes \mathcal {O}(G)^{\otimes n} \to \mathcal {O}(H) \otimes \mathcal {O}(G)^{\otimes n-1}$ given by

$$\begin{align*}d(f_0 \otimes f_1 \otimes \dots \otimes f_n) = \sum_{i=0}^{n-1} (-1)^i f_0 \otimes \dots \otimes f_i f_{i+1} \otimes \dots \otimes f_n + (-1)^n f_n(e) f_0 \otimes \dots \otimes f_{n-1}, \end{align*}$$

where $f_0 f_1$ is the product of $f_0$ and the restriction of $f_1$ to H. Thus, the cohomology of $\tilde {B}_{G,H}$ is the linear dual of the homology of $\tilde {B}_{G,H}'$ as an H-module.

The complex $\tilde {B}_{G,H}'$ is the standard complex computing the Hochschild homology

$$\begin{align*}\mathrm{HH}_*(\mathcal{O}(G), {}_{\mathrm{res}}\mathcal{O}(H)_{\epsilon}) = \operatorname{\mathrm{Tor}}^{\mathcal{O}(G) \otimes \mathcal{O}(G)}_*(\mathcal{O}(G), {}_{\mathrm{res}}\mathcal{O}(H)_{\epsilon}), \end{align*}$$

where the bimodule ${}_{\mathrm {res}}\mathcal {O}(H)_{\epsilon }$ has the underlying space $\mathcal {O}(H)$ with the bimodule structure $f.a.f' = f'(e) f a$ for $f, f' \in \mathcal {O}(G)$ and $a \in \mathcal {O}(H)$ . In other words, we are computing

$$\begin{align*}\operatorname{\mathrm{Tor}}^{\mathcal{O}(G\times G)}_*(\mathcal{O}(\Delta), \mathcal{O}(H\times\{e\})), \end{align*}$$

where $\Delta \subset G\times G$ is the diagonal. By [Reference Berthelot, Grothendieck, Ferrand, Jouanolou, Jussila, Kleiman and RaynaudBGI71, Proposition VII.2.5], this is the exterior algebra on $\operatorname {\mathrm {Tor}}^{\mathcal {O}(G\times G)}_1(\mathcal {O}(\Delta ), \mathcal {O}(H\times \{e\}))$ , and the latter is the conormal space of $H \subset G$ at the point e. Since this conormal space is the dual of $\mathfrak {g}/\mathfrak {h}$ , we obtain the assertion.

Remark 2.2. Proposition 2.1 and its proof are valid for any linear algebraic group G over $\mathbb {C}$ and any algebraic subgroup H, if we define $\mathcal {U}(H \times G^n)$ as the dual of $\mathcal {O}(H) \otimes \mathcal {O}(G)^{\otimes n}$ .

Corollary 2.3. For any reductive algebraic subgroup $H < G$ , the cohomology of $B_{G,H}$ is isomorphic to $(\bigwedge \mathfrak {g}/\mathfrak {h})^H$ .

Proof. As the factors $\operatorname {\mathrm {End}}(V_{\pi }) \otimes \operatorname {\mathrm {End}}(V_{\pi _1}) \otimes \dots \otimes \operatorname {\mathrm {End}}(V_{\pi _n})$ of $\tilde {B}^n_{G,H}$ decompose into direct sums of isotypical components, taking the H-invariant part commutes with taking cohomology.

We will mainly need the following particular case.

Corollary 2.4. The cohomology of $B_{G,G^{\sigma }}$ is isomorphic to $(\bigwedge \mathfrak {m}^{\mathbb {C}})^{\mathfrak {k}}$ .

Proof. This follows from the previous corollary since $G^{\sigma }$ is connected and $\mathfrak {g}^{\sigma }=\mathfrak {k}^{\mathbb {C}}$ .

Remark 2.5. Instead of the multiplier algebras we could use the universal enveloping algebras and define complexes $\tilde {B}_{\mathfrak {g},\mathfrak {h}}$ and $B_{\mathfrak {g},\mathfrak {h}}$ ; see Appendix A. The canonical maps $U(\mathfrak {g}) \to \mathcal {U}(G)$ and $U(\mathfrak {h}) \to \mathcal {U}(H)$ are injective homomorphisms compatible with the coproduct maps $\mathcal {U}(G) \to \mathcal {U}(G \times G)$ and $\mathcal {U}(H) \to \mathcal {U}(H \times H)$ . Thus, we get an inclusion $\tilde {B}_{\mathfrak {g},\mathfrak {h}}\to \tilde {B}_{G,H}$ , and if H is connected, we also get an inclusion $B_{\mathfrak {g},\mathfrak {h}}\to B_{G,H}$ . Corollary A.5 shows that these maps are quasi-isomorphisms.

2.2 Classification of associators and ribbon braids: non-Hermitian case

Assume the symmetric pair $\mathfrak {k}<\mathfrak {u}$ is non-Hermitian.

Consider a multiplier quasi-bialgebra such that $\Delta _h=\Delta $ modulo h. We claim that up to twisting by $(1,\mathcal {G})$ it has at most one quasi-coaction such that $\alpha =\Delta $ modulo h. Since by Lemma 1.1, we may assume that both $\Delta _h$ and $\alpha $ equal $\Delta $ , the following is an equivalent statement.

Theorem 2.6. Let $\mathfrak {k} = \mathfrak {u}^{\sigma } < \mathfrak {u}$ be a non-Hermitian symmetric pair, and be two associators defining quasi-coactions of , with the coaction homomorphisms $\alpha =\alpha ' = \Delta $ . Then there is an element such that $(\mathrm {id}\otimes \epsilon )(\mathcal {H})=1$ and $\Psi = \mathcal {H}_{0, 12} \Psi ' \mathcal {H}_{01, 2}^{-1} \mathcal {H}_{0,1}^{-1}$ .

Proof. Suppose that $\Psi ^{(k)} = \Psi ^{\prime (k)}$ for $k < n$ . We claim that there is $T \in \mathcal {U}(G^{\sigma } \times G)^{\mathfrak {k}}$ such that $(\mathrm {id}\otimes \epsilon )(T)=0$ and $\Psi $ and $\mathcal {H}_{0,12} \Psi ' \mathcal {H}_{01,2}^{-1} \mathcal {H}_{0,1}^{-1}$ have the same terms up to (and including) order n for $\mathcal {H} = 1 - h^n T$ . The lemma is then proved by inductively applying this claim and taking the product of the elements $1 - h^n T$ we thus get. Note only that by $\mathfrak {k}$ -invariance the elements $T_{0,1}$ and $T_{01,2}$ obtained at different steps commute with each other.

Take the difference of identities (1.3) for $\Psi $ and $\Psi '$ and consider the terms of order n. Since $\Psi $ and $\Psi '$ have the same terms up to order $n-1$ , we get

$$\begin{align*}(\Psi^{(n)}-\Psi^{\prime (n)})_{0, 12, 3} + (\Psi^{(n)}-\Psi^{\prime (n)})_{0, 1, 2}=(\Psi^{(n)}-\Psi^{\prime (n)})_{0, 1, 23} + (\Psi^{(n)}-\Psi^{\prime (n)})_{01, 2, 3}.\\[-15pt] \end{align*}$$

Since $\Psi $ and $\Psi '$ are $\mathfrak {k}$ -invariant by equation (1.2), it follows that $\Psi ^{(n)} - \Psi ^{\prime (n)}$ is a cocycle in $B_{G, G^{\sigma }}^2$ . As we are in the non-Hermitian case, by Corollary 2.4 and Lemma 1.2, we have $\Psi ^{(n)}-\Psi ^{\prime (n)}=d_{\mathrm {cH}}(T)$ for some $T\in \mathcal {U}(G^{\sigma } \times G)^{\mathfrak {k}}$ . As $(\mathrm {id}\otimes \epsilon \otimes \mathrm {id})(\Psi ^{(n)}-\Psi ^{\prime (n)})=0$ , we have $(\mathrm {id}\otimes \epsilon )(T)=0$ . Thus, T satisfies our claim.

Remark 2.7. Analogous results are true at the level of the universal enveloping algebras instead of the multiplier algebras. More precisely, given a quasi-bialgebra such that $\Delta _h=\Delta $ modulo h, up to twisting by $(1,\mathcal {G})$ there is at most one quasi-coaction of this quasi-bialgebra such that $\alpha =\Delta $ modulo h. This is proved along the same lines as Lemma 1.1 and Theorem 2.6, but now relying on Whitehead’s lemma for the semisimple Lie algebras $\mathfrak {g}$ and $\mathfrak {g}^{\sigma }$ to show that there are no nontrivial deformations of $\Delta \colon U(\mathfrak {g})\to U(\mathfrak {g})\otimes U(\mathfrak {g})$ and $\Delta \colon U(\mathfrak {g}^{\sigma })\to U(\mathfrak {g}^{\sigma })\otimes U(\mathfrak {g})$ , and using Corollary A.5 instead of Corollary 2.4.

Next, let us fix a $\sigma $ -invariant R-matrix for $(\Delta , \Phi )$ and look at compatible ribbon $\sigma $ -braids. Note that the left-hand side of equation (1.4) becomes $\mathcal {E} \Delta (T)$ in the present case.

Theorem 2.8. Let $\mathfrak {k} = \mathfrak {u}^{\sigma } < \mathfrak {u}$ be a non-Hermitian symmetric pair, and let and be $\sigma $ -invariant elements defining the structure of a quasi-triangular multiplier quasi-bialgebra . Assume further that we are given a quasi-coaction of the form by this quasi-bialgebra and that is a ribbon $\sigma $ -braid for $\mathcal {R}$ . Then $\mathcal {E}^{(0)}=1\otimes g$ for an element g in the centralizer $Z_U(K)$ , and any other ribbon $\sigma $ -braid, for the same $\Phi $ , $\Psi $ and $\mathcal {R}$ , and with the same order $0$ term, coincides with $\mathcal {E}$ . Furthermore, if $\mathcal {R}^{(1)}_{\phantom {1}}+\mathcal {R}^{(1)}_{21}\ne 0$ , then $g\in Z(U)$ .

The group $Z_U(K)$ is finite as remarked in Remark 1.3, so we have at most finitely many ribbon $\sigma $ -braids.

Proof. From equation (1.5), we get that $(\Delta \otimes \mathrm {id})(\mathcal {E}^{(0)})=\mathcal {E}_{02}^{(0)}$ . This implies that $\mathcal {E}^{(0)}=1\otimes g$ , with $g=(\epsilon \otimes \mathrm {id})(\mathcal {E}^{(0)})\in \mathcal {U}(G)$ . From equation (1.6) we then get $\Delta (g)=g\otimes g$ , hence $g\in G$ . But then equation (1.4) shows that $g\in Z_G(G^{\sigma })$ . Finally, as $\mathfrak {z}_{\mathfrak {g}}(\mathfrak {k})=\mathfrak {z}_{\mathfrak {u}}(\mathfrak {k})^{\mathbb {C}}=0$ , using the Cartan decomposition of G we see that $Z_G(K)=Z_U(K)$ , hence $g\in Z_U(K)$ .

Assume now that $\mathcal {E}'$ is another ribbon $\sigma $ -braid with ${\mathcal {E}'}{}^{(0)}=1\otimes g$ . We want to show that $\mathcal {E}'=\mathcal {E}$ . It will be convenient to first get rid of g. By multiplying both elements by $1\otimes g^{-1}$ on the right, we get new ribbon $\tilde {\sigma }$ -braids $\tilde {\mathcal {E}}$ and $\tilde {\mathcal {E}}'$ in with the order zero terms $1$ , where $\tilde {\sigma }=(\operatorname {\mathrm {Ad}} g)\circ \sigma $ .

We argue by induction on n that $\tilde {\mathcal {E}}^{(n)}=\tilde {\mathcal {E}}^{\prime (n)}$ . Suppose that we already know that $\tilde {\mathcal {E}}^{(k)} = \tilde {\mathcal {E}}^{\prime (k)}$ for $k < n$ . Comparing the terms of degree n in equations (1.5) and (1.6), we obtain

$$ \begin{align*} X_{01,2} &= X_{0,2},& X_{0,12} &= X_{0,2} + X_{0,1}\\[-15pt] \end{align*} $$

for $X = \tilde {\mathcal {E}}^{(n)} - \tilde {\mathcal {E}}^{\prime (n)}$ .

The first equality says that $X = 1 \otimes Y$ for $Y = (\epsilon \otimes \mathrm {id})(X) \in \mathcal {U}(G)$ . Then the second equality says $\Delta (Y) = Y_1 + Y_2$ , that is, Y is primitive, and we obtain $Y \in \mathfrak {g}$ .

Comparing the terms of degree n in equation (1.4), we see next that Y has to centralize $\mathfrak {k}$ . Hence $Y=0$ and $\tilde {\mathcal {E}}^{\prime (n)} = \tilde {\mathcal {E}}^{(n)}$ .

It remains to prove the last statement of the theorem. So assume $\mathcal {R}^{(1)}_{\phantom {1}}+\mathcal {R}^{(1)}_{21}\ne 0$ . Let us write $\tilde {B}_{G}$ , $B_{\mathfrak {g}}$ , etc., instead of $\tilde {B}_{G,\{e\}}$ , $B_{\mathfrak {g},0}$ .

By an analogue of [Reference DrinfeldDri89b, Proposition 3.1] for the multiplier algebras, by twisting $\Phi $ we may assume that $\Phi =1$ modulo $h^2$ . Such an analogue is proved in the same way as in [Reference DrinfeldDri89b] using that the embedding map $\tilde {B}_{\mathfrak {g}}\to \tilde {B}_{G}$ is a quasi-isomorphism by Corollary A.5.

Namely, consider the normalized skew-symmetrization map $\operatorname {\mathrm {Alt}}\colon \tilde {B}^3_{G}\to \tilde {B}^3_{G}$ . This map kills the coboundaries and transforms the cocycles of the form $X_1\otimes X_2\otimes X_3$ , $X_i\in \mathfrak {g}$ , into cohomologous ones by Remark A.4. But the classes of such cocycles span the entire space $\mathrm {H}^3(\tilde {B}_G)$ by the same remark and Corollary A.5. Therefore, if $T\in \tilde {B}^3_G$ is a cocycle killed by $\operatorname {\mathrm {Alt}}$ , then it is a coboundary. The hexagon relations imply that $\operatorname {\mathrm {Alt}}(\Phi ^{(1)})=0$ , so $\Phi ^{(1)}=d_{\mathrm {cH}}(T)$ for some $T\in \tilde {B}^2_G$ . We may assume that T is G- and $\sigma $ -invariant since $\Phi ^{(1)}$ has these invariance properties. Then the twisting by $\mathcal {F}=1-h T$ proves our claim.

Note that twisting does not change the element $\mathcal {R}^{(1)}_{\phantom {1}}+\mathcal {R}^{(1)}_{21}$ . The hexagon relations imply then that $\mathcal {R}^{(1)}\in \mathfrak {g}\otimes \mathfrak {g}$ (see the proof of [Reference DrinfeldDri89b, Proposition 3.1]), and since $\mathcal {R}$ commutes with the image of $\Delta $ , we get $\mathcal {R}^{(1)}\in (\mathfrak {g}\otimes \mathfrak {g})^{\mathfrak {g}}$ . Hence, $\mathcal {R}^{(1)}=-\lambda t^{\mathfrak {u}}$ for some $\lambda \ne 0$ , where $t^{\mathfrak {u}}$ is the normalized invariant $2$ -tensor defined by equation (1.1).

Next, identity (1.3) implies that $\Psi ^{(1)}$ is a $2$ -cocycle in $B_{G, G^{\sigma }}$ ; hence, by twisting we may assume that $\Psi =1$ modulo $h^2$ . We remind also that under twisting the ribbon twist-braids transform via formula (1.7).

Now, by looking at the first order terms in equation (1.5) for a $\tilde {\sigma }$ -braid $\tilde {\mathcal {E}}$ , with $\tilde {\mathcal {E}}^{(0)}=1$ , we get

$$\begin{align*}\tilde{\mathcal{E}}^{(1)}_{01,2}=-\lambda t^{\mathfrak{u}}_{2,1}+\tilde{\mathcal{E}}^{(1)}_{0,2}-(\mathrm{id}\otimes\mathrm{id}\otimes\tilde{\sigma})(\lambda t^{\mathfrak{u}}_{1,2}).\\[-10pt] \end{align*}$$

The tensor $t^{\mathfrak {u}}$ lies in $\mathfrak {k}\otimes \mathfrak {k}+\mathfrak {m}\otimes \mathfrak {m}$ . Denote the components of $t^{\mathfrak {u}}$ in $\mathfrak {k}\otimes \mathfrak {k}$ and $\mathfrak {m}\otimes \mathfrak {m}$ by $t^{\mathfrak {k}}$ and $t^{\mathfrak {m}}$ , resp. Then the above identity can be written as

$$\begin{align*}\tilde{\mathcal{E}}^{(1)}_{01,2}=-2\lambda t^{\mathfrak{k}}_{1,2}+\tilde{\mathcal{E}}^{(1)}_{0,2}+(\mathrm{id}\otimes\mathrm{id}\otimes\operatorname{\mathrm{Ad}} g)(\lambda t^{\mathfrak{m}}_{1,2})-\lambda t^{\mathfrak{m}}_{1,2}.\\[-10pt] \end{align*}$$

Applying $\epsilon $ to the $0$ th leg and letting $T=(\epsilon \otimes \mathrm {id})(\tilde {\mathcal {E}}^{(1)})$ , we obtain

(2.2) $$ \begin{align} \tilde{\mathcal{E}}^{(1)}=-2\lambda t^{\mathfrak{k}}+1\otimes T+(\mathrm{id}\otimes\operatorname{\mathrm{Ad}} g)(\lambda t^{\mathfrak{m}})-\lambda t^{\mathfrak{m}}.\\[-10pt]\nonumber \end{align} $$

But we must have $\tilde {\mathcal {E}}^{(1)}\in \mathcal {U}(G^{\sigma }\times G)$ . Since $t^{\mathfrak {m}}=\sum _j Y_j\otimes Y^j$ for a basis $(Y_j)_j$ in $\mathfrak {m}$ and the dual basis $(Y^j)_j$ , this is possible only when $\operatorname {\mathrm {Ad}} g$ acts trivially on $\mathfrak {m}$ . Hence, $g\in Z(U)$ .

Remark 2.9. Theorems 2.6 and 2.8 also hold for the type II symmetric pairs with appropriate modifications. Namely, consider $\tilde G = G \times G$ and its diagonal subgroup $\Delta (G) < \tilde G$ , which is the fixed point subgroup of the involution $\sigma (g, h) = (h, g)$ on $\tilde G$ . Then for the quasi-coactions of on the multiplier algebra , with the coaction map extending $G \ni g \mapsto (g, g, g) \in G \times \tilde G$ , and with associators and ribbon $\sigma $ -braids , one can easily prove analogues of these theorems. First, the proof of Theorem 2.6 carries over almost without a change. Indeed, its proof relies on Lemma 1.2 and Corollary 2.4, both of which have analogues for $G \simeq \Delta (G) < \tilde G$ . As for Theorem 2.8, we have $\mathfrak {z}_{\tilde {\mathfrak {g}}}(\mathfrak {g}) = 0$ for the diagonal inclusion $\mathfrak {g} < \tilde {\mathfrak {g}}$ , and $Z_{\tilde G}(G) = Z(U) \times Z(U)$ , which is enough to adapt the first half of the proof.

2.3 Associators from cyclotomic KZ equations

We want to extend the results of the previous subsection to the Hermitian case. Since $\mathrm {H}^2(B_{G,G^{\sigma }})$ is now one-dimensional by Lemma 1.2, we should expect a one-parameter family of nonequivalent associators. In this subsection, we define a candidate for such a family arising from the cyclotomic KZ-equations.

Thus, assume $\mathfrak {k}<\mathfrak {u}$ is a Hermitian symmetric pair. We have an element $Z\in \mathfrak {z}(\mathfrak {k})$ , unique up to a sign, such that

$$\begin{align*}(Z,Z)_{\mathfrak{g}}=-a_{\sigma}^{-2}. \end{align*}$$

This normalization is equivalent to $(\operatorname {\mathrm {ad}} Z)^2=-1$ on $\mathfrak {m}$ by Lemma 1.4. We fix such Z for the rest of this section. The operator $\operatorname {\mathrm {ad}} Z$ has eigenvalues $\pm i$ on $\mathfrak {m}^{\mathbb {C}}$ . Denote by $\mathfrak {m}_{\pm }\subset \mathfrak {m}^{\mathbb {C}}$ the corresponding eigenspaces.

We remind that we denote the components of the normalized invariant $2$ -tensor $t^{\mathfrak {u}}$ in $\mathfrak {k}\otimes \mathfrak {k}$ and $\mathfrak {m}\otimes \mathfrak {m}$ by $t^{\mathfrak {k}}$ and $t^{\mathfrak {m}}$ , resp. The tensor $t^{\mathfrak {m}}$ lies in $\mathfrak {m}_+\otimes \mathfrak {m}_- + \mathfrak {m}_-\otimes \mathfrak {m}_+$ . We denote the components of $t^{\mathfrak {m}}$ in $\mathfrak {m}_{\pm }\otimes \mathfrak {m}_{\mp }$ by $t^{\mathfrak {m}_{\pm }}$ . We thus have

(2.3) $$ \begin{align} t^{\mathfrak{m}} &= t^{\mathfrak{m}_+}+t^{\mathfrak{m}_-},& (\operatorname{\mathrm{ad}} Z \otimes \mathrm{id})(t^{\mathfrak{m}_{\pm}}) &= \pm i t^{\mathfrak{m}_{\pm}},& (\mathrm{id}\otimes\operatorname{\mathrm{ad}} Z)(t^{\mathfrak{m}_{\pm}}) &= \mp i t^{\mathfrak{m}_{\pm}}. \end{align} $$

Given $s \in \mathbb {C}$ , consider the following elements of :

(2.4) $$ \begin{align} A_{-1} &= \hbar(t^{\mathfrak{k}}_{12} - t^{\mathfrak{m}}_{12}),& A_1 &= \hbar t^{\mathfrak{u}}_{12},& A_0 &= \hbar(2 t^{\mathfrak{k}}_{01} + C^{\mathfrak{k}}_1) + s Z_1, \end{align} $$

where $C^{\mathfrak {k}}$ is the Casimir element of $\mathfrak {k}$ , the image of $t^{\mathfrak {k}}$ under the product map $U(\mathfrak {k})\otimes U(\mathfrak {k})\to U(\mathfrak {k})$ . These lead to the shifted modified $2$ -cyclotomic KZ $_2$ -equation [Reference Enriquez and EtingofEE05Reference De Commer, Neshveyev, Tuset and YamashitaDCNTY19]

(2.5) $$ \begin{align} G'(w) = \left( \frac{A_{-1}}{w + 1} + \frac{A_1}{w-1} + \frac{A_0}{w} \right) G(w). \end{align} $$

Remark 2.10. Consider a -valued character $\nu $ on $U(\mathfrak {k})$ such that $\nu (Z) = -(2 \hbar a_{\sigma }^2)^{-1}s$ . Then the slicing map $\varsigma _{\nu } = (\nu \otimes \mathrm {id})\Delta $ is an algebra homomorphism satisfying

$$\begin{align*}(\varsigma_{\nu} \otimes \mathrm{id})(2t^{\mathfrak{k}}) = 2 t^{\mathfrak{k}} + \hbar^{-1}(1 \otimes s Z) \end{align*}$$

and commuting with the right coaction $\Delta $ by $U(\mathfrak {g})$ . In particular, at least formally speaking, equation (2.5) is obtained from the case $s = 0$ by slicing. But since $\nu $ cannot be extended to , one should be careful with this construction.

The normalized monodromy of equation (2.5) from $w = 0$ to $w = 1$ is well-defined as long as the operator $\operatorname {\mathrm {ad}}(s Z)$ on $\mathcal {U}(G)$ does not have positive integers in its spectrum, cf. [Reference Neshveyev and TusetNT11, Proposition 3.1]. Since each matrix block $\operatorname {\mathrm {End}}(V_{\pi })$ in $\mathcal {U}(G)$ is generated by the image of $\mathfrak {g}$ , the eigenvalues of $\operatorname {\mathrm {ad}} Z$ are $i n$ for $n \in \mathbb {Z}$ by Lemma 1.4 and our choice of normalization. Therefore, $\Psi _{\mathrm {KZ},s}$ is well defined for all $s\not \in i\mathbb {Q}^{\times }$ . The element $\Psi _{\mathrm {KZ},s}$ together with the coproduct $\Delta \colon \mathcal {U}(G^{\sigma })\to \mathcal {U}(G^{\sigma }\times G)$ gives a quasi-coaction of on , where is Drinfeld’s KZ-associator for G.

In more detail, $\Psi _{\mathrm {KZ},s}$ is defined as follows. Under our restrictions on s, a standard argument (see, e.g., [Reference Neshveyev and TusetNT11, Proposition 3.1]) shows that there is a unique -valued solution $G_0$ of equation (2.5) on $(0,1)$ such that $G_0(w)w^{-A_0}$ extends to an analytic function in the unit disc with value $1$ at $w=0$ . Similarly, there is a unique solution $G_1$ of equation (2.5) such that $G_1(1-w)w^{-A_1}$ extends to an analytic function in the unit disc with value $1$ at $w=0$ . Then

$$\begin{align*}\Psi_{\mathrm{KZ},s}=G_1(w)^{-1}G_0(w) \end{align*}$$

for any $0<w<1$ . We can also write this as

(2.6) $$ \begin{align} \Psi_{\mathrm{KZ},s}=\lim_{w\to 1}(1-w)^{-A_1}G_0(w)=\lim_{w\to 1}(1-w)^{-A_1}G_0(w)w^{-A_0}. \end{align} $$

The case $s=0$ is special: In this case, it can be shown, using for example iterated integrals [Reference KasselKas95, Chapter XIX][Reference EnriquezEnr07], that $\Psi _{\mathrm {KZ},0}$ lives in the algebra rather than in its completion . Note also that this associator is well defined in the non-Hermitian case as well. We will denote it by $\Psi _{\mathrm {KZ}}$ .

Observe also that if $s\in \mathbb {R}$ , then $G_0$ is unitary; hence, $\Psi _{\mathrm {KZ},s}$ is unitary as well. Indeed, in this case $(G_0(w)^*)^{-1}$ has the defining properties of $G_0(w)$ , hence coincides with it.

Proposition 2.11 (cf. [Reference Enriquez and EtingofEE05, Proposition 4.7])

For every $s\not \in i\mathbb {Q}^{\times }$ , we have

$$\begin{align*}\Psi_{\mathrm{KZ},s}= 1 + \frac{h}{\pi i}\left((\log 2)t^{\mathfrak{u}}_{12}+\gamma t^{\mathfrak{m}}_{12}+ \psi\Big(\frac{1}{2}-\frac{is}{2}\Big)t^{\mathfrak{m}_+}_{12} + \psi\Big(\frac{1}{2}+\frac{is}{2}\Big)t^{\mathfrak{m}_-}_{12}\right)+ O(h^2), \end{align*}$$

where $\gamma $ is Euler’s constant and $\displaystyle \psi =\frac {\Gamma '}{\Gamma }$ is the digamma function.

Proof. If we restrict to a finite-dimensional block of $\mathcal {U}(G^{\sigma }\times G^2)$ , then $\operatorname {\mathrm {ad}}(s Z_1)$ has a finite number of eigenvalues there, so the corresponding component of $\Psi _{\mathrm {KZ},s}$ is well defined for all $s\not \in i N^{-1}\mathbb {Z}^{\times }$ for some N. As it is analytic in s in this domain, it therefore suffices to consider real s.

Put $H_0(w)=G_0(w)w^{-A_0}$ . Then $H_0$ satisfies the differential equation

$$\begin{align*}H^{\prime}_0(w)=\left( \frac{A_{-1}}{w + 1} + \frac{A_1}{w-1} \right)H_0(w)+\left[\frac{A_0}{w},H_0(w)\right] \end{align*}$$

and the initial condition $H_0(0)=1$ , and by equation (2.6) we have

$$\begin{align*}\Psi_{\mathrm{KZ},s}=\lim_{w\to 1}(1-w)^{-A_1}H_0(w). \end{align*}$$

Consider the expansion in h. For the order zero terms, we immediately get $\Psi ^{(0)}_{\mathrm {KZ},s}=H^{(0)}_0=1$ . Next, consider the order one terms. Let us write H for $\pi i H^{(1)}_0$ , so that $H_0=1+\hbar H+O(h^2)$ . Then

$$\begin{align*}H'(w)=\left( \frac{t^{\mathfrak{k}}_{12} - t^{\mathfrak{m}}_{12}}{w + 1} + \frac{t^{\mathfrak{u}}_{12}}{w-1} \right)+\left[\frac{s Z_1}{w},H(w)\right] \end{align*}$$

and $H(0)=0$ , while

$$\begin{align*}\pi i\Psi_{\mathrm{KZ},s}^{(1)}=\lim_{w\to1}(H(w)-\mathinner{\log(1-w)} t^{\mathfrak{u}}_{12}). \end{align*}$$

By equation (2.3), we have

$$ \begin{align*} H(w)&=\int^w_0\left(\frac{w}{u}\right)^{\operatorname{\mathrm{ad}} (s Z_1)}\left( \frac{t^{\mathfrak{k}}_{12} - t^{\mathfrak{m}}_{12}}{u + 1} + \frac{t^{\mathfrak{u}}_{12}}{u-1} \right) d u \\ &\ = \int^w_0 \left( t^{\mathfrak{k}}_{12} \Big(\frac1{u+1} + \frac1{u-1} \Big) + \Big( \left( \frac{w}{u} \right)^{i s} t^{\mathfrak{m}_+}_{12} + \left( \frac{w}{u} \right)^{-i s} t^{\mathfrak{m}_-}_{12} \Big) \Big( \frac{-1}{u+1} + \frac1{u-1} \Big) \right)d u. \end{align*} $$

Note that this integral is well defined for $0\le w<1$ as s is assumed to be real. We then get

$$\begin{align*}\pi i\Psi_{\mathrm{KZ},s}^{(1)}=b t^{\mathfrak{k}}_{12}+c(s)t^{\mathfrak{m}_+}_{12}+c(-s)t^{\mathfrak{m}_-}_{12}, \end{align*}$$

where

$$ \begin{align*} b&=\lim_{w\to 1}\left(\int^w_0\left( \frac{1}{u + 1} + \frac{1}{u-1} \right) d u-\log(1-w)\right)=\log 2,\\ c(s)&=\lim_{w\to 1}\left(\int^w_0\left(\frac{w}{u}\right)^{is}\left( \frac{1}{u - 1} - \frac{1}{u+1} \right) d u-\log(1-w)\right). \end{align*} $$

To compute $c(s)$ , we write $(u^2-1)^{-1}$ as a power series, integrate and get

$$\begin{align*}c(s)=\lim_{w\to 1}\bigg(-\sum^{\infty}_{n=0}\frac{w^{2n+1}}{n+\frac{1}{2}-\frac{is}{2}}-\log(1-w)\bigg). \end{align*}$$

Together with the Taylor expansion of $w^{-1}\log (1-w^2)$ and the standard formula

$$\begin{align*}\psi(z)+\gamma=\sum^{\infty}_{n=0}\left(\frac{1}{n+1}-\frac{1}{n+z}\right), \end{align*}$$

this gives

$$\begin{align*}c(s)=\psi\Big(\frac{1}{2}-\frac{is}{2}\Big)+\gamma+\lim_{w\to 1}(w^{-1}\log(1-w^2)-\log(1-w)) =\psi\Big(\frac{1}{2}-\frac{is}{2}\Big)+\gamma+\log2, \end{align*}$$

which completes the proof of the proposition.

Using the formula $\psi (1-z)-\psi (z)=\pi \cot (\pi z)$ , it will be convenient to rewrite the result as

(2.7) $$ \begin{align} \Psi_{\mathrm{KZ},s} &= 1 + \frac{h}{\pi i}\bigg((\log 2)t^{\mathfrak{u}}_{12}+\Big(\gamma+\frac{\psi\Big(\frac{1}{2}-\frac{is}{2}\Big)+ \psi\Big(\frac{1}{2}+\frac{is}{2}\Big)}{2}\Big)t^{\mathfrak{m}}_{12}\notag \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\frac{\pi i}{2}\tanh\Big(\frac{\pi s}{2}\Big)\Big(t^{\mathfrak{m}_+}_{12}-t^{\mathfrak{m}_-}_{12}\Big)\bigg)+ O(h^2). \end{align} $$

Now, take . Replacing $s\not \in i\mathbb {Q}^{\times }$ by $s+\mu $ in equation (2.4), we can construct yet another associator, which we denote by $\Psi _{\mathrm {KZ},s;\mu }$ . If $s\in \mathbb {R}$ and , then $\Psi _{\mathrm {KZ},s;\mu }$ is unitary for the same reason as for $\Psi _{\mathrm {KZ},s}$ .

Remark 2.12. Similarly to Remark 2.10, $\Psi _{\mathrm {KZ},s;\mu }$ could be obtained from $\Psi _{\mathrm {KZ},s}$ by slicing by a character $\nu $ of $U(\mathfrak {k})$ satisfying $\nu (Z) = -(2\hbar a_{\sigma }^2)^{-1}\mu $ . Since such a character does not always extend to $\mathcal {U}(G^{\sigma })$ , to make sense of this we should have allowed in the construction of $\Psi _{\mathrm {KZ},s}$ arbitrary finite-dimensional representations of $\mathfrak {g}^{\sigma }$ instead of those in $\operatorname {\mathrm {Rep}} G^{\sigma }$ . Alternatively, with $s\not \in i\mathbb {R}$ fixed, both $\Psi _{\mathrm {KZ},s;\mu }$ and $\Psi _{\mathrm {KZ},s+z}$ for small z are specializations of an associator in constructed by treating $\mu $ as a second formal parameter. But this implies that $\Psi _{\mathrm {KZ},s;\mu }$ is obtained from the Taylor expansion of $\Psi _{\mathrm {KZ},s+z}$ at $z=0$ by simply taking $\mu $ as the argument:

(2.8) $$ \begin{align} \Psi_{\mathrm{KZ},s;\mu}=\sum^{\infty}_{k=0}\frac{\mu^k}{k!}\frac{d^k\Psi_{\mathrm{KZ},s}}{d s^k}=\sum_{n,k=0}^{\infty}\frac{h^n\mu^k}{k!}\frac{d^k\Psi^{(n)}_{\mathrm{KZ},s}}{d s^k}. \end{align} $$

This also works for $s\in i(\mathbb {R}\setminus \mathbb {Q}^{\times })$ if we consider only the components of $\Psi _{\mathrm {KZ},s;\mu }$ in finite-dimensional blocks of $\mathcal {U}(G^{\sigma }\times G^2)$ , which are well defined and analytic in a neighborhood of s.

Corollary 2.13. For all $s\not \in i\mathbb {Q}^{\times }$ and , we have

$$ \begin{align*} \Psi_{\mathrm{KZ},s;\mu + \nu} - \Psi_{\mathrm{KZ},s; \mu} &= h^{1+\operatorname{\mathrm{ord}}(\nu)}\nu^{(\operatorname{\mathrm{ord}}(\nu))}\bigg(\frac{1}{4\pi}\Big(\psi'\Big(\frac{1}{2}+\frac{is}{2}\Big)- \psi'\Big(\frac{1}{2}-\frac{is}{2}\Big)\Big)t^{\mathfrak{m}}_{12}\\ &\qquad\qquad\qquad\qquad\qquad -\frac{\pi}{4}\operatorname{\mathrm{sech}}^2\Big(\frac{\pi s}{2}\Big)\Big(t^{\mathfrak{m}_+}_{12}-t^{\mathfrak{m}_-}_{12}\Big)\bigg)+ O(h^{2 + \operatorname{\mathrm{ord}}(\nu)}). \end{align*} $$

Proof. By equation (2.8), we have

$$\begin{align*}\Psi_{\mathrm{KZ},s;\mu + \nu} - \Psi_{\mathrm{KZ},s; \mu} = h^{1+\operatorname{\mathrm{ord}}(\nu)}\nu^{(\operatorname{\mathrm{ord}}(\nu))}\frac{d\Psi^{(1)}_{\mathrm{KZ},s}}{d s}+ O(h^{2 + \operatorname{\mathrm{ord}}(\nu)}). \end{align*}$$

Hence, the result follows from equation (2.7).

2.4 Detecting co-Hochschild classes

To see that the associators $\Psi _{\mathrm {KZ},s;\mu }$ are not all equivalent, we need to see that a perturbation of the parameter $\mu $ gives rise to a nontrivial $2$ -cocycle in $B_{G,G^{\sigma }}$ . We can actually see that this is the case from results in Appendix A, but let us present a concrete cycle to detect this.

Consider the tensor

(2.9) $$ \begin{align} \Omega = [t^{\mathfrak{u}}_{12}, t^{\mathfrak{u}}_{13}] = \sum_{i, j} [X_i, X_j] \otimes X^i \otimes X^j\in \Big({\textstyle\bigwedge}^3 \mathfrak{u}\Big)^{\mathfrak{u}} \end{align} $$

with $(X_i)_i$ and $(X^i)_i$ as in equation (1.1). Every element $X\in \mathfrak {g}$ defines a function on G such that $g\mapsto (X,(\operatorname {\mathrm {Ad}} g)(Z))_{\mathfrak {g}}$ . This way $\Omega $ defines an element of $\mathcal {O}(G^{\sigma })\otimes \mathcal {O}(G)\otimes \mathcal {O}(G)$ , which by slightly abusing notation we continue to denote by $\Omega $ . Thus, for $(g,h,k)\in G^{\sigma }\times G\times G$ ,

$$\begin{align*}\Omega(g,h,k)=\Big([(\operatorname{\mathrm{Ad}} h)(Z),(\operatorname{\mathrm{Ad}} k)(Z)],(\operatorname{\mathrm{Ad}} g)(Z)\Big)_{\mathfrak{g}}=\Big([(\operatorname{\mathrm{Ad}} h)(Z),(\operatorname{\mathrm{Ad}} k)(Z)],Z\Big)_{\mathfrak{g}} \end{align*}$$

since $G^{\sigma }$ stabilizes Z. This is a $2$ -cycle in the complex $\tilde B^{\prime }_{G,G^{\sigma }}$ from the proof of Proposition 2.1, as

$$\begin{align*}\Omega(g,g,h)-\Omega(g,h,h)+\Omega(g,h,e)=0 \end{align*}$$

for all $(g,h)\in G^{\sigma }\times G$ . Hence, the map $\langle \Omega ,\cdot \rangle \colon \mathcal {U}(G^{\sigma }\times G^2)\to \mathbb {C}$ defined by pairing with $\Omega $ passes to $\mathrm {H}^2(B_{G,G^{\sigma }})$ . Explicitly, for $T\in \mathcal {U}(G^{\sigma }\times G^2)$ , we have

(2.10) $$ \begin{align} \langle \Omega,T \rangle=\epsilon(T_0)\Big([(\operatorname{\mathrm{ad}} T_1)(Z),(\operatorname{\mathrm{ad}} T_2)(Z)],Z\Big)_{\mathfrak{g}}, \end{align} $$

where $\operatorname {\mathrm {ad}}$ denotes the extension of the adjoint representation of $\mathfrak {g}$ to $\mathcal {U}(G)$ .

Proposition 2.14. The elements $t^{\mathfrak {k}}_{12}$ , $t^{\mathfrak {m}_{\pm }}_{12}$ are $2$ -cocycles in $B_{G,G^{\sigma }}$ . Furthermore, $t^{\mathfrak {k}}_{12}$ and $t^{\mathfrak {m}_{\phantom {-}}}_{12}=t^{\mathfrak {m}_+}_{12}+t^{\mathfrak {m}_-}_{12}$ are coboundaries, while

$$\begin{align*}\langle\Omega,t^{\mathfrak{m}_{\pm}}_{12}\rangle=\pm \frac{i}{2}\dim \mathfrak{m}. \end{align*}$$

In particular, $t^{\mathfrak {m}_+}_{12}$ and $-t^{\mathfrak {m}_-}_{12}$ represent the same nontrivial class in $\mathrm {H}^2(B_{G,G^{\sigma }})$ .

Proof. It is easy to check that $d_{\mathrm {cH}}(1\otimes X\otimes Y)=0$ for all $X,Y\in \mathfrak {g}$ . As $t^{\mathfrak {k}}_{12}$ and $t^{\mathfrak {m}_{\pm }}_{12}$ are $\mathfrak {k}$ -invariant, they are therefore $2$ -cocycles in $B_{G,G^{\sigma }}$ .

We have

$$\begin{align*}d_{\mathrm{cH}}(C^{\mathfrak{k}}_1) = C^{\mathfrak{k}}_2 - \Delta(C^{\mathfrak{k}})_{12} + C^{\mathfrak{k}}_1 = -2 t^{\mathfrak{k}}_{12}, \end{align*}$$

so $t^{\mathfrak {k}}_{12}$ is a coboundary. Similarly, $d_{\mathrm {cH}}(C^{\mathfrak {u}}_1) = -2 t^{\mathfrak {u}}_{12}$ so that $ t^{\mathfrak {u}}_{12}$ is also a coboundary, and hence $t^{\mathfrak {m}}_{12}=t^{\mathfrak {u}}_{12}-t^{\mathfrak {k}}_{12}$ is a coboundary as well.

Next, take a basis $(Y_j)_j$ in $\mathfrak {m}_+$ and the dual basis $(Y^j)_j$ in $\mathfrak {m}_-$ . Using that $\operatorname {\mathrm {ad}} Z$ acts by the scalar $\pm i$ on $\mathfrak {m}_{\pm }$ , we then compute:

$$ \begin{align*} \langle \Omega,t^{\mathfrak{m}_+}_{12}\rangle&=\sum_j \Big( [(\operatorname{\mathrm{ad}} Y_j)(Z), (\operatorname{\mathrm{ad}} Y^j)(Z)], Z \Big)_{\mathfrak{g}}=\sum_j ([Y_j, Y^j],Z)_{\mathfrak{g}} =\sum_j (Y^j,[Z,Y_j])_{\mathfrak{g}}\\ &=i\sum_j(Y^j,Y_j)_{\mathfrak{g}}=i\dim_{\mathbb{C}}\mathfrak{m}_+=\frac{i}{2}\dim_{\mathbb{R}} \mathfrak{m}. \end{align*} $$

The value $\langle \Omega ,t^{\mathfrak {m}_-}_{12}\rangle $ is obtained similarly, but it also follows from the above, as $t^{\mathfrak {m}_-}_{12}=t^{\mathfrak {m}_{\phantom {-}}}_{12}-t^{\mathfrak {m}_+}_{12}$ and $t^{\mathfrak {m}_{\phantom {-}}}_{12}$ is a coboundary.

Remark 2.15. Let us give a different perspective on the above pairing and its nontriviality.

We can view the tensor (2.9) also as a function on $(U/K)^3$ in the same way as above. Let us call this function $\omega $ . Then it is again easy to check that $\omega $ is a $2$ -cycle in the Hochschild chain complex $(C_n(A,A)=A^{\otimes (n+1)},b)$ for $A=\mathcal {O}(U)^{\mathfrak {k}}\subset C(U/K)$ . Under the Hochschild–Kostant–Rosenberg map, this cycle corresponds to the differential $2$ -form associated with the Kostant–Kirillov–Souriau bracket on the coadjoint orbit of $(\cdot ,Z)_{\mathfrak {g}}$ , which in turn defines a nonzero class in $\mathrm {H}^2(U/K;\mathbb {C})\cong \mathbb {C}$ .

We have a left $\mathcal {U}(G)$ -module structure on $\mathcal {O}(U)$ given by right translations: $T.a=a_{(0)}\langle a_{(1)},T\rangle $ . Given $T\in B^n_{G,G^{\sigma }}$ , we can then define an n-cocycle $D_T$ in the Hochschild cochain complex $(C^n(A,A)=\operatorname {\mathrm {Hom}}(A^{\otimes n},A),\delta )$ by

$$\begin{align*}D_T(a_1, \dots, a_n) = \epsilon(T_0)(T_1.a_1)\cdots (T_n.a_n). \end{align*}$$

The Hochschild cochains act on the chains by contractions: Given $D \in C^m(A, A)$ , we have

$$\begin{align*}i_D \colon C_n(A,A) \to C_{n-m}(A,A), \quad a_0\otimes \dots \otimes a_n \mapsto a_0 D(a_1, \dots, a_{m})\otimes a_{m+1}\otimes \dots \otimes a_n, \end{align*}$$

with the convention $i_D = 0$ if $n < m$ .

Now, if $T\in B^n_{G,G^{\sigma }}$ and $c\in C_n(A,A)$ is U-invariant (with respect to left translations), then $i_{D_T}c\in A$ is U-invariant, hence a scalar. It can be checked that if $b c=0$ , then this scalar depends only on the cohomology class of T. Taking $c=\omega $ , we recover pairing (2.10): $i_{D_T}\omega =\langle \Omega ,T\rangle $ .

2.5 Classification of associators: Hermitian case

We are now ready to establish, in the Hermitian case, a universality result for the associators $\Psi _{\mathrm {KZ},s;\mu }$ for generic quasi-coactions of such that $\alpha =\Delta $ modulo h. Similarly to Section 2.2, it suffices to consider the case $\alpha =\Delta $ .

Theorem 2.16. Let $\mathfrak {k} = \mathfrak {u}^{\sigma } <\mathfrak {u}$ be a Hermitian symmetric pair. Assume we are given a quasi-coaction of such that the number $\langle \Omega ,\Psi ^{(1)}\rangle $ defined by equation (2.10) is

$$\begin{align*}\text{neither}\quad \pm\frac{i}{2}\dim\mathfrak{m}, \quad \text{nor}\quad \frac{i(\zeta-1)}{2(\zeta+1)}\dim\mathfrak{m} \end{align*}$$

for a root of unity $\zeta \ne \pm 1$ . Then there exist $s \not \in i\mathbb {Q}^{\times }$ , and such that $(\mathrm {id}\otimes \epsilon )(\mathcal {H})=1$ and $\Psi _{\mathrm {KZ},s;\mu } =\mathcal {H}_{0, 12}\Psi \mathcal {H}_{01, 2}^{-1} \mathcal {H}_{0,1}^{-1}$ . Furthermore,

  1. (i) The number s is unique up to adding $2i k$ ( $k\in \mathbb {Z}$ ), and once s is fixed, the element $\mu $ is uniquely determined;

  2. (ii) We can choose $s\in \mathbb {R}$ if and only if $\langle \Omega ,\Psi ^{(1)}\rangle $ is a purely imaginary number in the interval

    $$\begin{align*}\left(-\frac{i}{2}\dim\mathfrak{m},\frac{i}{2}\dim\mathfrak{m}\right); \end{align*}$$
  3. (iii) If $s\in \mathbb {R}$ and $\Psi $ is unitary, then and $\mathcal {H}$ can be chosen to be unitary.

We will later (Remark 2.20) slightly improve this result by showing that for any $\Psi $ the parameter $\mu $ is independent of the choice of s.

Proof. By Proposition 2.14 and our restrictions on $\Psi $ , we can choose $s \not \in i\mathbb {Q}^{\times }$ such that

(2.11) $$ \begin{align} -\frac{1}{2}\tanh\Big(\frac{\pi s}{2}\Big) \langle \Omega, t^{\mathfrak{m}_+}_{12}-t^{\mathfrak{m}_-}_{12}\rangle =-\frac{i}{2}\tanh\Big(\frac{\pi s}{2}\Big)\dim\mathfrak{m} = \langle \Omega,\Psi^{(1)}\rangle. \end{align} $$

We then start with $\mathcal {H}=1$ and $\mu =0$ and modify them by induction on n to have $\Psi _{\mathrm {KZ},s;\mu } =\mathcal {H}_{0, 12}\Psi \mathcal {H}_{01, 2}^{-1} \mathcal {H}_{0,1}^{-1}$ modulo $h^{n+1}$ .

Consider $n=1$ . By the proof of Theorem 2.6, $\Psi _{\mathrm {KZ},s;\mu }^{(1)}-\Psi ^{(1)}$ is a $2$ -cocycle in $B_{G,G^{\sigma }}$ . By Lemma 1.2 and Corollary 2.4, we have $\dim \mathrm {H}^2(B_{G,G^{\sigma }})=1$ . Hence, our choice of s, identity (2.7) and Proposition 2.14 imply that $\Psi _{\mathrm {KZ},s;\mu }^{(1)}-\Psi ^{(1)}$ is a coboundary so that $\Psi ^{(1)}-\Psi ^{(1)}_{\mathrm {KZ},s;\mu }=d_{\mathrm {cH}}(T)$ for some $T\in \mathcal {U}(G^{\sigma }\times G)^{\mathfrak {k}}$ . Letting $\mathcal {H}^{(1)}=T$ , we then get $\Psi _{\mathrm {KZ},s;\mu } = \mathcal {H}_{0, 12} \Psi \mathcal {H}_{01, 2}^{-1} \mathcal {H}_{0,1}^{-1}$ modulo $h^2$ .

For the induction step, assume we have $\Psi _{\mathrm {KZ},s;\mu } =\mathcal {H}_{0, 12}\Psi \mathcal {H}_{01, 2}^{-1} \mathcal {H}_{0,1}^{-1}$ modulo $h^{n+1}$ for some $n\ge 1$ . Then, again by the proof of Theorem 2.6,

$$\begin{align*}\Psi_{\mathrm{KZ},s;\mu}^{(n+1)}-(\mathcal{H}_{0, 12}\Psi \mathcal{H}_{01, 2}^{-1} \mathcal{H}_{0,1}^{-1})^{(n+1)} \end{align*}$$

is a $2$ -cocycle in $B_{G,G^{\sigma }}$ . On the other hand, by Corollary 2.13, for any $a\in \mathbb {C}$ , we have

$$\begin{align*}\Psi_{\mathrm{KZ},s;\mu+h^n a}^{(n+1)}-\Psi_{\mathrm{KZ},s;\mu}^{(n+1)}=-a\frac{\pi}{4}\operatorname{\mathrm{sech}}^2\Big(\frac{\pi s}{2}\Big)\Big(t^{\mathfrak{m}_+}_{12}-t^{\mathfrak{m}_-}_{12}\Big)+b t^{\mathfrak{m}}_{12} \end{align*}$$

for some $b\in \mathbb {C}$ , and $\Psi _{\mathrm {KZ},s;\mu +h^n a}^{(k)}=\Psi _{\mathrm {KZ},s;\mu }^{(k)}$ for $k\le n$ . As $t^{\mathfrak {m}_+}_{12}-t^{\mathfrak {m}_-}_{12}$ represents a nontrivial cohomology class, the value of $\operatorname {\mathrm {sech}}$ is nonzero for our s and $t^{\mathfrak {m}}_{12}$ is cohomologically trivial, we see that with different choices of a the above difference can represent arbitrary classes in $\mathrm {H}^2(B_{G,G^{\sigma }})\cong \mathbb {C}$ . In particular, we can find $a\in \mathbb {C}$ such that

(2.12) $$ \begin{align} (\mathcal{H}_{0, 12}\Psi \mathcal{H}_{01, 2}^{-1} \mathcal{H}_{0,1}^{-1})^{(n+1)}-\Psi_{\mathrm{KZ},s;\mu+h^n a}^{(n+1)}=d_{\mathrm{cH}}(T) \end{align} $$

for some $T\in \mathcal {U}(G^{\sigma }\times G)^{\mathfrak {k}}$ . Replacing $\mathcal {H}$ by $(1+h^{n+1}T)\mathcal {H}$ and $\mu ^{(n)}$ by $\mu ^{(n)}+a$ , we then get $\Psi _{\mathrm {KZ},s;\mu } =\mathcal {H}_{0, 12}\Psi \mathcal {H}_{01, 2}^{-1} \mathcal {H}_{0,1}^{-1}$ modulo $h^{n+2}$ , proving the induction step.

As at the step n of our induction process we only modify $\mu ^{(n-1)}$ and $\mathcal {H}^{(k)}$ for $k\ge n$ , in the limit we get the required $\mu $ and $\mathcal {H}$ . It remains to prove (i)–(iii).

(iii): Assume $s\in \mathbb {R}$ and that $\Psi $ is unitary. In this case, we slightly modify the above inductive procedure to make sure that at every step we have unitarity of $\mathcal {H}$ and that .

Consider $n=1$ . We found $\mathcal {H}$ such that $\Psi _{\mathrm {KZ},s;\mu } = \mathcal {H}_{0, 12} \Psi \mathcal {H}_{01, 2}^{-1} \mathcal {H}_{0,1}^{-1}$ modulo $h^2$ . The unitarity of $\Psi _{\mathrm {KZ},s;\mu }$ and $\Psi $ implies then that the same identity holds for the unitary $\mathcal {H}(\mathcal {H}^*\mathcal {H})^{-1/2}$ instead of $\mathcal {H}$ , cf. the proof of [Reference Neshveyev and TusetNT11, Proposition 2.3].

For the induction step, we assume that we have $\Psi _{\mathrm {KZ},s;\mu } =\mathcal {H}_{0, 12}\Psi \mathcal {H}_{01, 2}^{-1} \mathcal {H}_{0,1}^{-1}$ modulo $h^{n+1}$ for some $n\ge 1$ , and unitary $\mathcal {H}$ . Then we take the unique $a\in \mathbb {C}$ such that

$$\begin{align*}(\mathcal{H}_{0, 12}\Psi \mathcal{H}_{01, 2}^{-1} \mathcal{H}_{0,1}^{-1})^{(n+1)}-\Psi_{\mathrm{KZ},s;\mu}^{(n+1)}+a\frac{\pi}{4}\operatorname{\mathrm{sech}}^2 \Big(\frac{\pi s}{2}\Big)\Big(t^{\mathfrak{m}_+}_{12}-t^{\mathfrak{m}_-}_{12}\Big) \end{align*}$$

is a coboundary. By taking adjoints and using that $(t^{\mathfrak {m}_+})^*=t^{\mathfrak {m}_-}$ , we also get that

$$\begin{align*}\Big((\mathcal{H}_{0, 12}\Psi \mathcal{H}_{01, 2}^{-1} \mathcal{H}_{0,1}^{-1})^{(n+1)}-\Psi_{\mathrm{KZ},s;\mu}^{(n+1)}\Big)^*-\bar a\frac{\pi}{4}\operatorname{\mathrm{sech}}^2\Big(\frac{\pi s}{2}\Big)\Big(t^{\mathfrak{m}_+}_{12}-t^{\mathfrak{m}_-}_{12}\Big) \end{align*}$$

is a coboundary. Hence, in order to conclude that