Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik-Zamolodchikov equations and Letzter-Kolb coideals
Kenny De Commer, Sergey Neshveyev, Lars Tuset, Makoto Yamashita
CCOMPARISON OF QUANTIZATIONS OF SYMMETRIC SPACES: CYCLOTOMICKNIZHNIK–ZAMOLODCHIKOV EQUATIONS AND LETZTER–KOLB COIDEALS
KENNY DE COMMER, SERGEY NESHVEYEV, LARS TUSET, AND MAKOTO YAMASHITA
Abstract.
We establish an equivalence between two approaches to quantization of irreducible symmetricspaces of compact type within the framework of quasi-coactions, one based on the Enriquez–Etingofcyclotomic Knizhnik–Zamolodchikov (KZ) equations and the other on the Letzter–Kolb coideals. Thisequivalence can be upgraded to that of ribbon braided quasi-coactions, and then the associated reflectionoperators ( K -matrices) become a tangible invariant of the quantization. As an application we obtain aKohno–Drinfeld type theorem on type 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-Hermitiansymmetric spaces are significantly different. In particular, in the latter case a quasi-coaction is essentiallyunique, while in the former we show that there is a one-parameter family of mutually nonequivalentquasi-coactions. Contents
Introduction 21. Preliminaries 61.1. Conventions 61.2. Simple Lie groups 71.3. Multiplier algebras 71.4. Quasi-coactions and ribbon twist-braids 81.5. Twisting 81.6. Symmetric pairs 92. Classification of quasi-coactions and ribbon braids 102.1. Co-Hochschild cohomology for multiplier algebras 102.2. Classification of associators and ribbon braids: non-Hermitian case 112.3. Associators from cyclotomic KZ equations 132.4. Detecting co-Hochschild classes 152.5. Classification of associators: Hermitian case 162.6. Classification of ribbon braids 183. Interpolated subgroups 203.1. Root vectors and Poisson structure 203.2. Satake form 213.3. A family of coisotropic subgroups 243.4. Coactions of quantized multiplier algebras 283.5. Regularity of ribbon braids 314. Letzter–Kolb coideals 334.1. Quantized universal enveloping algebra and Letzter–Kolb coideal subalgebras 344.2. Untwisting by Drinfeld twist 354.3. Parameter case and Cayley transform 364.4. Multiplier algebra model of Letzter–Kolb coideals 384.5. K -matrix of Balagović–Kolb 405. Comparison theorems 435.1. Twisting of ribbon twist-braids 435.2. Comparison theorem: non-Hermitian case 445.3. Comparison theorem: Hermitian case 45 Date : September 13, 2020; minor changes October 5, 2020.The work of K.DC. was supported by the FWO grants G025115N and G032919N. The work of S.N. and M.Y. waspartially supported by the NFR funded project 300837 “Quantum Symmetry”. M.Y. also acknowledges support by Grantfor Basic Science Research Projects from The Sumitomo Foundation and JSPS Kakenhi 18K13421 at an early stage ofcollaboration. a r X i v : . [ m a t h . QA ] O c t .4. A Kohno–Drinfeld type theorem 475.5. Example: AIII case 49Appendix A. Co-Hochschild cohomology 54Appendix B. Spherical vectors 57Appendix C. Coideals as deformations 59References 60 Introduction
This paper is about quantization of symmetric spaces of compact type. It will be sufficient toconcentrate on the irreducible simply connected symmetric spaces of type I , that is, the spaces of theform
U/U σ for a compact simply connected simple Lie group U with an involutive automorphism σ .Our approach is motivated by the groundbreaking work of Drinfeld [Dri89], in which he gave a newalgebraic proof of Kohno’s theorem [Koh87] on equivalence of the braid group representations that appearas deformations of representations of the symmetric group on tensor powers of some representation of g = u 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 ( g )of the universal enveloping algebra U ( g ), on the other.Drinfeld developed a framework of quasi-triangular quasi-bialgebras , which captures both types ofrepresentations. He showed that a deformation of U ( g ) among such quasi-bialgebras is controlled bythe co-Hochschild cohomology of the coalgebra U ( g ), up to a natural notion of equivalence derived fromtensor categorical considerations. This cohomology is the exterior algebra V g , and the part giving thedeformation parameter is the one-dimensional space ( V g ) g . Moreover, this parameter is detected by theeigenvalues of the square of the braiding.In the course of developing the theory Drinfeld also clarified the geometric structures behind suchdeformations. Namely, the first order terms of the deformations correspond to Poisson–Lie group structureson U , or structures of a Lie bialgebra on u . The two types of representations of the braid groups arise fromdifferent models of quantizations of Poisson–Lie groups, and Drinfeld’s result says that such quantizationsare 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 aquasi-isomorphism of differential graded Lie algebras.Having understood quantizations of Poisson–Lie groups, one natural next direction is to look atquantizations of the Poisson homogeneous spaces. The first important step towards a classification ofsuch spaces was again made by Drinfeld [Dri93]: for the standard Poisson–Lie group structure on U ,they correspond to the real Lagrangian subalgebras of g . A complete classification of these spaces (withconnected stabilizers) was then given by Karolinsky [Kar96].The first classification result for quantizations of Poisson homogeneous spaces was obtained by Podleś[Pod87]. He classified the actions of Woronowicz’s compact quantum group SU q (2) [Wor87] with thesame spectral pattern as that of SU(2) acting on (the functions on) the 2-sphere S . In other words, heconsidered coactions of the C ∗ -bialgebra C (SU q (2)), which is a deformation of the algebra of continuousfunctions on SU(2) and is dual to (an analytic version of) U h ( sl ). Podleś showed that there is aone-parameter family of isomorphism classes of such coactions. From the geometric point of view this isexplained by the fact that the covariant Poisson structures on S form a Poisson pencil [She91].Tensor categorical counterparts of Hopf algebra coactions are module categories. Although the precisecorrespondence, through a Tannaka–Krein type duality, came later [Ost03, DCY13, Nes14], in the contextof quantization of Poisson homogeneous spaces there is already a rich accumulation of results obtainedfrom various angles, all related to the reflection equation .This equation was introduced by Cherednik [Che84] to study quantum integrable systems on thehalf-line. While braiding (Yang–Baxter operator) represents scattering of two particles colliding in a1-dimensional system, a solution of the reflection equation (reflection operator) represents the interactionof a particle with a boundary. Adding this operator to a braided tensor category (where the Yang–Baxteroperators live) gives rise to a new category with a larger space of morphisms, which admits a monoidalproduct of the braided tensor category from one side, thus yielding a module category [tDHO98], or moreprecisely, a braided module category [Bro13]. atrix solutions of the reflection equation for the universal R -matrix of quasi-triangular Hopf algebraslead to coideal subalgebras , as originally pointed out by Noumi [Nou96] and further clarified by Kolb–Stokman [KS09]. In this direction, the best understood class is that of quantum symmetric pairs , that is,the coideals which are deformations of U ( g θ ) for a conjugate θ of σ such that g θ is maximally noncompactrelative to the Cartan subalgebra defining the deformation U h ( g ). Following Koornwinder’s work [Koo93]on the dual coideals of the Podleś spheres, Letzter [Let99] developed a systematic way of constructingsuch coideal subalgebras U t h ( g θ ) < U h ( g ) for finite type Lie algebras, which was refined and extended byKolb to Kac–Moody Lie algebras [Kol14]. Next, a universal K -matrix for U t h ( g θ ), which gives reflectionoperators in the representations of U t h ( g θ ), was defined by Kolb and Balagović [Kol08, BK19] expanding onthe earlier work of Bao and Wang [BW18] on the (quasi-split) type AIII and AIV cases. The constructionrelied on a coideal analogue of Lusztig’s bar involution [BW18, BK15]. Kolb [Kol20] further showed,developing on the ideas from [tDHO98, Bro13], that these structures give rise to ribbon twist-braidedmodule categories .On the dual side, a deformation quantization of U/U σ from the reflection equation was developed byGurevich, Donin, Mudrov, and others [GS99, DGS99, DM03b, DM03a]. Here one sees a close connectionto the theory of dynamical r -matrices [Fel95, EV98].There is a parallel theory of module categories over the Drinfeld category, that is, the tensor categoryof finite dimensional g -modules with the associator defined by the monodromy of the KZ-equations. Thebasic idea is to add an extra pole in these equations, then the reflection operator appears as a suitablynormalized monodromy around it. Conceptually, the usual KZ-equations give flat connections on theconfiguration space of points in the complement of type A hyperplane configurations, and the modifiedequations are obtained by looking at the complement of type B hyperplane configurations. Followingearly works of Leibman [Lei94] and Golubeva–Leksin [GL00] on monodromy of such equations, Enriquez[Enr07] introduced cyclotomic KZ-equations . He also defined quasi-reflection algebras , a particular classof quasi-coactions of quasi-bialgebras, which can be considered as type B analogues of quasi-triangularquasi-bialgebras. This formalism turned out to have powerful applications to quantization of Poissonhomogeneous spaces, where the associator of a quasi-coaction gives rise to a quantization of a dynamical r -matrix [EE05].Based on these developments, and guided by the categorical duality between module categories andHopf algebraic coactions, we proposed a conjecture on equivalence between the following structures[DCNTY19]: • a category of finite dimensional representations of g σ , considered as a ribbon twist-braided modulecategory over the Drinfeld category, with the associator and ribbon twist-braid defined by thecyclotomic KZ-equations; • a category of finite dimensional modules over a Letzter–Kolb coideal U t h ( g θ ), considered as a ribbontwist-braided module category over the category of U h ( g )-modules, with the ribbon twist-braiddefined 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 tobe a real number and the categories carried unitary structures. In this paper, we give a proof of thecorresponding conjecture in the formal setting using the framework of quasi-coactions.It should be mentioned that Brochier [Bro12] has already proved an interesting equivalence betweentwo quasi-coactions on U h ( h ), where h < g is the Cartan subalgebra and one of the quasi-coactions comesfrom the cyclotomic KZ-equations associated with a finite order automorphism σ such that g σ = h . Inhis setting, the extra deformation parameter space is the formal group generated by the Cartan algebra.The construction of the equivalence follows the strategy of [Dri89], this time relying on the co-Hochschildcohomology studied by Calaque [Cal06].Now, let us sketch what we concretely carry out: • Show that the quasi-coactions of Drinfeld’s quasi-bialgebra induced by the cyclotomic KZ-equationsare generically universal among the quasi-coactions deforming ∆ on U ( g σ ). • Give a complete classification of the corresponding ribbon twist-braids and show that the correspond-ing 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 σ by studying coisotropicsubgroups which are conjugates (‘Cayley transforms’) of U σ . Make a concrete comparison with the Letzter–Kolb coideals and the Balagović–Kolb braided modulecategorical structures.In the first step, the main idea is to reduce the problem to vanishing of obstructions in a suitableversion of the co-Hochschild cohomology. This strategy is quite standard, see [Dri89, Bro12], but whilethese papers relied on the braiding/ribbon braids to have a good control of the cohomology, we workwith the cohomology classes directly, analogously to Donin–Shnider’s approach [DS97] to Lie bialgebraquantization, and the identification of the ribbon twist-braids comes only towards the end. The relevantco-Hochschild cohomology turns out to be isomorphic to V m C for m C = g (cid:9) g σ , and the deformation ofa quasi-coaction is controlled by the invariant part of the second cohomology, that is, ( V m C ) g σ . Upto complexification, this space can be interpreted as the space of U -invariant bivectors on U/U σ , hencethere is a direct connection to equivariant deformation quantization. This is where one sees the formalityprinciple in action.At this point we encounter an important dichotomy between the Hermitian and the non-Hermitian cases. Although we already discussed it in [DCNTY19] based on the parameters t for the coideals U t h ( g θ ),the following observation is perhaps more illuminating: the dimension of ( V m C ) g σ is either zero or one,and is equal to that of the center of g σ . In the Hermitian case, and only in this case, this dimension isone and the corresponding homogeneous space U/U σ has an invariant Hermitian structure, induced byan element of the center of g σ (hence the name).In the non-Hermitian case, the triviality of the center eliminates cohomological obstructions, quicklyleading to rigidity of the algebra structure and coaction homomorphisms on U ( g σ ). Our results in thiscase can be summarized as follows. Theorem A (Section 2.2 and Theorem 2.18) . Let u σ < u be a non-Hermitian irreducible symmetricpair. Suppose that α : U ( g σ ) (cid:74) h (cid:75) → U ( g σ ) ⊗ U ( g ) (cid:74) h (cid:75) and Ψ ∈ U ( g σ ) ⊗ U ( g ) ⊗ (cid:74) h (cid:75) define a quasi-coactionof Drinfeld’s quasi-bialgebra ( U ( g ) (cid:74) h (cid:75) , ∆ , Φ KZ ) that deforms ∆ : U ( g σ ) → U ( g σ ) ⊗ U ( g ) , and let ( α , Ψ ) be another such pair. Then ( α, Ψ) and ( α , Ψ ) are obtained from each other by twisting. Moreover,the quasi-coaction ( U ( g σ ) (cid:74) h (cid:75) , α, Ψ) admits a unique ribbon σ -braid E with prescribed constant term E (0) ∈ ⊗ Z ( U ) . In the above formulation the ribbon twist-braid is allowed to live in a certain completion of U ( g σ ) ⊗ U ( g ) (cid:74) h (cid:75) . Namely, consider the multiplier algebra of the algebra of finitely supported functions on thedual of U σ [VD96], which is the direct product of full matrix algebras U ( G σ ) = Y π End( V π ) , where π runs over the irreducible finite dimensional representations of g σ which appear in finite dimensionalrepresentations of g . We can further define U ( G σ × G n ) = Y π,π ,...,π n End( V π ) ⊗ End( V π ) ⊗ · · · ⊗ End( V π n ) , where π , . . . π n run over the irreducible finite dimensional representations of g . Then we take E as anelement of U ( G σ × G ) (cid:74) h (cid:75) .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 equivalentto the ones arising from the cyclotomic KZ-equations with prescribed coefficients [EE05, DCNTY19]: theassociator Ψ KZ ,s ; µ , for parameters s ∈ C \ i Q × and µ ∈ h C (cid:74) h (cid:75) , is given as the normalized monodromyfrom w = 0 to w = 1 of the differential equation H ( w ) = (cid:18) (cid:126) ( t k − t m ) w + 1 + (cid:126) t u w − (cid:126) (2 t k + C k ) + ( s + µ ) Z w (cid:19) H ( w ) . Here we put (cid:126) = hπi , and the coefficients are given as follows: t u , t k , t m are the canonical 2-tensors of u , k = u σ , and m = u (cid:9) k respectively, C k is the Casimir element of k associated to t k , and Z is a normalizedelement of z ( k ).If s = 0, then Ψ KZ ,s ; µ makes sense in U ( g σ ) ⊗ U ( g ) ⊗ (cid:74) h (cid:75) , but otherwise we can only say that Ψ KZ ,s ; µ is in U ( G σ × G ) (cid:74) h (cid:75) . It is therefore convenient to start working with the multiplier algebras throughout insteadof the universal enveloping algebras. Fortunately, the concepts of quasi-bialgebras and quasi-coactionshave straightforward formulations in this setting, and from the categorical point of view this formalism isactually even more natural when dealing with semisimple module categories. Then ( U ( G σ ) (cid:74) h (cid:75) , ∆ , Ψ KZ ,s ; µ )is a quasi-coaction of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ), and our results can be summarized as follows. heorem B (Theorems 2.16 and 2.19) . Let u σ < u be an irreducible Hermitian symmetric pair, and let ω be an invariant symplectic form on U/U σ . There is a countable subset A ⊂ C with the following property:if α : U ( G σ ) (cid:74) h (cid:75) → U ( G σ × G ) (cid:74) h (cid:75) and Ψ ∈ U ( G σ × G ) (cid:74) h (cid:75) define a quasi-coaction of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ) that deforms ∆ : U ( G σ ) → U ( G σ × G ) , and the first order term Ψ (1) of Ψ satisfies h ω, Ψ (1) i ∈ C \ A ,then there is a pair ( s, µ ) , unique up to translation by (2 i Z , , such that ( U ( G σ ) (cid:74) h (cid:75) , α, Ψ) is equivalentto ( U ( G σ ) (cid:74) h (cid:75) , ∆ , Ψ KZ ,s ; µ ) . Moreover, ( U ( G σ ) (cid:74) h (cid:75) , α, Ψ) admits a unique ribbon σ -braid E with prescribedconstant term E (0) ∈ ⊗ exp( − πisZ ) Z ( U ) . We resolve the cohomological obstruction to equivalence by looking at the expansion of Ψ KZ ,s ; µ ,where we follow Enriquez and Etingof’s work [EE05] on quantization of dynamical r -matrices. Up to acoboundary, Ψ KZ ,s ; µ has the expansionΨ KZ ,s ; µ ∼ − h (cid:18) π ( s + µ )2 (cid:19) X α ∈ Φ +nc ( α, α )2 1 ⊗ ( X α ⊗ X − α − X − α ⊗ X α ) + · · · , where Φ +nc is the set of positive roots in m C with respect to a choice of Cartan subalgebra in g σ , and X ± α is a normalized root vector for ± α , see Sections 2.3 and 2.5 for details. This shows that, under aperturbation of µ , the associator changes in the term one order higher than the perturbation, with aprecise control of the cohomology class (formal Poisson structure) of the difference in that term. Thisleads to the universality of quasi-coactions with the associators Ψ KZ ,s ; µ and can be interpreted as ‘poorman’s formality’ for equivariant deformation quantization.We next apply these results to the Letzter–Kolb coideals. Since our classification is formulated in theframework of multiplier algebras, we show that the coideals indeed give rise to such structures, essentiallyby taking a completion. It should be stressed that the formalism of multiplier algebras is importantnot only for making sense of Ψ KZ ,s ; µ . The second and even more important reason is that it allows usto check that the coactions defined by the Letzter–Kolb coideals are twistings of ∆. The point is thatsince g σ is not semisimple in the Hermitian case, the standard arguments based on Whitehead’s firstlemma are not applicable. By working with the multiplier algebras, which are built out of semisimplealgebras, we can circumvent the nonvanishing of Lie algebraic cohomological obstructions. We still needto use Letzter’s result [Let00] on existence of spherical vectors for this, which means that we have toconsider ∗ -coideals U t h ( g θ ).Next, in the Hermitian case, we have to verify the condition on the first order term Ψ (1) . For thiswe study Poisson homogeneous structures on U/U σ . More precisely, we have to compare two Poissonstructures, corresponding to two ways we obtain the quasi-coactions. On the one hand, from the cyclotomicKZ-equations we obtain a Poisson pencil [DG95], where one takes the sum of the left action of thestandard r -matrix r on U/U σ and a scalar multiple of the Kostant–Kirillov–Souriau bracket, which agreeswith the bracket defined by the right action of r . On the other hand, from the coideals we obtain thereduction of the Sklyanin bracket to quotients by coisotropic subgroups .Starting from the model σ = θ in the maximally noncompact position, where the subgroup iscoisotropic [FL04], we take a distinguished one-parameter family of subgroups U θ φ that are conjugateto U θ by interpolated Cayley transforms, and show that the associated fixed point subgroups U θ φ remaincoisotropic. At the level of Lie algebras, this construction interpolates between the maximally noncompactsubalgebra g θ and the maximally compact one g ν (which contains h ). Moreover, the Lie algebras g θ φ turnout to be the classical limits of the Letzter–Kolb ∗ -coideals U t h ( g θ ). By a detailed analysis of the Cayleytransforms, we are able to find the relation between the parameters φ and t , as well as to compute thecohomology classes of Ψ (1) for the associators we get. In a bit imprecise form these results are summarizedas follows. Theorem C (Theorems 5.4, 5.5, 5.8, and 5.10) . There is a parameter set T ∗ (consisting of one point t = 0 in the non-Hermitian case) defining ∗ -coideals U t h ( g θ ) and satisfying the following properties. Forevery t ∈ T ∗ , the coideal U t h ( g θ ) gives rise to a coaction of a multiplier bialgebra which is equivalent tothe quasi-coaction ( U ( G θ t ) (cid:74) h (cid:75) , ∆ , Ψ KZ ,s ; µ ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ) , where G θ t < G is a subgroup conjugateto G θ , while s ∈ R and µ ∈ h R (cid:74) h (cid:75) are uniquely determined parameters (equal to in the non-Hermitiancase), with s given by an explicit formula. Under this equivalence, the Balagović–Kolb ribbon twist-braidscorrespond 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 B braid groups arising from the coideals and the cyclotomic KZ-equations are equivalent. formula for the parameter µ in Theorem C can in principle be obtained by comparing the eigenvaluesof the reflection operators in the two pictures. In the general case this step might be somewhat involved,but at least for the AIII case (which corresponds to the symmetric pairs s ( u p ⊕ u N − p ) < su N ) this canbe done thanks to the classification of reflection operators by Mudrov [Mud02].So far we have discussed the case of irreducible symmetric spaces of type I, i.e., U/U σ with U simple.However, the type II case, corresponding to U itself as a symmetric space, or the quotient of U × U bythe diagonal subgroup, can be handled in essentially the same way as the non-Hermitian type I cases. Inparticular, Theorems A and C can be adapted to this case. This implies that an analogue of Theorem Cholds in general for Letzter–Kolb ∗ -coideals of U h ( g ) with g semisimple.Let us now briefly summarize the contents of the paper. In Section 1 we recall basic definitions andintroduce conventions which are used throughout the paper.In Section 2 we prove our main conceptual results on classification of quasi-coactions and ribbontwist-braids. As explained above, the non-Hermitian case is done by a more or less standard cohomologicalargument, while in the Hermitian case we look into the structure of the associators arising from thecyclotomic KZ-equations.In Section 3 we focus on the Hermitian case and look at conjugates of u σ < u in the maximallycompact position by interpolated Cayley transforms. We show that these conjugates generate coisotropicsubgroups and relate them to models arising from the cyclotomic KZ-equations, with an explicit formulafor the first order term.In Section 4 we explain how the quantized universal enveloping algebra and the Letzter–Kolb coidealsfit 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 comparisontheorems. We finish the section with a detailed analysis of the AIII case.There are three appendices, in which we collect some technical but not fundamentally new results usedin the paper.Let us close the introduction with some further problems. First of all, a general formula for µ inTheorem C would be nice to find, especially if this can be done in a unified way rather than via acase-by-case analysis. Second, the analytic version of the conjecture, as originally proposed in [DCNTY19],remains to be settled, together with a comparison with the ‘Vogan picture’ introduced there. On thegeometric side, one would like to extend the above results in the Hermitian case to all coadjoint orbitsof U . Acknowledgements : K.DC. thanks A. Brochier for discussions in an early stage of this project. S.N. isgrateful to P.A. Østvær for a reference. M.Y. thanks D. Jordan and A. Appel for stimulating discussions.1.
Preliminaries
Conventions.
We treat h as a formal variable, and put h ∗ = h when we consider ∗ -algebraicstructures. We put q = e h and (cid:126) = hπi , the latter is mostly reserved for the KZ-equations. We denote the space of formal power series withcoefficients in A by A (cid:74) h (cid:75) = (cid:26) a = ∞ X n =0 h n a ( n ) (cid:12)(cid:12)(cid:12)(cid:12) a ( n ) ∈ A (cid:27) , and the space of Laurent series by A [ h − , h (cid:75) = (cid:26) a = ∞ X n = k h n a ( n ) (cid:12)(cid:12)(cid:12)(cid:12) a ( n ) ∈ A, k ∈ Z (cid:27) . For a ∈ A [ h − , h (cid:75) , we denote the smallest n such that a ( n ) = 0 by ord( a ).We denote the h -adically completed tensor product of C (cid:74) h (cid:75) -modules by ˆ ⊗ . In particular, we have( A (cid:74) h (cid:75) ) ˆ ⊗ ( B (cid:74) h (cid:75) ) = ( A ⊗ B ) (cid:74) h (cid:75) .When A = C and a ∈ C (cid:74) h (cid:75) has constant term a (0) >
0, we take its n th root b = a n to be the uniquesolution of b n = a such that b (0) is positive. A similar convention is used for log. .2. Simple Lie groups.
Throughout the entire paper u denotes a compact simple Lie algebra and g denotes its complexification. The connected and simply connected Lie groups corresponding to g and u are denoted by G and U .We denote by ( · , · ) g the unique invariant symmetric bilinear form on g such that, for any Cartansubalgebra h < g , its dual form on h ∗ has the property that ( α, α ) = 2 for every short root α . Let t u ∈ u ⊗ be the corresponding invariant tensor: t u = X i X i ⊗ X i , (1.1)where ( X i ) i is a basis in g and ( X i ) i is the dual basis.Recall that ( · , · ) g is negative definite on u . Therefore, if we define an antilinear involution ∗ on g byletting X ∗ = − X for X ∈ u , then ( X, Y ∗ ) g becomes an (Ad U )-invariant Hermitian scalar product on g .We denote the category of finite dimensional algebraic representations of the linear algebraic group G (equivalently, finite dimensional representations of g ) by Rep G . It is equivalent to the category of finitedimensional unitary representations of U . We write π ∈ Rep G to say that π is a finite dimensionalrepresentation of G , its underlying space is denoted by V π . We also fix a set Irr G of representatives ofthe isomorphism classes of irreducible representations.We will often have to extend the scalars to C (cid:74) h (cid:75) . Denote the category we get by (Rep G ) (cid:74) h (cid:75) . Thus, theobjects of (Rep G ) (cid:74) h (cid:75) are the G -modules over C (cid:74) h (cid:75) that are isomorphic to the modules of the form V π (cid:74) h (cid:75) for π ∈ Rep G .1.3. Multiplier algebras.
For n = 1 , , . . . , we put U ( G n ) = Y π i ∈ Irr
G,i =1 ,...,n End( V π ) ⊗ · · · ⊗ End( V π n )We view G and g as subsets of U ( G ) = U ( G ).Since for every irreducible π ∈ Rep G there is a unique up to a scalar factor U -invariant Hermitianscalar product on V π , we have a canonical involution ∗ on U ( G n ). There is also a unique homomorphism∆ : U ( G ) → U ( G )characterized by the identities ( π ⊗ π )(∆( T )) S = Sπ ( T ) for all intertwiners S : V π → V π ⊗ V π . Then∆( g ) = g ⊗ g for g ∈ G . This characterizes the elements of G among the nonzero elements of U ( G ).Similarly, the identity ∆( X ) = X ⊗ ⊗ X for X ∈ g characterizes g inside U ( G ).Denote by O ( G ) the Hopf algebra of regular functions (matrix coefficients of finite dimensionalrepresentations) on G . We occasionally write O ( U ) instead of O ( G ) when we think of it as a functionalgebra on U . As a vector space, U ( G ) is the dual of O ( G ). Concretely, if π is irreducible, T ∈ End( V π ), v ∈ V π , ‘ ∈ V ∗ π , then for the matrix coefficient a v,‘ ∈ O ( G ), a v,‘ ( g ) = ‘ ( π ( g ) v ), we have h a v,‘ , T i = ‘ ( T v ) , and h f, T i = 0 for the matrix coefficients f of the irreducible representations π inequivalent to π . Similarly, U ( G n ) is the linear dual of O ( G ) ⊗ n . With respect to this duality the bialgebra structures are related by h f ⊗ f , ∆( T ) i = h f f , T i , h ∆( f ) , T ⊗ T i = h f, T T i for f i ∈ O ( G ) and T i ∈ U ( G ).We can do the same constructions for any reductive linear algebraic group H over C . We then alsodefine U ( H × G n ) = Y π ∈ Irr
H,π i ∈ Irr
G,i =1 ,...,n End( V π ) ⊗ End( V π ) ⊗ · · · ⊗ End( V π n )for 0 ≤ n < ∞ . In a more invariant form, U ( H × G n ) is the linear dual of O ( H × G n ).Assume in addition that H is an algebraic subgroup of G . Then the embedding H → G extends to anembedding of U ( H n +1 ) into U ( H × G n ). In particular, the comultiplication ∆ : U ( H ) → U ( H ) can beviewed as a homomorphism U ( H ) → U ( H × G ).Note that in general H is not simply connected. In Lie algebraic terms the category Rep H consistsof the finite dimensional representations of h that are subrepresentations of the finite dimensionalrepresentations of g restricted to h . .4. Quasi-coactions and ribbon twist-braids.
The notion of a quasi-bialgebra [Dri89] has a straight-forward adaptation to the setting of multiplier algebras, cf. [NT11, Section 2]. We will be interestedin multiplier quasi-bialgebras of the form ( U ( G ) (cid:74) h (cid:75) , ∆ h , (cid:15) h , Φ). Thus, ∆ h is a nondegenerate homo-morphism U ( G ) (cid:74) h (cid:75) → U ( G ) (cid:74) h (cid:75) , meaning that the images of the idempotents ∆ h (id V π ) ( π ∈ Irr G ) inEnd( V π ⊗ V π ) (cid:74) h (cid:75) add up to 1, (cid:15) h : U ( G ) (cid:74) h (cid:75) → C (cid:74) h (cid:75) is a nondegenerate homomorphism, and Φ ∈ U ( G ) (cid:74) h (cid:75) is an invertible element (with Φ (0) = 1) satisfying the same identities as in [Dri89, Section 1].The assumption of nondegeneracy for the counit (cid:15) h implies that it is determined by its restrictions to theblocks End( V π ) (cid:74) h (cid:75) of U ( G ) (cid:74) h (cid:75) . Since there are no nonzero ( C (cid:74) h (cid:75) -linear) homomorphisms End( V ) (cid:74) h (cid:75) → C (cid:74) h (cid:75) for dim V > V = 1, we conclude that (cid:15) h coincides with the standard counit (cid:15) on U ( G ) (cid:74) h (cid:75) . From now on we will therefore omit (cid:15) h from the notationfor a multiplier quasi-bialgebra.Given a reductive algebraic subgroup H of G , a quasi-coaction of ( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ) on U ( H ) (cid:74) h (cid:75) is givenby a nondegenerate homomorphism α : U ( H ) (cid:74) h (cid:75) → U ( H × G ) (cid:74) h (cid:75) and an associator Ψ ∈ U ( H × G ) (cid:74) h (cid:75) satisfying Ψ (0) = 1, (id ⊗ (cid:15) ) α = id , Ψ( α ⊗ id) α ( T ) = (id ⊗ ∆ h ) α ( T )Ψ ( T ∈ U ( H )) , (1.2)the mixed pentagon equation Φ , , Ψ , , Ψ , , = Ψ , , Ψ , , , (1.3)with Ψ , , = ( α ⊗ id)(Ψ), Ψ , , = (id U ( H ) ⊗ ∆ h ⊗ id)(Ψ), etc., and the normalization conditions(id ⊗ (cid:15) ⊗ id)(Ψ) = (id ⊗ id ⊗ (cid:15) )(Ψ) = 1 . A multiplier quasi-bialgebra ( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ) defines a tensor category ((Rep G ) (cid:74) h (cid:75) , ⊗ h , Φ), where thetensor product ⊗ h on (Rep G ) (cid:74) h (cid:75) is defined using ∆ h and the associativity isomorphism is given by theaction of Φ. A quasi-coaction as above defines then the structure of a right ((Rep G ) (cid:74) h (cid:75) , ⊗ h , Φ) -modulecategory on (Rep H ) (cid:74) h (cid:75) . Namely, the functor (cid:12) α : (Rep H ) (cid:74) h (cid:75) × (Rep G ) (cid:74) h (cid:75) → (Rep H ) (cid:74) h (cid:75) defining themodule category structure is induced by α , while the associativity morphisms are defined by the actionof Ψ. See [DCNTY19, Section 1] for more details, but note that in [DCNTY19] we worked in the analyticsetting, meaning that q = e h was a real number and Φ ∈ U ( G ), Ψ ∈ U ( H × G ).Next, let R ∈ U ( G ) (cid:74) h (cid:75) be an R -matrix (with R (0) = 1) for ( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ), that is, R ∆ h ( · ) =∆ op h ( · ) R and R satisfies the hexagon relations . Let β be an automorphism of the quasi-triangular multiplierquasi-bialgebra ( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ , R ). A ribbon β -braid is given by an invertible element E ∈ U ( H × G ) (cid:74) h (cid:75) satisfying E (id ⊗ β ) α ( T ) = α ( T ) E ( T ∈ U ( H ) (cid:74) h (cid:75) ) , (1.4)( α ⊗ id)( E ) = Ψ − R Ψ E (id ⊗ id ⊗ β )(Ψ − R Ψ) , (1.5)(id ⊗ ∆ h )( E ) = R Ψ E (id ⊗ id ⊗ β )(Ψ − R Ψ) E (id ⊗ β ⊗ β )(Ψ − ) . (1.6)When β is the identity map, we just say “ribbon braid” instead of “ribbon id-braid”. We want tostress that, as opposed to Φ, Ψ and R , we do not require E (0) = 1. A quadruple ( U ( H ) (cid:74) h (cid:75) , α, Ψ , E )satisfying (1.4) and (1.5) is a version of a quasi-reflection algebra [Enr07]. In categorical terms, the actionof E on M (cid:12) α N defines the structure of a ribbon β -braided module category on ((Rep H ) (cid:74) h (cid:75) , (cid:12) α , Ψ). Seeagain [DCNTY19, Section 1] for more details.1.5.
Twisting.
We can transform a quasi-coaction ( U ( H ) (cid:74) h (cid:75) , α, Ψ) of ( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ) into a new oneas follows. Suppose that we are given elements
F ∈ U ( G ) (cid:74) h (cid:75) and G ∈ U ( H × G ) (cid:74) h (cid:75) such that F (0) = 1, G (0) = 1 and ( (cid:15) ⊗ id)( F ) = (id ⊗ (cid:15) )( F ) = 1 , (id ⊗ (cid:15) )( G ) = 1 . Then the twisting of the quasi-coaction by ( F , G ) is the quasi-coaction ( U ( H ) (cid:74) h (cid:75) , α G , Ψ F , G ) of the multiplierquasi-bialgebra ( U ( G ) (cid:74) h (cid:75) , ∆ h, F , Φ F ), where∆ h, F = F ∆ h ( · ) F − , Φ F = (1 ⊗ F )(id ⊗ ∆ h )( F )Φ(∆ h ⊗ id)( F − )( F − ⊗ ,α G = G α ( · ) G − , Ψ F , G = (1 ⊗ F )(id ⊗ ∆ h )( G )Ψ( α ⊗ id)( G − )( G − ⊗ . Twisting defines an equivalence relation on the quasi-coactions. In categorical terms it means thatwe pass from ((Rep G ) (cid:74) h (cid:75) , ⊗ h , Φ) to the equivalent tensor category ((Rep G ) (cid:74) h (cid:75) , ⊗ h, F , Φ F ), with the ensor product defined by ∆ h, F , and, up to this equivalence, the ((Rep G ) (cid:74) h (cid:75) , ⊗ h , Φ)-module category((Rep H ) (cid:74) h (cid:75) , (cid:12) α , Ψ) is equivalent to the ((Rep G ) (cid:74) h (cid:75) , ⊗ h, F , Φ F )-module category((Rep H ) (cid:74) h (cid:75) , (cid:12) α G , Ψ F , G ) . As the following result shows, twisting often allows one to push all the information on a quasi-coactioninto the associators.
Lemma 1.1.
Assume H is a reductive algebraic subgroup of G and ( U ( H ) (cid:74) h (cid:75) , α, Ψ) is a quasi-coactionof ( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ) such that both α and ∆ h equal ∆ modulo h . Then this quasi-coaction is a twisting ofa quasi-coaction ( U ( H ) (cid:74) h (cid:75) , ∆ , Ψ ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ ) for some Ψ and Φ .Proof. Take irreducible representations π and π of G . Consider the homomorphisms f = ( π ⊗ π )∆ and f h = ( π ⊗ π )∆ h from U ( G ) (cid:74) h (cid:75) into End( V π ⊗ V π ) (cid:74) h (cid:75) . The assumption of nondegeneracy for ∆ h impliesthat there exists a finite set F ⊂ Irr G such that f h factors through L π ∈ F End( V π ) (cid:74) h (cid:75) . By taking F large enough we may assume that the same is true for f . Since the algebra L π ∈ F End( V π ) is semisimple,there are no nontrivial deformations of any given homomorphism L π ∈ F End( V π ) → End( V π ⊗ V π ).Hence there exists F π ,π ∈ End( V π ⊗ V π ) (cid:74) h (cid:75) such that F (0) π ,π = 1 and f h = (Ad F π ,π ) f . Then F = ( F π ,π ) π ,π ∈ Irr G ∈ U ( G ) (cid:74) h (cid:75) satisfies F (0) = 1 and ∆ h = F ∆( · ) F − . Furthermore, since thecounit of ( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ) is (cid:15) , we could take F π ,π = 1 if either π or π were trivial representations.In this case F would additionally satisfy ( (cid:15) ⊗ id)( F ) = (id ⊗ (cid:15) )( F ) = 1.In a similar way we can find G ∈ U ( H × G ) (cid:74) h (cid:75) such that G (0) = 1, ( ι ⊗ (cid:15) )( G ) = 1 and α = G ∆( · ) G − .Then the twisting by ( F − , G − ) gives the required quasi-coaction. (cid:3) Next, given a quasi-coaction ( U ( H ) (cid:74) h (cid:75) , α, Ψ) of ( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ), assume in addition we have anautomorphism β of ( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ). If F satisfies ( β ⊗ β )( F ) = F , then β remains an automorphismof ( U ( G ) (cid:74) h (cid:75) , ∆ h, F , Φ F ). Assume also that R ∈ U ( G ) (cid:74) h (cid:75) is an R -matrix for ( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ) that isfixed under β . Then R F = F RF − is an R -matrix for ( U ( G ) (cid:74) h (cid:75) , ∆ h, F , Φ F ), again fixed by β . Given aribbon β -braid E for the original quasi-coaction we get a ribbon β -braid E G for the twisted quasi-coaction( U ( H ) (cid:74) h (cid:75) , α G , Ψ F , G ) of ( U ( G ) (cid:74) h (cid:75) , ∆ h, F , Φ F , R F ) defined by E G = GE (id ⊗ β )( G ) − . (1.7)The condition ( β ⊗ β )( F ) = F can be relaxed, we will return to this in Section 5.1.1.6. Symmetric pairs.
Let k be a proper Lie subalgebra of u . We say that k < u is a symmetric pair ,or more precisely, an irreducible symmetric pair of type I , if there is a (necessarily unique) involutiveautomorphism σ of u such that k = u σ . Whenever convenient we extend σ to U ( G ), in particular, to g .Let K = U σ . The compact group K is connected by [Hel01, Theorem VII.8.2]. Using the Cartandecomposition of G we can also conclude that G σ is connected.Given such a symmetric pair, put m = { X ∈ u | σ ( X ) = − X } , which is the orthogonal complement of k in u with respect to the invariant inner product. We also write m C = m ⊗ R C for its complexification.We say that a symmetric pair k < u is Hermitian , if
U/K is a Hermitian symmetric space. Suchsymmetric pairs are equivalently characterized by either of the following conditions, see [Bor98, Proposi-tion VI.1.3]: • the center z ( k ) is nontrivial (and 1-dimensional); • the space m has a (unique up to a sign) k -invariant complex structure.The following closely related characterization will be crucial for us. Lemma 1.2.
For any symmetric pair k < u , we have dim( V m ) k = 1 if k < u is Hermitian, and dim( V m ) k = 0 otherwise. We always have m k = 0 .Proof. Since U is simple by assumption, U/K is an irreducible symmetric space, so K acts irreducibly on m = T [ e ] ( U/K ). As
U/K is not one-dimensional, this cannot be the trivial action, and we get m k = 0.Next, since m has a k -invariant inner product, the space ( V m ) k is isomorphic to the space of k -invariantskew-adjoint operators on m . Assume we are given such a nonzero operator A . Then A = − A ∗ A isself-adjoint, with negative eigenvalues. Hence A is diagonalizable, and by irreducibility of the actionof k on m we conclude that A must be a strictly negative scalar. Therefore by rescaling A we get a -invariant complex structure on m . Since there is a unique up to a sign such structure in the Hermitiancase and no such structure in the non-Hermitian case, we get the result. (cid:3) An irreducible symmetric pair of type II is an inclusion that is isomorphic to the diagonal inclusion of u into u ⊕ u (with a simple compact Lie algebra u ). This corresponds to the involution σ ( X, Y ) = (
Y, X )on u ⊕ u . For such a pair we can put m = { ( X, − X ) | X ∈ u } . Since both m u and ( V m ) u are trivial,such pairs behave in many respects similarly to the non-Hermitian type I pairs. We will therefore mostlyfocus 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 structureon m is defined by an element of z ( k ). The correct normalization is given by the following. Lemma 1.3.
Assuming that k < u is a Hermitian symmetric pair, let Z ∈ z ( k ) be a vector such that ( Z, Z ) g = − . Then on m we have (ad Z ) = − a σ id , where a σ = r h ∨ c dim m ,c ∈ { , , } is the ratio of the square lengths of long and short roots of g and h ∨ is the dual Coxeternumber of g .Proof. As (ad Z ) | m is k -invariant and skew-adjoint, by the proof of the previous lemma we have (ad Z ) = − a on m for some scalar a ≥
0. Hence for the Killing form B Kill on g we have B Kill ( Z, Z ) = Tr((ad Z ) ) = − a dim m . The Killing form and the normalized bilinear form ( · , · ) g are related by B Kill = 2 h ∨ c ( · , · ) g ,see [Kac90, Chapter 6, Exercise 2]. Combining this with ( Z, Z ) g = −
1, we get that a = a σ . (cid:3) Corollary 1.4.
The k -invariant complex structures on m are given by ± a σ (ad Z ) | m . For the involutiveautomorphism σ such that k = u σ we have σ = exp (cid:18) πa σ ad Z (cid:19) . In particular, we see that K is the stabilizer of Z in U with respect to the adjoint action. As theadjoint and coadjoint representations are equivalent, this leads to yet another known characterizationof the Hermitian symmetric pairs: a symmetric pair k < 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 k = u σ < u denotes a symmetric pair. Our goal is to classify using theco-Hochschild cohomology a class of quasi-coactions of ( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ) on U ( G σ ) (cid:74) h (cid:75) .2.1. Co-Hochschild cohomology for multiplier algebras.
The co-Hochschild cochains will play acentral role in this paper. Let H be a reductive algebraic subgroup of G . Put ˜ B nG,H = U ( H × G n ) for0 ≤ n < ∞ , and define a differential ˜ B nG,H → ˜ B n +1 G,H by d cH ( T ) = T , ,...,n +1 − T , ,...,n +1 + · · · + ( − n T , ,...,n ( n +1) + ( − n +1 T , ,...,n , (2.1)where T ,...,jj +1 ,...,n +1 = (id U ( H × G j − ) ⊗ ∆ ⊗ id U ( G n − j ) )( T ) and T , ,...,n = T ⊗
1. The group H actsdiagonally by conjugation on U ( H × G n ), the differential d cH is equivariant with respect to this action.We put B nG,H = ( ˜ B nG,H ) H . Proposition 2.1.
The cohomology of ˜ B G,H is isomorphic to the exterior algebra V g / h as a graded H -module.Proof. The complex ˜ B G,H is the algebraic linear dual of ˜ B G,H = ( O ( H ) ⊗O ( G ) ⊗ n ) ∞ n =0 with the differential d : O ( H ) ⊗ O ( G ) ⊗ n → O ( H ) ⊗ O ( G ) ⊗ n − given by d ( f ⊗ f ⊗ · · · ⊗ f n ) = n − X i =0 ( − i f ⊗ · · · ⊗ f i f i +1 ⊗ · · · ⊗ f n + ( − n f n ( e ) f ⊗ · · · ⊗ f n − , where f f is the product of f and the restriction of f to H . Thus, the cohomology of ˜ B G,H is thelinear dual of the homology of ˜ B G,H as an H -module. he complex ˜ B G,H is the standard complex computing the Hochschild homologyHH ∗ ( O ( G ) , res O ( H ) (cid:15) ) = Tor O ( G ) ⊗O ( G ) ∗ ( O ( G ) , res O ( H ) (cid:15) ) , where the bimodule res O ( H ) (cid:15) has the underlying space O ( H ) with the bimodule structure f.a.f = f ( e ) f a for f, f ∈ O ( G ) and a ∈ O ( H ). In other words, we are computingTor O ( G × G ) ∗ ( O (∆) , O ( H × { e } )) , where ∆ ⊂ G × G is the diagonal. By [BGI71, Proposition VII.2.5], this is the exterior algebra onTor O ( G × G )1 ( O (∆) , O ( H × { e } )), and the latter is the conormal space of H ⊂ G at the point e . Since thisconormal space is the dual of g / h , we obtain the assertion. (cid:3) Remark . Proposition 2.1 and its proof are valid for any linear algebraic group G over C and anyalgebraic subgroup H , if we define U ( H × G n ) as the dual of O ( H ) ⊗ O ( G ) ⊗ n . Corollary 2.3.
For any reductive algebraic subgroup
H < G , the cohomology of B G,H is isomorphic to ( V g / h ) H .Proof. As the factors End( V π ) ⊗ End( V π ) ⊗ · · · ⊗ End( V π n ) of ˜ B nG,H decompose into direct sums ofisotypical components, taking the H -invariant part commutes with taking cohomology. (cid:3) We will mainly need the following particular case.
Corollary 2.4.
The cohomology of B G,G σ is isomorphic to ( V m C ) k .Proof. This follows from the previous corollary, since G σ is connected and g σ = k C . (cid:3) Remark . Instead of the multiplier algebras we could use the universal enveloping algebras and definecomplexes ˜ B g , h and B g , h , see Appendix A. The canonical maps U ( g ) → U ( G ) and U ( h ) → U ( H ) areinjective homomorphisms compatible with the coproduct maps U ( G ) → U ( G × G ) and U ( H ) → U ( H × H ).Thus we get an inclusion ˜ B g , h → ˜ B G,H , and if H is connected, we also get an inclusion B g , h → 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 symmet-ric pair k < u is non-Hermitian.Consider a multiplier quasi-bialgebra ( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ) such that ∆ h = ∆ modulo h . We claim that upto twisting by (1 , G ) it has at most one quasi-coaction ( U ( G σ ) (cid:74) h (cid:75) , α, Ψ) such that α = ∆ modulo h . Sinceby Lemma 1.1 we may assume that both ∆ h and α equal ∆, the following is an equivalent statement. Theorem 2.6.
Let k = u σ < u be a non-Hermitian symmetric pair, and Ψ , Ψ ∈ U ( G σ × G ) (cid:74) h (cid:75) be twoassociators defining quasi-coactions of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ) , with the coaction homomorphisms α = α = ∆ .Then there is an element H ∈ h U ( G σ × G ) k (cid:74) h (cid:75) such that (id ⊗ (cid:15) )( H ) = 1 and Ψ = H , Ψ H − , H − , .Proof. Suppose that Ψ ( k ) = Ψ ( k ) for k < n . We claim that there is T ∈ U ( G σ × G ) k such that(id ⊗ (cid:15) )( T ) = 0 and Ψ and H , Ψ H − , H − , have the same terms up to (and including) order n for H = 1 − h n T . The lemma is then proved by inductively applying this claim and taking the product ofthe elements 1 − h n T we thus get. Note only that by k -invariance the elements T , and T , obtained atdifferent steps commute with each other.Take the difference of identities (1.3) for Ψ and Ψ and consider the terms of order n . Since Ψ and Ψ have the same terms up to order n −
1, we get(Ψ ( n ) − Ψ ( n ) ) , , + (Ψ ( n ) − Ψ ( n ) ) , , = (Ψ ( n ) − Ψ ( n ) ) , , + (Ψ ( n ) − Ψ ( n ) ) , , . Since Ψ and Ψ are k -invariant by (1.2), it follows that Ψ ( n ) − Ψ ( n ) is a cocycle in B G,G σ . As we arein the non-Hermitian case, by Corollary 2.4 and Lemma 1.2, we have Ψ ( n ) − Ψ ( n ) = d cH ( T ) for some T ∈ U ( G σ × G ) k . As (id ⊗ (cid:15) ⊗ id)(Ψ ( n ) − Ψ ( n ) ) = 0, we have (id ⊗ (cid:15) )( T ) = 0. Thus T satisfies ourclaim. (cid:3) Remark . Analogous results are true at the level of the universal enveloping algebras instead of themultiplier algebras. More precisely, given a quasi-bialgebra ( U ( g ) (cid:74) h (cid:75) , ∆ h , Φ) such that ∆ h = ∆ modulo h ,up to twisting by (1 , G ) there is at most one quasi-coaction ( U ( g σ ) (cid:74) h (cid:75) , α, Ψ) of this quasi-bialgebra suchthat α = ∆ modulo h . This is proved along the same lines as Lemma 1.1 and Theorem 2.6, but now relyingon Whitehead’s lemma for the semisimple Lie algebras g and g σ to show that there are no nontrivialdeformations of ∆ : U ( g ) → U ( g ) ⊗ U ( g ) and ∆ : U ( g σ ) → U ( g σ ) ⊗ U ( g ), and using Corollary A.5 insteadof Corollary 2.4. ext let us fix a σ -invariant R -matrix R ∈ U ( G ) (cid:74) h (cid:75) for (∆ , Φ) and look at compatible ribbon σ -braids.Note that the left hand side of (1.4) becomes E ∆( T ) in the present case. Theorem 2.8.
Let k = u σ < u be a non-Hermitian symmetric pair, and let Φ ∈ U ( G ) G (cid:74) h (cid:75) and R ∈U ( G ) G (cid:74) h (cid:75) be σ -invariant elements defining the structure of a quasi-triangular multiplier quasi-bialgebra ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ , R ) . Assume further that we are given a quasi-coaction of the form ( U ( G σ ) (cid:74) h (cid:75) , ∆ , Ψ) bythis quasi-bialgebra, and that E ∈ U ( G σ × G ) (cid:74) h (cid:75) is a ribbon σ -braid for R . Then E (0) = 1 ⊗ g for anelement g in the centralizer Z U ( K ) of K in U , and any other ribbon σ -braid, for the same Φ , Ψ and R ,and with the same order term, coincides with E . Furthermore, if R (1) + R (1)21 = 0 , then g ∈ Z ( U ) . Since z u ( k ) = 0 in the non-Hermitian case, the group Z U ( K ) is finite, so we have at most finitely manyribbon σ -braids. Proof.
From (1.5) we get that (∆ ⊗ id)( E (0) ) = E (0)02 . This implies that E (0) = 1 ⊗ g , with g = ( (cid:15) ⊗ id)( E (0) ) ∈U ( G ). From (1.6) we then get ∆( g ) = g ⊗ g , hence g ∈ G . But then (1.4) shows that g ∈ Z G ( G σ ).Finally, as z g ( k ) = z u ( k ) C = 0, using the Cartan decomposition of G we see that Z G ( K ) = Z U ( K ), hence g ∈ Z U ( K ).Assume now that E is another ribbon σ -braid with E (0) = 1 ⊗ g . We want to show that E = E . Itwill be convenient to first get rid of g . By multiplying both elements by 1 ⊗ g − on the right we get newribbon ˜ σ -braids ˜ E and ˜ E in U ( G σ × G ) (cid:74) h (cid:75) with the order zero terms 1, where ˜ σ = (Ad g ) ◦ σ .We argue by induction on n that ˜ E ( n ) = ˜ E ( n ) . Suppose that we already know that ˜ E ( k ) = ˜ E ( k ) for k < n . Comparing the terms of degree n in (1.5) and (1.6), we obtain X , = X , , X , = X , + X , for X = ˜ E ( n ) − ˜ E ( n ) .The first equality says that X = 1 ⊗ Y for Y = ( (cid:15) ⊗ id)( X ) ∈ U ( G ). Then the second equality says∆( Y ) = Y + Y , that is, Y is primitive, and we obtain Y ∈ g .Comparing the terms of degree n in (1.4), we see next that Y has to centralize k . Hence Y = 0 and˜ E ( n ) = ˜ E ( n ) .It remains to prove the last statement of the theorem. So assume R (1) + R (1)21 = 0. Let us write ˜ B G , B g , etc., instead of ˜ B G, { e } , B g , .By an analogue of [Dri89, Proposition 3.1] for the multiplier algebras, by twisting Φ we may assumethat Φ = 1 modulo h . Such an analogue is proved in the same way as in [Dri89] using that the embeddingmap ˜ B g → ˜ B G is a quasi-isomorphism by Corollary A.5.Namely, consider the normalized skew-symmetrization map Alt : ˜ B G → ˜ B G . This map kills thecoboundaries and transforms the cocycles of the form X ⊗ X ⊗ X , X i ∈ g , into cohomologous ones byRemark A.4. But the classes of such cocycles span the entire space H ( ˜ B G ) by the same remark andCorollary A.5. Therefore if T ∈ ˜ B G is a cocycle killed by Alt, then it is a coboundary. The hexagonrelations imply that Alt(Φ (1) ) = 0, so Φ (1) = d cH ( T ) for some T ∈ ˜ B G . We may assume that T is G - and σ -invariant, since Φ (1) has these invariance properties. Then the twisting by F = 1 − hT proves our claim.Note that twisting does not change the element R (1) + R (1)21 . The hexagon relations imply then that R (1) ∈ g ⊗ g (see the proof of [Dri89, Proposition 3.1]), and since R commutes with the image of ∆, weget R (1) ∈ ( g ⊗ g ) g . Hence R (1) = − λt u for some λ = 0, where t u is the normalized invariant 2-tensordefined by (1.1).Next, identity (1.3) implies that Ψ (1) is a 2-cocycle in B G,G σ , hence by twisting we may assume thatΨ = 1 modulo h . We remind also that under twisting the ribbon twist-braids transform via formula (1.7).Now, by looking at the first order terms in (1.5) for a ˜ σ -braid ˜ E , with ˜ E (0) = 1, we get˜ E (1)01 , = − λt u , + ˜ E (1)0 , − (id ⊗ id ⊗ ˜ σ )( λt u , ) . The tensor t u lies in k ⊗ k + m ⊗ m . Denote the components of t u in k ⊗ k and m ⊗ m by t k and t m , resp.Then the above identity can be written as˜ E (1)01 , = − λt k , + ˜ E (1)0 , + (id ⊗ id ⊗ Ad g )( λt m , ) − λt m , . Applying (cid:15) to the 0th leg and letting T = ( (cid:15) ⊗ id)( ˜ E (1) ), we obtain˜ E (1) = − λt k + 1 ⊗ T + (id ⊗ Ad g )( λt m ) − λt m . (2.2)But we must have ˜ E (1) ∈ U ( G σ × G ). Since t m = P j Y j ⊗ Y j for a basis ( Y j ) j in m and the dualbasis ( Y j ) j , this is possible only when Ad g acts trivially on m . Hence g ∈ Z ( U ). (cid:3) emark . Theorems 2.6 and 2.8 also hold for the type II symmetric pairs with appropriate modifications.Namely, consider ˜ G = G × G and its diagonal subgroup ∆( G ) < ˜ G , which is the fixed point subgroup ofthe involution σ ( g, h ) = ( h, g ) on ˜ G . Then for the quasi-coactions of ( U ( ˜ G ) (cid:74) h (cid:75) , ∆ , Φ) on the multiplieralgebra U ( G ) (cid:74) h (cid:75) , with the coaction map extending G g ( g, g, g ) ∈ G × ˜ G , and with associatorsΨ ∈ U ( G × ˜ G ) (cid:74) h (cid:75) and ribbon σ -braids E ∈ U ( G × ˜ G ) (cid:74) h (cid:75) , one can easily prove analogues of thesetheorems. First, the proof of Theorem 2.6 carries over almost without a change. Indeed, its proof relieson Lemma 1.2 and Corollary 2.4, both of which have analogues for G ’ ∆( G ) < ˜ G . As for Theorem 2.8,we have z ˜ g ( g ) = 0 for the diagonal inclusion g < ˜ g , and Z ˜ G ( G ) = Z ( U ) × Z ( U ), which is enough to adaptthe first half of the proof.2.3. Associators from cyclotomic KZ equations.
We want to extend the results of the previoussubsection to the Hermitian case. Since H ( B G,G σ ) is now one-dimensional by Lemma 1.2, we shouldexpect a one-parameter family of nonequivalent associators. In this subsection we define a candidate forsuch a family arising from the cyclotomic KZ-equations.Thus, assume k < u is a Hermitian symmetric pair. We have an element Z ∈ z ( k ), unique up to a sign,such that ( Z, Z ) g = − a − σ . This normalization is equivalent to (ad Z ) = − m by Lemma 1.3. We fix such Z for the rest of thissection. The operator ad Z has eigenvalues ± i on m C . Denote by m ± ⊂ m C the corresponding eigenspaces.We remind that we denote the components of the normalized invariant 2-tensor t u in k ⊗ k and m ⊗ m by t k and t m , resp. The tensor t m lies in m + ⊗ m − + m − ⊗ m + . We denote the components of t m in m ± ⊗ m ∓ by t m ± . We thus have t m = t m + + t m − , (ad Z ⊗ id)( t m ± ) = ± it m ± , (id ⊗ ad Z )( t m ± ) = ∓ it m ± . (2.3)Given s ∈ C , consider the following elements of U ( G σ × G ) (cid:74) h (cid:75) : A − = (cid:126) ( t k − t m ) , A = (cid:126) t u , A = (cid:126) (2 t k + C k ) + sZ , (2.4)where C k is the Casimir element of k , the image of t k under the product map U ( k ) ⊗ U ( k ) → U ( k ). Theselead to the shifted modified -cyclotomic KZ -equation [EE05, DCNTY19] G ( w ) = (cid:18) A − w + 1 + A w − A w (cid:19) G ( w ) . (2.5) Remark . Consider a C [ h − , h (cid:75) -valued character ν on U ( k ) such that ν ( Z ) = − (2 (cid:126) a σ ) − s . Then theslicing map ς ν = ( ν ⊗ id)∆ is an algebra homomorphism U ( k ) → U ( k )[ h − , h (cid:75) satisfying( ς ν ⊗ id)(2 t k ) = 2 t k + (cid:126) − (1 ⊗ sZ )and commuting with the right coaction ∆ by U ( g ). In particular, at least formally speaking, (2.5) isobtained from the case s = 0 by slicing. But since ν cannot be extended to U ( k ) (cid:74) h (cid:75) , one should be carefulwith this construction.The normalized monodromy Ψ KZ ,s ∈ U ( G σ × G ) (cid:74) h (cid:75) of (2.5) from w = 0 to w = 1 is well-defined as longas the operator ad( sZ ) on U ( G ) does not have positive integers in its spectrum, cf. [NT11, Proposition 3.1].Since each matrix block End( V π ) in U ( G ) is generated by the image of g , the eigenvalues of ad Z are in for n ∈ Z by Lemma 1.3 and our choice of normalization. Therefore Ψ KZ ,s is well-defined for all s i Q × .The element Ψ KZ ,s together with the coproduct ∆ : U ( G σ ) → U ( G σ × G ) gives a quasi-coaction of( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ) on U ( G σ ) (cid:74) h (cid:75) , where Φ KZ = Φ( (cid:126) t u , (cid:126) t u ) ∈ U ( G ) (cid:74) h (cid:75) is Drinfeld’s KZ-associator for G .In more detail, Ψ KZ ,s is defined as follows. Under our restrictions on s , a standard argument (see, e.g.,[NT11, Proposition 3.1]) shows that there is a unique U ( G σ × G ) (cid:74) h (cid:75) -valued solution G of (2.5) on (0 , G ( w ) w − A extends to an analytic function in the unit disc with value 1 at w = 0. Similarly,there is a unique solution G of (2.5) such that G (1 − w ) w − A extends to an analytic function in theunit disc with value 1 at w = 0. Then Ψ KZ ,s = G ( w ) − G ( w )for any 0 < w <
1. We can also write this asΨ KZ ,s = lim w → (1 − w ) − A G ( w ) = lim w → (1 − w ) − A G ( w ) w − A . (2.6)The case s = 0 is special: in this case, it can be shown, using for example iterated integrals[Kas95, Chapter XIX; Enr07], that Ψ KZ , lives in the algebra U ( g σ ⊗ g ⊗ ) (cid:74) h (cid:75) rather than in its completion ( G σ × G ) (cid:74) h (cid:75) . Note also that this associator is well-defined in the non-Hermitian case as well. We willdenote it by Ψ KZ .Observe also that if s ∈ R , then G is unitary, hence Ψ KZ ,s is unitary as well. Indeed, in this case( G ( w ) ∗ ) − has the defining properties of G ( w ), hence coincides with it. Proposition 2.11 (cf. [EE05, Proposition 4.7]) . For every s i Q × , we have Ψ KZ ,s = 1 + hπi (cid:18) (log 2) t u + γt m + ψ (cid:16) − is (cid:17) t m + + ψ (cid:16)
12 + is (cid:17) t m − (cid:19) + O ( h ) , where γ is Euler’s constant and ψ = Γ Γ is the digamma function.Proof. If we restrict to a finite dimensional block of U ( G σ × G ), then ad( sZ ) has a finite number ofeigenvalues there, so the corresponding component of Ψ KZ ,s is well-defined for all s iN − Z × for some N .As it is analytic in s in this domain, it therefore suffices to consider real s .Put H ( w ) = G ( w ) w − A . Then H satisfies the differential equation H ( w ) = (cid:18) A − w + 1 + A w − (cid:19) H ( w ) + (cid:20) A w , H ( w ) (cid:21) and the initial condition H (0) = 1, and by (2.6) we haveΨ KZ ,s = lim w → (1 − w ) − A H ( w ) . Consider the expansion in h . For the order zero terms we immediately get Ψ (0)KZ ,s = H (0)0 = 1. Next,consider the order one terms. Let us write H for πiH (1)0 , so that H = 1 + (cid:126) H + O ( h ). Then H ( w ) = (cid:18) t k − t m w + 1 + t u w − (cid:19) + (cid:20) sZ w , H ( w ) (cid:21) and H (0) = 0, while πi Ψ (1)KZ ,s = lim w → ( H ( w ) − log(1 − w ) t u ) . By (2.3), we have H ( w ) = Z w (cid:16) wu (cid:17) ad( sZ ) (cid:18) t k − t m u + 1 + t u u − (cid:19) du = Z w (cid:18) t k (cid:16) u + 1 + 1 u − (cid:17) + (cid:16)(cid:16) wu (cid:17) is t m + + (cid:16) wu (cid:17) − is t m − (cid:17)(cid:16) − u + 1 + 1 u − (cid:17)(cid:19) du. Note that this integral is well-defined for 0 ≤ w < s is assumed to be real. We then get πi Ψ (1)KZ ,s = bt k + c ( s ) t m + + c ( − s ) t m − , where b = lim w → (cid:18)Z w (cid:18) u + 1 + 1 u − (cid:19) du − log(1 − w ) (cid:19) = log 2 ,c ( s ) = lim w → (cid:18)Z w (cid:16) wu (cid:17) is (cid:18) u − − u + 1 (cid:19) du − log(1 − w ) (cid:19) . To compute c ( s ), we write ( u − − as a power series, integrate and get c ( s ) = lim w → (cid:18) − ∞ X n =0 w n +1 n + − is − log(1 − w ) (cid:19) . Together with the Taylor expansion of w − log(1 − w ) and the standard formula ψ ( z ) + γ = ∞ X n =0 (cid:18) n + 1 − n + z (cid:19) this gives c ( s ) = ψ (cid:16) − is (cid:17) + γ + lim w → ( w − log(1 − w ) − log(1 − w )) = ψ (cid:16) − is (cid:17) + γ + log 2 , which completes the proof of the proposition. (cid:3) sing the formula ψ (1 − z ) − ψ ( z ) = π cot( πz ) it will be convenient to rewrite the result asΨ KZ ,s = 1 + hπi (cid:18) (log 2) t u + (cid:16) γ + ψ (cid:16) − is (cid:17) + ψ (cid:16) + is (cid:17) (cid:17) t m − πi (cid:16) πs (cid:17)(cid:0) t m + − t m − (cid:1)(cid:19) + O ( h ) . (2.7)Now, take µ ∈ h C (cid:74) h (cid:75) . Replacing s i Q × by s + µ in (2.4), we can construct yet another associator,which we denote by Ψ KZ ,s ; µ . If s ∈ R and µ ∈ h R (cid:74) h (cid:75) , then Ψ KZ ,s ; µ is unitary for the same reason asfor Ψ KZ ,s . Remark . Similarly to Remark 2.10, Ψ KZ ,s ; µ could be obtained from Ψ KZ ,s by slicing by a character ν of U ( k ) satisfying ν ( Z ) = − (2 (cid:126) a σ ) − µ . Since such a character does not always extend to U ( G σ ), tomake sense of this we should have allowed in the construction of Ψ KZ ,s arbitrary finite dimensionalrepresentations of g σ instead of those in Rep G σ . Alternatively, with s i R fixed, both Ψ KZ ,s ; µ andΨ KZ ,s + z for small z are specializations of an associator in U ( G σ × G ) (cid:74) h, µ (cid:75) constructed by treating µ as a second formal parameter. But this implies that Ψ KZ ,s ; µ is obtained from the Taylor expansion ofΨ KZ ,s + z at z = 0 by simply taking µ as the argument:Ψ KZ ,s ; µ = ∞ X k =0 µ k k ! d k Ψ KZ ,s ds k = ∞ X n,k =0 h n µ k k ! d k Ψ ( n )KZ ,s ds k . (2.8)This also works for s ∈ i ( R \ Q × ) if we consider only the components of Ψ KZ ,s ; µ in finite dimensionalblocks of U ( G σ × G ), which are well-defined and analytic in a neighborhood of s . Corollary 2.13.
For all s i Q × and µ, ν ∈ h C (cid:74) h (cid:75) , we have Ψ KZ ,s ; µ + ν − Ψ KZ ,s ; µ = h ν ) ν (ord( ν )) (cid:18) π (cid:16) ψ (cid:16)
12 + is (cid:17) − ψ (cid:16) − is (cid:17)(cid:17) t m − π (cid:16) πs (cid:17)(cid:0) t m + − t m − (cid:1)(cid:19) + O ( h ν ) ) . Proof.
By (2.8), we haveΨ KZ ,s ; µ + ν − Ψ KZ ,s ; µ = h ν ) ν (ord( ν )) d Ψ (1)KZ ,s ds + O ( h ν ) ) . Hence the result follows from (2.7). (cid:3)
Detecting co-Hochschild classes.
To see that the associators Ψ KZ ,s ; µ are not all equivalent, weneed to see that a perturbation of the parameter µ gives rise to a nontrivial 2-cocycle in B G,G σ . We canactually see that this is the case from results in Appendix A, but let us present a concrete cycle to detectthis.Consider the tensor Ω = [ t u , t u ] = X i,j [ X i , X j ] ⊗ X i ⊗ X j ∈ (cid:0)V u (cid:1) u (2.9)with ( X i ) i and ( X i ) i as in (1.1). Every element X ∈ g defines a function on G such that g ( X, (Ad g )( Z )) g . This way Ω defines an element of O ( G σ ) ⊗ O ( G ) ⊗ O ( G ), which by slightly abusingnotation we continue to denote by Ω. Thus, for ( g, h, k ) ∈ G σ × G × G ,Ω( g, h, k ) = (cid:0) [(Ad h )( Z ) , (Ad k )( Z )] , (Ad g )( Z ) (cid:1) g = (cid:0) [(Ad h )( Z ) , (Ad k )( Z )] , Z (cid:1) g , since G σ stabilizes Z . This is a 2-cycle in the complex ˜ B G,G σ from the proof of Proposition 2.1, asΩ( g, g, h ) − Ω( g, h, h ) + Ω( g, h, e ) = 0for all ( g, h ) ∈ G σ × G . Hence the map h Ω , ·i : U ( G σ × G ) → C defined by pairing with Ω passesto H ( B G,G σ ). Explicitly, for T ∈ U ( G σ × G ) we have h Ω , T i = (cid:15) ( T ) (cid:0) [(ad T )( Z ) , (ad T )( Z )] , Z (cid:1) g , (2.10)where ad denotes the extension of the adjoint representation of g to U ( G ). roposition 2.14. The elements t k , t m ± are -cocycles in B G,G σ . Furthermore, t k and t m = t m + + t m − are coboundaries, while h Ω , t m ± i = ± i m . In particular, t m + and − t m − represent the same nontrivial class in H ( B G,G σ ) .Proof. It is easy to check that d cH (1 ⊗ X ⊗ Y ) = 0 for all X, Y ∈ g . As t k and t m ± are k -invariant, theyare therefore 2-cocycles in B G,G σ .We have d cH ( C k ) = C k − ∆( C k ) + C k = − t k , so t k is a coboundary. Similarly, d cH ( C u ) = − t u , so that t u is also a coboundary, and hence t m = t u − t k is a coboundary as well.Next, take a basis ( Y j ) j in m + and the dual basis ( Y j ) j in m − . Using that ad Z acts by the scalar ± i on m ± , we then compute: h Ω , t m + i = X j (cid:0) [(ad Y j )( Z ) , (ad Y j )( Z )] , Z (cid:1) g = X j ([ Y j , Y j ] , Z ) g = X j ( Y j , [ Z, Y j ]) g = i X j ( Y j , Y j ) g = i dim C m + = i R m . The value h Ω , t m − i is obtained similarly, but it also follows from the above, as t m − = t m − t m + and t m is a coboundary. (cid:3) Remark . 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 ) in the same way as above. Let us callthis function ω . Then it is again easy to check that ω is a 2-cycle in the Hochschild chain complex( C n ( A, A ) = A ⊗ ( n +1) , b ) for A = O ( U ) k ⊂ C ( U/K ). Under the Hochschild–Kostant–Rosenberg map thiscycle corresponds to the differential 2-form associated with the Kostant–Kirillov–Souriau bracket on thecoadjoint orbit of ( · , Z ) g , which in turn defines a nonzero class in H ( U/K ; C ) ∼ = C .We have a left U ( G )-module structure on O ( U ) given by right translations: T.a = a (0) h a (1) , T i .Given T ∈ B nG,G σ , we can then define an n -cocycle D T in the Hochschild cochain complex ( C n ( A, A ) =Hom( A ⊗ n , A ) , δ ) by D T ( a , . . . , a n ) = (cid:15) ( T )( T .a ) · · · ( T n .a n ) . The Hochschild cochains act on the chains by contractions: given D ∈ C m ( A, A ), we have i D : C n ( A, A ) → C n − m ( A, A ) , a ⊗ · · · ⊗ a n a D ( a , . . . , a m ) ⊗ a m +1 ⊗ · · · ⊗ a n , with the convention i D = 0 if n < m .Now, if T ∈ B nG,G σ and c ∈ C n ( A, A ) is U -invariant (with respect to left translations), then i D T c ∈ A is U -invariant, hence a scalar. It can be checked that if bc = 0, then this scalar depends only on thecohomology class of T . Taking c = ω , we recover pairing (2.10): i D T ω = h Ω , T i .2.5. Classification of associators: Hermitian case.
We are now ready to establish, in the Hermitiancase, a universality result for the associators Ψ KZ ,s ; µ for generic quasi-coactions ( U ( G σ ) (cid:74) h (cid:75) , α, Ψ) of( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ) such that α = ∆ modulo h . Similarly to Section 2.2, it suffices to consider the case α = ∆. Theorem 2.16.
Let k = u σ < u be a Hermitian symmetric pair. Assume we are given a quasi-coaction ( U ( G σ ) (cid:74) h (cid:75) , ∆ , Ψ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ) such that the number h Ω , Ψ (1) i defined by (2.10) isneither ± i m , nor i ( ζ − ζ + 1) dim m for a root of unity ζ = ± . Then there exist s i Q × , µ ∈ h C (cid:74) h (cid:75) and H ∈ h U ( G σ × G ) k (cid:74) h (cid:75) such that (id ⊗ (cid:15) )( H ) = 1 and Ψ KZ ,s ; µ = H , Ψ H − , H − , . Furthermore, (i) the number s is unique up to adding ik ( k ∈ Z ), and once s is fixed, the element µ is uniquelydetermined; (ii) we can choose s ∈ R if and only if h Ω , Ψ (1) i is a purely imaginary number in the interval ( − i m , i m );(iii) if s ∈ R and Ψ is unitary, then µ ∈ h R (cid:74) h (cid:75) and H can be chosen to be unitary. e will later (Remark 2.20) slightly improve this result, by showing that for any Ψ the parameter µ isindependent of the choice of s . Proof.
By Proposition 2.14 and our restrictions on Ψ, we can choose s i Q × such that −
12 tanh (cid:16) πs (cid:17) h Ω , t m + − t m − i = − i (cid:16) πs (cid:17) dim m = h Ω , Ψ (1) i . (2.11)We then start with H = 1 and µ = 0 and modify them by induction on n to have Ψ KZ ,s ; µ = H , Ψ H − , H − , modulo h n +1 .Consider n = 1. By the proof of Theorem 2.6, Ψ (1)KZ ,s ; µ − Ψ (1) is a 2-cocycle in B G,G σ . By Lemma 1.2and Corollary 2.4, we have dim H ( B G,G σ ) = 1. Hence our choice of s , identity (2.7) and Proposition 2.14imply that Ψ (1)KZ ,s ; µ − Ψ (1) is a coboundary, so that Ψ (1) − Ψ (1)KZ ,s ; µ = d cH ( T ) for some T ∈ U ( G σ × G ) k .Letting H (1) = T , we then get Ψ KZ ,s ; µ = H , Ψ H − , H − , modulo h .For the induction step, assume we have Ψ KZ ,s ; µ = H , Ψ H − , H − , modulo h n +1 for some n ≥ ( n +1)KZ ,s ; µ − ( H , Ψ H − , H − , ) ( n +1) is a 2-cocycle in B G,G σ . On the other hand, by Corollary 2.13, for any a ∈ C , we haveΨ ( n +1)KZ ,s ; µ + h n a − Ψ ( n +1)KZ ,s ; µ = − a π (cid:16) πs (cid:17)(cid:0) t m + − t m − (cid:1) + bt m for some b ∈ C , and Ψ ( k )KZ ,s ; µ + h n a = Ψ ( k )KZ ,s ; µ for k ≤ n . As t m + − t m − represents a nontrivial cohomologyclass, the value of sech is nonzero for our s , and t m is cohomologically trivial, we see that with differentchoices of a the above difference can represent arbitrary classes in H ( B G,G σ ) ∼ = C . In particular, we canfind a ∈ C such that ( H , Ψ H − , H − , ) ( n +1) − Ψ ( n +1)KZ ,s ; µ + h n a = d cH ( T ) (2.12)for some T ∈ U ( G σ × G ) k . Replacing H by (1 + h n +1 T ) H and µ ( n ) by µ ( n ) + a , we then get Ψ KZ ,s ; µ = H , Ψ H − , H − , modulo h n +2 , proving the induction step.As at the step n of our induction process we only modify µ ( n − and H ( k ) for k ≥ n , in the limit weget the required µ and H . It remains to prove (i)–(iii).(iii): Assume s ∈ R and that Ψ is unitary. In this case we slightly modify the above inductive procedureto make sure that at every step we have unitarity of H and that µ ∈ h R (cid:74) h (cid:75) .Consider n = 1. We found H such that Ψ KZ ,s ; µ = H , Ψ H − , H − , modulo h . The unitarityof Ψ KZ ,s ; µ and Ψ implies then that the same identity holds for the unitary H ( H ∗ H ) − / instead of H ,cf. the proof of [NT11, Proposition 2.3].For the induction step, we assume that we have Ψ KZ ,s ; µ = H , Ψ H − , H − , modulo h n +1 for some n ≥ µ ∈ h R (cid:74) h (cid:75) and unitary H . Then we take the unique a ∈ C such that( H , Ψ H − , H − , ) ( n +1) − Ψ ( n +1)KZ ,s ; µ + a π (cid:16) πs (cid:17)(cid:0) t m + − t m − (cid:1) is a coboundary. By taking adjoints and using that ( t m + ) ∗ = t m − , we also get that (cid:0) ( H , Ψ H − , H − , ) ( n +1) − Ψ ( n +1)KZ ,s ; µ (cid:1) ∗ − ¯ a π (cid:16) πs (cid:17)(cid:0) t m + − t m − (cid:1) is a coboundary. Hence in order to conclude that a ∈ R it suffices to show that if we have two unitaries Ψ and Ψ in 1 + h U ( G σ × G ) (cid:74) h (cid:75) such that Ψ = Ψ modulo h n +1 , then the element Ψ ( n +1)1 − Ψ ( n +1)2 isskew-adjoint. But this is clear from the identity(Ψ − Ψ ) ∗ = − Ψ ∗ (Ψ − Ψ )Ψ ∗ . Then we take T satisfying (2.12) and replace µ ( n ) by µ ( n ) + a and H by(1 + h n +1 T ) (cid:0) (1 + h n +1 T ∗ )(1 + h n +1 T ) (cid:1) − / H , similarly to the step n = 1.(i), (ii): If Ψ KZ ,s ; µ = H , Ψ H − , H − , , then Ψ (1) − Ψ (1)KZ ,s = d cH ( H (1) ). Hence (2.11) is not onlysufficient but also necessary for the existence of H and µ . Therefore s is determined uniquely up toadding 2 ik ( k ∈ Z ). This also makes (ii) obvious.Next, assume Ψ KZ ,s ; µ = H , Ψ KZ ,s ; µ H − , H − , for some s i Q × , µ, µ ∈ h C (cid:74) h (cid:75) and H ∈ h U ( G σ × G ) k (cid:74) h (cid:75) . We have to show that µ = µ . Assume this is not the case. et n ≥ µ ( n ) = µ ( n ) . By Corollary 2.13 we have Ψ KZ ,s ; µ = Ψ KZ ,s ; µ modulo h n +1 . We claim that we can modify H so that we still have Ψ KZ ,s ; µ = H , Ψ KZ ,s ; µ H − , H − , ,but H = 1 modulo h n +1 .We will modify H by induction on k ≤ n to get Ψ KZ ,s ; µ = H , Ψ KZ ,s ; µ H − , H − , and H = 1modulo h k +1 . Assume we have these two properties for some k < n . Then d cH ( H ( k +1) ) = Ψ ( k +1)KZ ,s ; µ − Ψ ( k +1)KZ ,s ; µ = 0 . As H ( B G,G σ ) = 0 by Lemma 1.2 and Corollary 2.4, there exists a central element S ∈ U ( G σ ) suchthat H ( k +1) = S − S . Putting H = exp( − h k +1 ( S − S )), we see that Ψ KZ ,s ; µ commutes with H , ,hence we have H , Ψ KZ ,s ; µ H , H , = Ψ KZ ,s ; µ . It follows that by replacing H by H H we getΨ KZ ,s ; µ = H , Ψ KZ ,s ; µ H − , H − , and H = 1 modulo h k +2 . Thus our claim is proved.It follows now that Ψ ( n +1)KZ ,s ; µ − Ψ ( n +1)KZ ,s ; µ = d cH ( H ( n +1) ). Since t m + − t m − is not a coboundary, thiscontradicts Corollary 2.13. Hence µ = µ . (cid:3) Remark . In view of Corollary A.5, a similar result should in principle be true at the level of theuniversal enveloping algebras as well. However, since we only know that Ψ KZ ,s ; µ ∈ U ( G σ × G ) (cid:74) h (cid:75) (for s = 0), in the first place one has to show that Ψ KZ ,s ; µ belongs to U ( g σ ) ⊗ U ( g ) ⊗ (cid:74) h (cid:75) , at least up to sometwist.2.6. Classification of ribbon braids.
We complement our classification of associators by describingcompatible ribbon twist-braids, both in the Hermitian and non-Hermitian cases.In the following we always take the universal R -matrix R KZ = exp( − ht u ) ∈ U ( G ) (cid:74) h (cid:75) for ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ). Theorem 2.18. If k = u σ < u is a non-Hermitian symmetric pair, then the ribbon σ -braids for thequasi-coaction ( U ( G σ ) (cid:74) h (cid:75) , ∆ , Ψ KZ ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ , R KZ ) are the elements E KZ g = exp( − h (2 t k + C k )) g , where g ∈ Z ( U ) .Proof. The fact that exp( − h (2 t k + C k )) is a ribbon σ -braid is essentially proved in [Enr07], see [DCNTY19,Theorem 3.8]. Hence the elements exp( − h (2 t k + C k )) g ( g ∈ Z ( U )) are ribbon σ -braids as well. Theclaim that these are the only ribbon σ -braids follows from Theorem 2.8. (cid:3) A similar result holds in the Hermitian case, but the proof is more involved.
Theorem 2.19. If k = u σ < u is a Hermitian symmetric pair, s i Q × and µ ∈ h C (cid:74) h (cid:75) , then the ribbon σ -braids for the quasi-coaction ( U ( G σ ) (cid:74) h (cid:75) , ∆ , Ψ KZ ,s ; µ ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ , R KZ ) are the elements E KZ ,s ; µ g = exp( − h (2 t k + C k ) − πi ( s + µ ) Z ) g , where g ∈ Z ( U ) .Proof. The fact that the element E = E KZ ,s ; µ = exp( − h (2 t k + C k ) − πi ( s + µ ) Z ) is a ribbon σ -braid,and hence that the elements E (1 ⊗ g ) ( g ∈ Z ( U )) are ribbon σ -braids as well, follows again from the proofof [DCNTY19, Theorem 3.8].Let E be another ribbon σ -braid. The same argument as in the proof of Theorem 2.8 shows that E (0) = 1 ⊗ g for an element g ∈ Z G ( G σ ). We now use the same strategy as in the proof of the laststatement of that theorem to get more restrictions on g . Namely, we replace E by E (1 ⊗ g − ) to get aribbon ˜ σ -braid, where ˜ σ = (Ad g ) ◦ σ , and then twist E and Ψ KZ ,s ; µ further by an element H to get ridof the terms t u and t m in Ψ (1)KZ ,s ; µ , see (2.7) and recall that by Corollary 2.13 identity (2.7) is still validfor Ψ KZ ,s ; µ . Note for a future use that by the proof of Proposition 2.14 we can take H of the form H = 1 + h ( aC u + bC k ) (2.13)for appropriate constants a and b . Thus, our new associator Ψ = H , Ψ KZ ,s ; µ H − , H − , satisfiesΨ = 1 − h (cid:16) πs (cid:17) ( t m + − t m − ) + O ( h ) . (2.14) ooking at the order one terms in (1.5) and applying (cid:15) to the 0th leg, instead of (2.2) we now get E (1) = − t k + 1 ⊗ T + (id ⊗ Ad g )( t m ) − t m + tanh (cid:16) πs (cid:17) ( t m + − t m − ) + tanh (cid:16) πs (cid:17) (id ⊗ Ad g )( t m + − t m − ) , where T = ( (cid:15) ⊗ id)( E (1) ) and we used that t m ± = t m ∓ . As E (1) ∈ U ( G σ × G ), this means thatAd g − id ∓ tanh (cid:16) πs (cid:17) id ∓ tanh (cid:16) πs (cid:17) Ad g = 0 on m ± . (2.15)Hence Ad g = e ± πs id on m ± , which implies that g = exp( − πisZ ) g for some g ∈ Z ( G ) = Z ( U ).Without loss of generality we may assume that g = e , and then we want to prove that our original E coincides with E . It is more convenient to modify E in the same way as E , that is, by replacing it by HE (1 ⊗ g − )(id ⊗ ˜ σ )( H ) − = exp( − h (2 t k + C k ) − πiµZ ) , where we used that H has the form (2.13).Thus, our new setup is that we have two ribbon ˜ σ -braids E and E , with ˜ σ = (Ad exp( − πisZ )) ◦ σ ,with respect to an associator Ψ satisfying (2.14), E (0) = E (0) = 1, E (1) = − (2 t k + C k + πiµ (1) Z ) , (2.16)and the goal is to show that E = E .We will prove by induction on n that E ( n ) = E ( n ) . Consider the case n = 1. Put p = tanh (cid:16) πs (cid:17) and t = t m + − t m − . By the argument in the proof of Theorem 2.8, we have E (1) = E (1) + 1 ⊗ Y for some Y ∈ z g ( k ) = C Z .Put also T = E (2) − E (2) .Using (2.14), formula (1.5) for E , modulo h and terms depending only on R KZ and Ψ, becomes(1 + h E (1) + h E (2) ) , = (1 + h ( pt , − t u , ))(1 + h E (1) + h E (2) ) , (id ⊗ id ⊗ ˜ σ )(1 − h ( pt , + t u , )) , where we again used that t = − t . We have a similar formula for E . Taking the difference andcomparing the coefficients of h , we obtain T , = T , + ( pt , − t u , ) Y − Y (id ⊗ id ⊗ ˜ σ )( pt , + t u , ) . Using the identity ( pt − t m ) − (id ⊗ ˜ σ )( pt + t m ) = 0 , (2.17)which is an equivalent form of (2.15), we can write this as T , = T , − t k , Y + [ pt , − t m , , Y ] . (2.18)Similarly, formula (1.6) for E , modulo h and terms depending only on R KZ and Ψ, becomes(1 + h E (1) + h E (2) ) , = (cid:16) h (cid:16) p t , − t u , (cid:17)(cid:17) (1 + h E (1) + h E (2) ) , × (id ⊗ id ⊗ ˜ σ )(1 − h ( pt , + t u , ))(1 + h E (1) + h E (2) ) , (id ⊗ ˜ σ ⊗ ˜ σ ) (cid:16) h p t , (cid:17) , and we have a similar identity for E . Taking the difference and comparing the coefficients of h , weobtain T , = T , + T , + (cid:16) p t , − t u , (cid:17) ( Y + Y ) − Y (id ⊗ id ⊗ ˜ σ )( pt , + t u , ) − (id ⊗ id ⊗ ˜ σ )( pt , + t u , ) Y + ( Y + Y )(id ⊗ ˜ σ ⊗ ˜ σ ) (cid:16) p t , (cid:17) + Y E (1)0 , + E (1)0 , Y + Y Y . Using that ˜ σ ⊗ ˜ σ is the identity on m ± ⊗ m ∓ and that ad Y = − ad Y on m ± , we can write this as T , = T , + T , + ( pt , − t u , )( Y + Y ) − Y (id ⊗ id ⊗ ˜ σ )( pt , + t u , ) − (id ⊗ id ⊗ ˜ σ )( pt , + t u , ) Y + Y E (1)0 , + E (1)0 , Y + Y Y , and then, using again (2.17), we get T , = T , + T , − t k , ( Y + Y ) + [ pt , − t m , , Y ] + Y E (1)0 , + E (1)0 , Y + Y Y . (2.19)Subtracting (2.19) from (2.18), we obtain d cH ( T ) = 2 t k Y − E (1)01 Y − E (1)02 Y − Y Y . y (2.16), this means that d cH ( T ) = (2 t k + 2 t k + C k + πiµ (1) Z ) Y + (2 t k + C k + πiµ (1) Z ) Y − Y Y . As Y ∈ C Z , we can also write this as d cH ( T ) = (2 t k + 2 t k + C k ) Y + (2 t k + C k + 2 πiµ (1) Z ) Y − Y Y . On the other hand, a straightforward computation shows that the right hand side of the above identityis the coboundary of 2 t k Y + C k Y + C k Y + 2 πiµ (1) Z Y − Y Y . As H ( B G,G σ ) = 0 by Lemma 1.2 and Corollary 2.4, it follows that there exists S in the center of U ( G σ )such that T = S − S + 2 t k , Y + C k Y + C k Y + 2 πiµ (1) Z Y − Y Y . The only consequence of the above identity that we need is that T ∈ U ( G σ × G σ ). By looking at (2.18)we see that this implies[ pt − t m , ⊗ Y ] = ( p − t m + , ⊗ Y ] − ( p + 1)[ t m − , ⊗ Y ] ∈ U ( G σ × G σ ) . As Y is a scalar multiple of Z , and ad Z acts by nonzero operators on m ± , this is possible only if Y = 0.This shows that E (1) = E (1) .The induction step is similar. Assuming that E ( k ) = E ( k ) for k < n for some n ≥
2, we have E ( n ) = E ( n ) + 1 ⊗ Y for an element Y ∈ z g ( k ) = C Z . Then, with T = E ( n +1) − E ( n +1) , comparingthe coefficients of h n +1 in (1.5) we get the same identity (2.18). If we do the same for (1.6), the onlydifference from (2.19) is that we do not get the term Y Y at the end. But this has almost no effect onthe rest of the argument, we just have to remove the terms Y Y and Y Y in the subsequent identities.Thus we get Y = 0. (cid:3) Remark . Theorem 2.19 implies that in the Hermitian case the ribbon twist-braids contain completeinformation about the associators. Namely, assume that Ψ KZ ,s ; µ = H , Ψ KZ ,s ; µ H − , H − , for some H ∈ h U ( G σ × G ) k (cid:74) h (cid:75) . By (1.7) and Theorem 2.19 it follows that H exp( − h (2 t k + C k ) − πi ( s + µ ) Z )(id ⊗ σ )( H ) − = exp( − h (2 t k + C k ) − πi ( s + µ ) Z ) g for some g ∈ Z ( U ). Since ( (cid:15) ⊗ id)( H ) is a k -invariant element of U ( G ) and σ is an inner automorphismdefined by an element of K , we have σ (cid:0) ( (cid:15) ⊗ id)( H ) (cid:1) = ( (cid:15) ⊗ id)( H ), and then by applying (cid:15) to the 0th legwe get exp( − hC k − πi ( s + µ ) Z ) = exp( − hC k − πi ( s + µ ) Z ) g. This implies that s = s + 2 ik for some k ∈ Z , and µ = µ .3. Interpolated subgroups
In this section we fix an involutive automorphism ν of u such that u ν < u is an irreducible Hermitiansymmetric pair. We are going to fix a Cartan subalgebra in u ν and then apply the results of the previoussection to particular conjugates of ν and true coactions of a quantization of ( U ( G ) , ∆). Along the waywe will study a distinguished family of coisotropic subgroups of U that are conjugates of U ν . The mainresult is Theorem 3.17, where we in particular relate the induced Poisson structures on the associatedhomogeneous spaces to a Poisson pencil for the Kostant–Kirillov–Souriau bracket which appears from thecyclotomic KZ-equations.Throughout this section we use the subscript ν for the Lie algebra constructions we had for σ . Thus, m ν = { X ∈ u | ν ( X ) = − X } , Z ν ∈ z ( g ν ).3.1. Root vectors and Poisson structure.
Let us quickly review a standard Poisson–Lie groupstructure on U making U ν a Poisson–Lie subgroup (for Hermitian symmetric pairs).Let t be a Cartan subalgebra of u containing z ( u ν ). Then t is contained in u ν , and its complexification h is a Cartan subalgebra of g .Recall that a root α is called compact if g α ⊂ g ν , and noncompact otherwise. As g ν is the centralizerof z ( u ν ), a root is compact if and only if it vanishes on z ( u ν ).As in Section 2.3, we fix Z ν ∈ z ( u ν ) such that ( Z ν , Z ν ) g = − a − ν . Let us fix an ordered basis in i t ,with − iZ ν being the first element of the basis, and consider the corresponding lexicographic order on theroots. Then, in this order, any noncompact positive root is bigger than any compact root. Furthermore,the noncompact positive roots are totally positive in the sense of [HC55], meaning that if γ is a noncompact ositive root, α , . . . , α k are compact roots, and m , . . . , m k are integers such that γ = γ + P ki =1 m i α i isa root, then γ is positive.We denote by Φ the set of all roots, and by Φ + that of positive roots. We further denote by Φ +nc (resp. Φ − nc ) the set of positive (resp. negative) noncompact roots. Let Π = { α i } i ∈ I be the set of simplepositive roots. Recall that we denote by m ν ± ⊂ m C ν the eigenspaces of ad Z ν corresponding to theeigenvalues ± i . It is clear from our choice of the ordering that m ν + = M α ∈ Φ +nc g α , m ν − = M α ∈ Φ − nc g α . Since 2 i is not an eigenvalue of ad Z ν , we have [ m ν + , m ν + ] = 0. It follows that p = g ν + m ν + is aparabolic subalgebra of g . As g ν acts irreducibly on g / p ∼ = m ν − , this parabolic subalgebra is maximal.Hence it corresponds to a maximal proper subset of Π. This set must consist of compact roots, and henceits complement consists of one noncompact root. We denote this unique noncompact simple positive rootby α o . It corresponds to the black vertex in a standard Vogan diagram of ν .For every root α , let H α ∈ h be the element dual to the coroot α ∨ = α,α ) α , so that we have α ( H β ) = ( α, β ∨ ) ( α, β ∈ Φ) , where ( · , · ) is the scalar product dual to the restriction of ( · , · ) g to h . For positive roots α , we choose rootvectors X α ∈ g α such that the antilinear involution for u satisfies( X α , X ∗ α ) g = 2( α, α ) . Then [ X α , X ∗ α ] = H α , and we put X − α = X ∗ α .Let Y α = i X α + X − α ) ∈ u , Z α = 12 ( X α − X − α ) ∈ u ( α ∈ Φ + ) , and consider the antisymmetric tensor r = X α> ( α, α )( Y α ⊗ Z α − Z α ⊗ Y α ) ∈ u ⊗ . Note that we can also write r = i X α> ( α, α )2 ( X − α ⊗ X α − X α ⊗ X − α ) ∈ g ⊗ . (3.1)Then ir ± t u (with t u defined by (1.1)) satisfies the classical Yang–Baxter equation, and u becomes a Liebialgebra with the cobracket δ r ( X ) = [ r, ∆( X )] . Thus, U becomes a Poisson–Lie group with the Sklyanin Poisson bracket { f , f } Sk = m ( r ( l,l ) − r ( r,r ) )( f ⊗ f ) , where m is the product map, and for X ∈ u and f ∈ C ∞ ( U ) we put( X ( l ) f )( g ) = ddt f ( e tX g ) | t =0 , ( X ( r ) f )( g ) = ddt f ( ge tX ) | t =0 . Note that if we as usual view u and U as sitting inside U ( G ), then we can write the Sklyanin bracket as { f , f } Sk ( g ) = h f ⊗ f , [ r, g ⊗ g ] i ( g ∈ U, f , f ∈ O ( U )) . The Lie algebra u ν is the intersection of u with a parabolic subalgebra of g , namely, u ν = u ∩ p . It iswell-known that this implies that U ν is a Poisson–Lie subgroup of U .3.2. Satake form.
We will have to conjugate U ν in order to go beyond the Poisson–Lie subgroups,making it closer to the coisotropic subgroup associated with symmetric pairs [FL04]. In order to do so, letus review the Satake form of involutions.For X ⊂ I , we denote by Π X the subset { α j | j ∈ X } ⊂ Π. Assume θ is a nontrivial involution on u such that its extension to g leaves h globally invariant. We write Θ ∈ End( h ∗ ) for the endomorphismdual to θ | h . Definition 3.1.
We say that θ is in maximally split form , or in Satake form , with respect to ( h , b + ), orthat ( h , b + ) is a split pair for θ , if there exists X ⊂ I satisfying the following conditions:(1) Φ + ∩ Θ(Φ + ) = Φ + ∩ Z Π X , θ = id on g α for all α ∈ Φ + ∩ Z Π X .The above set X is then uniquely determined, representing the black vertices in the correspondingSatake diagram. Then there exist unimodular w α ∈ C such that θ ( X α ) = w α X Θ( α ) ( α ∈ Φ) . Moreover, there exists a unique Dynkin diagram involution τ θ such thatΘ( α ) = − w X τ θ ( α ) ( α ∈ Φ) , (3.2)with w X the longest element of the Weyl group associated to X . This involution leaves the set X globallyinvariant. See [KW92] and [Kol14, Appendix A] for details.Put I ns = { i ∈ I \ X | τ θ ( i ) = i and ( α i , α j ) = 0 for all j ∈ X } , which corresponds to the τ θ -stable white vertices not connected to any black vertices in the Satakediagram. We then put I S = { i ∈ I ns | ( α ∨ j , α i ) ∈ Z for all j ∈ I ns } . We also put I C = { i ∈ I \ X | τ θ ( i ) = i and ( α i , Θ( α i )) = 0 } . It can be shown that this is the set of white vertices not fixed by τ θ such that, if ¯ α i is the correspondingrestricted root, then 2¯ α i is not a restricted root, see [Ara62; Hel01, p. 530].We will use the following definition from [DCNTY19]. Definition 3.2.
We say that a Hermitian symmetric pair u θ < u is • of S-type , if I S = ∅ , • of C-type , if there exists i ∈ I \ ( X ∪ I C ) such that τ θ ( i ) = i .See [DCNTY19, Appendix C] for a concrete classification of the Hermitian symmetric pairs into thesetypes. Recall that the restricted root system is always of type C or BC in the Hermitian case [Moo64,Theorem 2]. By a case-by-case analysis (see, e.g., [OV90, Reference Chapter, table 9; Hel01, Chapter X,Table VI; Kna02, Appendix C]), one sees that we are in the S-type case exactly when the restricted rootsystem is of type C, while we are in the C-type case when the restricted root system is of type BC.Moreover, by [DCNTY19, Lemma C.2], in the C-type case there exists exactly one τ θ -orbit of the form { o = o = τ θ ( o ) } in I \ ( X ∪ I C ). We call the roots α o and α o distinguished . By the same lemma, in theS-type case the set I S consists of one root α o , which we again call distinguished. Note that we use thesame label o for one of the distinguished roots as for a noncompact root in Section 3.1. This will bejustified in Proposition 3.5.We now recall the construction of a cascade of orthogonal roots in the setting of Section 3.1. Let γ bethe largest in our lexicographic order (hence necessarily noncompact) root of g (0) = g , and let g (1) be thecentralizer of H γ . If g (1) g ν , then let γ be the largest root such that X γ ∈ g (1) , and let g (2) be thecentralizer of { H γ , H γ } . Continuing this until we have g ( s ) ⊂ g ν , we obtain a strictly decreasing sequenceof Lie subalgebras ( g ( i ) ) si =1 and noncompact positive roots γ , . . . , γ s . Furthermore, these roots form amaximal family of mutually orthogonal noncompact roots, and they are mutually strongly orthogonal,i.e., γ i ± γ j is never a root, see [Kna02, Lemma 7.143].Let h + ⊂ h consist of all H ∈ h with γ i ( H ) = 0 for all i . Let h − be the complex linear span of { H γ i | i = 1 , . . . , s } . Then clearly h = h + ⊕ h − , and hence for the dual spaces we have h ∗ = ( h + ) ∗ ⊕ ( h − ) ∗ .Using this decomposition we will often think of the γ i as elements of ( h − ) ∗ .On the other hand, let e h − be the complex linear span of { X γ i − X − γ i | i = 1 , . . . , s } . Then also h + ⊕ e h − is a Cartan subalgebra of g . This is a version of Harish-Chandra’s construction of maximallysplit Cartan subalgebras.To relate the two Cartan subalgebras, we consider the (partial, unitary) Cayley transform Ad g , where g = exp (cid:16) πi s X i =1 ( X − γ i + X γ i ) (cid:17) ∈ U, cf. [Kna02, Section VI.7]. Then (Ad g )( iH γ i ) = X γ i − X − γ i (see Lemma 3.9 below), so (Ad g )( h − ) = e h − ,while Ad g acts as the identity on h + . We then have the following concrete presentation of maximallysplit involutions. roposition 3.3. The involution θ = (Ad g ) − ◦ ν ◦ (Ad g ) is in maximally split form with respect to ( h , b + ) , with the associated set X = { i ∈ I | H α i ∈ h + } . In order to prove this, we will need some detailed information on the restricted roots. We will followclosely the treatment of Harish-Chandra in [HC56].For λ, µ ∈ h ∗ , let us write λ ∼ µ when they restrict to the same functional on h − . For each i , let C i denote the set of compact positive roots α such that α ∼ γ i . Similarly, let P i be the set of noncompactpositive roots γ such that γ ∼ γ i .Next, for i < j , let C ij denote the set of compact positive roots α such that α ∼ ( γ i − γ j ). Let P ij denote the set of noncompact positive roots γ such that γ ∼ ( γ i + γ j ).Finally, let P = { γ , . . . , γ s } , and C denote the set of positive roots α such that α ∼
0, that is, H α ∈ h + , or equivalently, α is orthogonal to γ , . . . , γ s . The set C consists of compact roots, as { γ , . . . , γ s } is a maximal family of mutually orthogonal noncompact positive roots. Proposition 3.4.
The set Φ + is partitioned by the subsets P , C , ( P i ) si =1 , ( C i ) si =1 , ( P ij ) ≤ i Proof of Proposition 3.3. First of all observe that by construction θ = id on h + and θ = − id on h − . Thisalready implies that Φ + ∩ Z Π X ⊂ { α > | Θ( α ) = α } ⊂ Φ + ∩ Θ(Φ + ) . Next, by Proposition 3.4, every positive root restricts to 0, γ i , γ i or ( γ i ± γ j ) for some i , j with i < j . From this we see that the intersection of the restrictions of Φ + and Θ(Φ + ) is at most { } . Inparticular, if α ∈ Φ + ∩ Θ(Φ + ), then α restricts to 0. Decompose such an α into a combination of thesimple roots and restrict to h − . Since no nontrivial sum with nonnegative integral coefficients of thevectors γ i , γ i and ( γ i ± γ j ) ( i < j ) is zero, it follows that α decomposes into a combination of thesimple roots that restrict to 0, that is, α ∈ Φ + ∩ Z Π X . This proves property (1) in Definition 3.1.To establish property (2), take α ∈ Φ + ∩ Z Π X , that is, α is a positive root restricting to 0. This rootmust be compact, since { γ , . . . , γ s } is a maximal family of mutually orthogonal noncompact roots, andit is strongly orthogonal to γ i by Proposition 3.4 (i). Therefore g α ⊂ g ν and g α centralizers X − γ i + X γ i .Hence θ acts trivially on g α . (cid:3) Refining the observation after Definition 3.2, we have the following. Proposition 3.5. With the above notation, the roots γ , . . . , γ s are all of the same length. The restrictionmap α α | h − defines a bijection between the τ θ -orbits in Π \ Π X and a basis of the restricted root system.This basis and the distinguished roots are concretely described as follows. • S-type: The restricted root system is of type C s , consisting of {± ( γ i ± γ j ) } i The restricted root system is of type BC s , consisting of {± ( γ i ± γ j ) } i In the S-type case we have Z ν = i s X j =1 H γ j , as well as a ν = 2 / √ s for u s ⊂ sp s and a ν = p /s in all other cases.Proof. Since in the S-type case the compact positive roots restrict to 0 or ( γ i − γ j ) ( i < j ), such rootsvanish on P sj =1 H γ j . Therefore P sj =1 H γ j ∈ z ( g ν ). As we must have γ j ( Z ν ) = i for any j , we get theformula for Z ν in the formulation.A case-by-case verification shows that γ s = α o is a short root in all cases except for u s ⊂ sp s , whilein the last case it is a long root of length 2. Since the roots γ , . . . , γ s are all of the same length and( Z ν , Z ν ) g = − a − ν , we then get the formula for a ν . (cid:3) A family of coisotropic subgroups. Now we are ready to introduce a one-parameter family ofinvolutive automorphisms interpolating between ν and θ = (Ad g ) − ◦ ν ◦ (Ad g ), which define coisotropicsubgroups of U .For φ ∈ R , let g φ = exp (cid:16) πiφ s X i =1 ( X − γ i + X γ i ) (cid:17) , θ φ = (Ad g φ ) ◦ θ ◦ (Ad g φ ) − , so that θ = θ and θ = ν . Definition 3.7. We will write K φ for U θ φ = g φ − U ν g − φ , and denote its Lie algebra by k φ .Our first goal is to understand how the r -matrix (3.1) transforms under g φ . Proposition 3.8. For all φ ∈ R , we have (Ad g φ ) ⊗ ( r ) − cos (cid:16) πφ (cid:17) r ∈ u ν ⊗ u + u ⊗ u ν . For the proof we need to introduce a more convenient basis for computations. We will write e α = − iX α to adapt to the conventions of [Bou07, Tit66]. Note that then e ∗ α = − e − α and [ e α , e − α ] = − H α . Let us write [ e α , e β ] = N α,β e α + β when α , β , and α + β are roots. We then have ¯ N α,β = N − α, − β and N α,β N − α,α + β = − p ( q + 1) , | N α,β | = q + 1 , (3.3) here p , resp. q , is the largest integer such that β + pα , resp. β − qα , is a root [Bou07, Section VIII.2.4].(In fact, if we were more careful in choosing root vectors, we could arrange N α,β to be real, with the signof N α,β described in [Tit66].)Recall from Proposition 3.4 that Φ + is partitioned by the subsets P , C , ( P i ) si =1 , ( C i ) si =1 , ( P ij ) ≤ i The map Ad g φ acts as follows: x i x i , y i cos (cid:16) πφ (cid:17) y i − sin (cid:16) πφ (cid:17) H γ i , H γ i sin (cid:16) πφ (cid:17) y i + cos (cid:16) πφ (cid:17) H γ i . Proof. Since γ i is strongly orthogonal to γ j for j = i , we have h s X j =1 ( e − γ j + e γ j ) , x i i = 0 , h s X j =1 ( e − γ j + e γ j ) , y i i = 2 H γ i , h s X j =1 ( e − γ j + e γ j ) , H γ i i = − y i . Since g φ = exp (cid:16) − πφ s X j =1 ( e − γ j + e γ j ) (cid:17) , exp (cid:18) − λλ (cid:19) = (cid:18) cos λ − sin λ sin λ cos λ (cid:19) , (3.4)we get the result. (cid:3) As Ad g φ acts trivially on the orthogonal complement of { H γ , . . . , H γ s } in h , this lemma alreadydescribes the action of Ad g φ on the Cartan subalgebra. Lemma 3.10. If α ∈ C , then Ad g φ acts trivially on e ± α and H α .Proof. This follows from Proposition 3.4 (i). (cid:3) Next, on the root vectors of P i and C i we have the following description of Ad g φ . Lemma 3.11. Assume ≤ i ≤ s , and γ ∈ P i and α ∈ C i are such that γ + α = γ i . Then Ad g φ acts asfollows: e γ cos (cid:16) πφ (cid:17) e γ − ¯ N γ i , − γ sin (cid:16) πφ (cid:17) e − α , e − α N γ i , − γ sin (cid:16) πφ (cid:17) e γ + cos (cid:16) πφ (cid:17) e − α , and we have | N γ i , − γ | = 1 . Note that we also get formulas for the action on e − γ and e α by taking adjoints. Proof. From Proposition 3.4 we see that γ + γ i , γ − γ i and γ ± γ j for j = i are not roots. Hence h s X j =1 ( e − γ j + e γ j ) , e γ i = [ e − γ i , e γ ] = N − γ i ,γ e − α = ¯ N γ i , − γ e − α , and N − γ i ,γ N γ i , − α = − | N − γ i ,γ | = 1 by (3.3). Similarly, γ i + α and α ± γ j for j = i are not roots,hence h s X j =1 ( e − γ j + e γ j ) , ¯ N γ i , − γ e − α i = [ e γ i , ¯ N γ i , − γ e − α ] = − e γ . The lemma follows by again using (3.4). (cid:3) Consider now γ ∈ P ij , and put α = γ − γ j ∈ C ij . By Proposition 3.4 (iii) we also have roots γ ∈ P ij and α ∈ C ij such that γ i − α = γ = γ j + α . Put ε ( γ ) = N − γ i ,γ N − γ j ,γ , ε ( α ) = N − γ i ,α N γ j ,α , and take the elements x γ = e γ − ε ( γ ) e − γ , y γ = e γ + ε ( γ ) e − γ , x α = e α − ε ( α ) e − α , y α = e α + ε ( α ) e − α . Lemma 3.12. If γ ∈ P ij and α ∈ C ij are such that γ − α = γ j , then the map Ad g φ acts as follows: x γ x γ , x α x α , y γ cos (cid:16) πφ (cid:17) y γ − ¯ N γ j , − γ sin (cid:16) πφ (cid:17) y α , y α N γ j , − γ sin (cid:16) πφ (cid:17) y γ + cos (cid:16) πφ (cid:17) y α , and we have | ε ( γ ) | = | ε ( α ) | = | N γ j , − γ | = 1 . roof. First of all observe that γ i − γ = α , γ i − γ = α, γ j − γ = − α, γ j − γ = − α . (3.5)From Proposition 3.4 we see that − γ i − γ and 2 γ i − γ are not roots, hence | N γ i , − γ | = 1 by the secondidentity in (3.3). For similar reasons the numbers N γ i , − γ , N γ j , − γ and N γ j , − γ are of modulus one, andby the first identity in (3.3) we have N γ i , − γ N − γ i ,α = N γ i , − γ N − γ i ,α = N γ j , − γ N − γ j , − α = N γ j , − γ N − γ j , − α = − . (3.6)We claim that also the following identity holds: N − γ j , − α N − γ i ,γ = N − γ i ,α N − γ j ,γ . (3.7)Indeed, the expressions on both sides are precisely the coefficients of e − γ in [ e − γ j , [ e − γ i , e γ ]] and[ e − γ i , [ e − γ j , e γ ]]. By the Jacobi identity,[ e − γ j , [ e − γ i , e γ ]] − [ e − γ i , [ e − γ j , e γ ]] = [ e γ , [ e − γ i , e − γ j ]] . But we have [ e − γ i , e − γ j ] = 0 by strong orthogonality. Thus our claim is proved.Now, a simple computation using (3.5)–(3.7) gives h s X k =1 ( e − γ k + e γ k ) , x γ i = h s X k =1 ( e − γ k + e γ k ) , x α i = 0 , h s X k =1 ( e − γ k + e γ k ) , y γ i = 2 ¯ N γ j , − γ y α , h s X k =1 ( e − γ k + e γ k ) , ¯ N γ j , − γ y α i = − y γ . The lemma follows again from (3.4). (cid:3) Proof of Proposition 3.8. We have r = − i X α> ( α, α )2 ( e − α ⊗ e α − e α ⊗ e − α ) . We will use the partition of Φ + into the subsets P , C , ( P i ) si =1 , ( C i ) si =1 , ( P ij ) ≤ i For every φ ∈ R , the subgroup K φ of Definition 3.7 is a coisotropic subgroup of ( U, r ) ,that is, δ r ( k φ ) ⊂ k φ ⊗ u + u ⊗ k φ . It is a Poisson–Lie subgroup if and only if φ is an odd integer.Proof. By definition we have k φ = (Ad g φ − )( u ν ). Since U ν is a Poisson–Lie subgroup of ( U, r ), K φ is aPoisson–Lie subgroup of ( U, (Ad g φ − ) ⊗ ( r )). As r − cos (cid:16) π (1 − φ )2 (cid:17) (Ad g φ − ) ⊗ ( r ) ∈ k φ ⊗ u + u ⊗ k φ , this shows that K φ is coisotropic in ( U, r ).Assume now that K φ is a Poisson–Lie subgroup of ( U, r ) for some φ . Since K φ has the same rank as U ,it follows that k φ must contain the Cartan subalgebra t (see, e.g., [Sto03, Proposition 2.1]). Therefore θ φ = (Ad g φ − ) ◦ ν ◦ (Ad g φ − ) − acts trivially on h . From Lemma 3.9 we see that this is the case if andonly if sin( π (1 − φ )2 ) = 0, that is, φ is an odd integer.Assume that indeed φ = 2 n + 1 for some n ∈ Z . In the S-type case, when the sets P i and C i are emptyfor 1 ≤ i ≤ s , we see from Lemmas 3.10 and 3.12 that k C φ = g ν , so K φ = U ν is a Poisson-Lie subgroup.Consider the C-type case. If n is even, so that cos( π ( φ − ) = ± 1, we see from Lemmas 3.10–3.12 that k C φ = g ν , so K φ = U ν is again a Poisson–Lie subgroup.Assume now that n is odd. Then sin( π ( φ − ) = ± 1, and we see from Lemmas 3.10–3.12 that k C φ isspanned by h , X ± α ( α ∈ C ), X ± γ ( γ ∈ P i , 1 ≤ i ≤ s ) and X ± α ( α ∈ C ij ). Moreover, by Proposition 3.5,the nondistinguished simple roots in Π \ Π X lie in the sets C i,i +1 , while the distinguished roots satisfy α o ∈ P s and α o ∈ C s . We conclude that we have X ± α ∈ k C φ for the nondistinguished simple roots α , X ± α o ∈ k C φ and X ± α o k C φ . It follows that if q ⊂ g is the parabolic subalgebra defined by the subsetΠ \ { α o } of simple roots, then q ∩ u ⊂ k φ . We have a Dynkin diagram involution τ θ mapping Π \ { α o } onto Π \ { α o } . Then the corresponding automorphism of g maps q onto p = g ν + m ν + . Since u ν = p ∩ u is a maximal proper Lie subalgebra of u , it follows that q ∩ u is a maximal proper Lie subalgebra of u .Hence q ∩ u = k φ , and therefore K φ is a Poisson–Lie subgroup of ( U, r ). (cid:3) Remark . We see from the above argument, or directly from Lemmas 3.9–3.12, that (Ad g φ )( u ν ) = u ν if and only if φ ∈ Z in the S-type case and φ ∈ Z in the C-type case.We finish this subsection by exhibiting generators of k C φ . Proposition 3.15. If φ ∈ R \ (1 + 2 Z ) , then the Lie algebra k C φ is generated by the following elements: H ∈ h θ , X ± α for α ∈ Π X , X α + θ ( X α ) for the nondistinguished roots α ∈ Π \ Π X , plus the followingelements: • S-type: X α o + θ ( X α o ) − s o H α o , where s o = i tan( πφ ) ; • C-type: X α o + c o θ ( X α o ) and X α o + c − o θ ( X α o ) , where c o = − cot( π ( φ − .Proof. We denote by g φ the Lie algebra generated by the elements in the formulation. For φ = 0,the generators of g are the adjoints of the generators of g θ from [Kol14, Lemma 2.8]. Since g θ is ∗ -invariant, we therefore get g = g θ , that is, the proposition is true for φ = 0. In order to prove it for all φ ∈ R \ { Z } it suffices to show that g φ = (Ad g φ )( g ).We will check how Ad g φ acts on the generators of g . By definition, Ad g φ is the identity map on h θ = h + . By Lemma 3.10 it is also the identity map on X ± α for α ∈ Π X ⊂ C . ext, consider a nondistinguished root α ∈ Π \ Π X . Then α ∈ C i,i +1 for some 1 ≤ i ≤ s − 1. As θ = (Ad g − ) ◦ ν ◦ (Ad g ), from Lemma 3.12 we see that θ ( x α ) = x α and θ ( y α ) = − y α . It follows that θ ( e α ) = − ε ( α ) e − α , and therefore X α + θ ( X α ) = ie α + iθ ( e α ) = ix α . By Lemma 3.12, Ad g φ acts trivially on this element.It remains to understand what happens with the generators corresponding to the distinguished roots.Consider the S-type case. Then the distinguished root is α o = γ s . By Lemma 3.9 we have θ ( x s ) = − x s and θ ( y s ) = y s , hence θ ( e γ s ) = − e − γ s . Therefore X α o + θ ( X α o ) = iy s . By Lemma 3.9 we then get(Ad g φ )( X α o + θ ( X α o )) = cos (cid:16) πφ (cid:17) ( X α o + θ ( X α o )) − i sin (cid:16) πφ (cid:17) H γ s , which is exactly the remaining generator of g φ multiplied by cos( πφ ).Consider now the C-type case. In this case the distinguished roots are α o ∈ P s and α o ∈ C s . Generally,if γ ∈ P s and α ∈ C s are such that γ + α = γ s , then by Lemma 3.11 we have θ ( e γ ) = − ¯ N γ s , − γ e − α .Applying the same lemma again we get(Ad g φ )( X γ + θ ( X γ )) = (cid:16) cos (cid:16) πφ (cid:17) − sin (cid:16) πφ (cid:17)(cid:17) X γ + (cid:16) cos (cid:16) πφ (cid:17) + sin (cid:16) πφ (cid:17)(cid:17) θ ( X γ ) . For γ = α o the right hand side is, up to the factor cos( πφ ) − sin( πφ ) = −√ π ( φ − ), the generatorof g φ corresponding to α o . We similarly get(Ad g φ )( X α + θ ( X α )) = √ (cid:16) π ( φ − (cid:17) X α − √ (cid:16) π ( φ − (cid:17) θ ( X α ) , so again we see that for α = α o the right hand side is, up to a factor, the corresponding generator of g φ .Thus the identity g φ = (Ad g φ )( g ) is proved. (cid:3) Definition 3.16. Denote by G φ the subgroup G θ φ = (Ad g φ − )( G ν ) of G .3.4. Coactions of quantized multiplier algebras. Let us relate the computation of the previoussubsection to the associators from Section 2.3.Given a reductive algebraic subgroup H (which will be G φ ) of G , consider a coaction ( U ( H ) (cid:74) h (cid:75) , α ) of amultiplier Hopf algebra ( U ( G ) (cid:74) h (cid:75) , ∆ h ). By Lemma 1.1, if ∆ h and α both equal ∆ modulo h , they can betwisted to ∆. We will assume that this can be done by elements satisfying extra properties. Specifically,assume there exist F ∈ U ( G × G ) (cid:74) h (cid:75) and G ∈ U ( H × G ) (cid:74) h (cid:75) such that G (0) = 1 , (id ⊗ (cid:15) )( G ) = 1 , α = G ∆( · ) G − , (3.8) F (0) = 1 , ( (cid:15) ⊗ id)( F ) = (id ⊗ (cid:15) )( F ) = 1 , ∆ h = F ∆( · ) F − , (3.9) F = 1 + h ir O ( h ) , (3.10)(id ⊗ ∆)( F − )(1 ⊗ F − )( F ⊗ ⊗ id)( F ) = Φ KZ . (3.11)We remind that Φ KZ = Φ( (cid:126) t u , (cid:126) t u ) ∈ U ( g ) ⊗ (cid:74) h (cid:75) is Drinfeld’s KZ-associator for G . Then by twisting by( F − , G − ) we get a quasi-coaction ( U ( H ) (cid:74) h (cid:75) , ∆ , Ψ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ) and we can try to apply theresults of Section 2. Theorem 3.17. Let u ν < u be a Hermitian symmetric pair, and let G φ be as in Definition 3.16 forsome φ ∈ R \ (1 + 2 Z ) . Assume we are given a coaction α : U ( G φ ) (cid:74) h (cid:75) → U ( G φ × G ) (cid:74) h (cid:75) of a multiplierHopf algebra ( U ( G ) (cid:74) h (cid:75) , ∆ h ) . Assume also that there exist F ∈ U ( G × G ) (cid:74) h (cid:75) and G ∈ U ( G φ × G ) (cid:74) h (cid:75) satisfying conditions (3.8) – (3.11) . Then there exist unique s φ ∈ R and µ ∈ h C (cid:74) h (cid:75) such that the coaction isobtained by twisting the quasi-coaction ( U ( G φ ) (cid:74) h (cid:75) , ∆ , Ψ KZ ,s φ ; µ ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ) . The parameter s φ is determined by sin (cid:16) πφ (cid:17) = tanh (cid:16) πs φ (cid:17) . (3.12) If, in addition, F and G are chosen to be unitary, then µ ∈ h R (cid:74) h (cid:75) . ere Ψ KZ ,s φ ; µ is defined for the Hermitian symmetric pair k φ < u as in Section 2.3, using the element Z φ = (Ad g φ − )( Z ν ) of z ( k φ ). We will give examples of coactions satisfying the assumptions of the theoremin Section 4.We will need the following lemma for the uniqueness part. Lemma 3.18 (cf. [Dri90, Proposition 3.2]) . Assume we are given a homomorphism ∆ h : U ( G ) (cid:74) h (cid:75) →U ( G × G ) (cid:74) h (cid:75) and two elements F , F ∈ U ( G × G ) (cid:74) h (cid:75) satisfying (3.9) and the identity (id ⊗ ∆)( F − )(1 ⊗ F − )( F ⊗ ⊗ id)( F ) = (id ⊗ ∆)( F )(1 ⊗ F )( F ⊗ ⊗ id)( F ) , (3.13) defining a G -invariant element of U ( G ) (cid:74) h (cid:75) . Then there exists a unique central element u ∈ U ( G ) (cid:74) h (cid:75) such that u = 1 modulo h and F = F ( u ⊗ u )∆( u ) − . If, in addition, F and F are unitary, then u is also unitary.Proof. To be able to use an inductive construction for u , it suffices to show that if F = F modulo h n +1 ,then there exists a central element T ∈ U ( G ) such that F ( n +1) = F ( n +1) + T ⊗ ⊗ T − ∆( T ) . By considering the order n + 1 terms in (3.13) we get that the element S = F ( n +1) − F ( n +1) satisfies(id ⊗ ∆)( S ) + 1 ⊗ S − S ⊗ − (∆ ⊗ id)( S ) = 0 . This means that S is a 2-cocycle in the complex ˜ B G = ˜ B G,e from Section 2.1. Furthermore, F − F commutes with the image of ∆, hence S also commutes with the image of ∆. Therefore S is a 2-cocyclein the complex B G = ˜ B GG . By Proposition 2.1, the cohomology of ˜ B G is V g , and then the cohomology of B G is ( V g ) g . In particular, H ( B G ) = 0, which implies the existence of T .Assume now that we have two central elements u and u with the required properties. Consider thecentral element v = u u − ∈ U ( G ). Then v = 1 modulo h and ∆( v ) = v ⊗ v . Assume v = 1 and takethe smallest n ≥ v ( n ) = 0. Then ∆( v ( n ) ) = v ( n ) ⊗ ⊗ v ( n ) , hence v ( n ) ∈ g . But as v iscentral, we must have v ( n ) ∈ z ( g ) = 0, which is a contradiction.Finally, if F and F are unitary, then ( u − ) ∗ has the defining properties of u , hence ( u − ) ∗ = u . (cid:3) Proof of Theorem 3.17. Taking s φ ∈ R defined by (3.12), let us first prove the existence of µ . By twistingthe coaction α by ( F − , G − ) we obtain a quasi-coaction ( U ( G φ ) (cid:74) h (cid:75) , ∆ , Ψ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ), whereΨ = (id ⊗ ∆)( G − )(1 ⊗ F − )( α ⊗ id)( G )( G ⊗ 1) = (id ⊗ ∆)( G − )(1 ⊗ F − )( G ⊗ ⊗ id)( G ) , and hence Ψ (1) = − ir d cH ( G (1) ) . (3.14)Let us use the subscript φ for the constructions we had in Section 2 applied to the pair k φ < u . ByTheorem 2.16, in order to prove the existence of µ , it suffices to compute h Ω φ , Ψ (1) i .Since Ω φ is a cycle in the chain complex ˜ B G,G φ , the term d cH ( G (1) ) in (3.14) does not contribute tothe pairing. By Proposition 3.8, we have r − cos (cid:16) π − φ ) (cid:17) (Ad g φ − ) ⊗ ( r ) ∈ g φ ⊗ g + g ⊗ g φ . We also have ir = t m ν + − t m ν − modulo g ν ⊗ g ν , hence (Ad g φ − ) ⊗ ( ir ) = t m φ + − t m φ − modulo g φ ⊗ g φ .Therefore ir − sin (cid:16) πφ (cid:17) ( t m φ + − t m φ − ) ∈ g φ ⊗ g + g ⊗ g φ . As g φ centralizes Z φ , any cochain in 1 ⊗ g φ ⊗ g + 1 ⊗ g ⊗ g φ pairs trivially with Ω φ . Hence h Ω φ , Ψ (1) i = − 12 sin (cid:16) πφ (cid:17) h Ω φ , t m φ + − t m φ − i . By Theorem 2.16 and identity (2.11), it follows that ( U ( G φ ) (cid:74) h (cid:75) , ∆ , Ψ) is obtained by twisting the quasi-coaction ( U ( G φ ) (cid:74) h (cid:75) , ∆ , Ψ KZ ,s φ ; µ ) for some µ ∈ h C (cid:74) h (cid:75) , and if, in addition, F and G are unitary, we canchoose µ ∈ h R (cid:74) h (cid:75) .Assume now that the coaction α : U ( G φ ) (cid:74) h (cid:75) → U ( G φ × G ) (cid:74) h (cid:75) is obtained by twisting the quasi-coaction( U ( G φ ) (cid:74) h (cid:75) , ∆ , Ψ KZ ,s ; µ ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ) for some other s ∈ R and µ ∈ h C (cid:74) h (cid:75) . Let ( F , G ) bea pair defining this twisting. By Lemma 3.18, we have F = F ( u ⊗ u )∆( u ) − for a central element u ∈ U ( G ) (cid:74) h (cid:75) such that u = 1 modulo h . But then the pairs ( F , G ) and ( F , (1 ⊗ u − ) G ) define the ame twistings. In other words, without loss of generality we may assume that F = F . Then the quasi-coaction ( U ( G φ ) (cid:74) h (cid:75) , ∆ , Ψ KZ ,s ; µ ) is obtained from ( U ( G φ ) (cid:74) h (cid:75) , ∆ , Ψ KZ ,s φ ; µ ) by twisting with (1 , G G ).By Theorem 2.16 this implies that s = s φ and µ = µ . (cid:3) Remark . Using isomorphisms and twistings that are not trivial modulo h , we can pass from G φ to its conjugate by an element g ∈ U . Namely, the conjugation by Ad g in the 0th leg transforms thequasi-coaction ( U ( G φ ) (cid:74) h (cid:75) , ∆ , Ψ KZ ,s φ ; µ ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ) into the isomorphic quasi-coaction( U ( gG φ g − ) (cid:74) h (cid:75) , (Ad g ) ◦ ∆ ◦ (Ad g ) − , (Ad g ) (Ψ KZ ,s φ ; µ ))of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ), and then the twisting by 1 ⊗ g ∈ U ( gG φ g − × G ) (cid:74) h (cid:75) gives the quasi-coaction( U ( gG φ g − ) (cid:74) h (cid:75) , ∆ , Ψ KZ ,s φ ; µ ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ), where Ψ KZ ,s φ ; µ now denotes the associator definedby the symmetric pair (Ad g )( k φ ) < u .Before moving on to the next part, let us explain some geometric structures motivating the abovecomputations.Starting from the KZ-equations, after fixing a twist F satisfying (3.9)–(3.11), an associator Ψ ∈U ( G ν × G ) (cid:74) h (cid:75) defines an associative product ∗ Ψ h on O ( U/U ν ) (cid:74) h (cid:75) = O ( U ) U ν (cid:74) h (cid:75) , see [EE05, Section 6].Moreover, the algebra ( O ( U/U ν ) (cid:74) h (cid:75) , ∗ Ψ h ) becomes a comodule algebra over the quantized function algebra O h ( U ), the restricted dual Hopf algebra of ( U ( G ) (cid:74) h (cid:75) , ∆ h ). This structure corresponds to the modulecategory ((Rep G ν ) (cid:74) h (cid:75) , Ψ) under the Tannaka–Krein type duality for module categories and coactions.To be more precise, given an element Ψ ∈ U ( G ν × G ) G ν (cid:74) h (cid:75) such that ( U ( G ν ) (cid:74) h (cid:75) , ∆ , Ψ) is a quasi-coaction of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ), we define f ∗ Ψ h f by h f ∗ Ψ h f , T i = h f ⊗ f , F ∆( T )( (cid:15) ⊗ id)(Ψ) i ( T ∈ U ( G )) . As we have F (1) − F (1)21 = ir , the corresponding Poisson bracket { f , f } Ψ = lim h → ih ( f ∗ Ψ h f − f ∗ Ψ h f )is characterized by h{ f , f } Ψ , T i = (cid:10) f ⊗ f , r ∆( T ) − i ∆( T )( (cid:15) ⊗ id)(Ψ (1) − Ψ (1)0 , , ) (cid:11) . If we have Ψ = H , Ψ H − , H − , with H ∈ h U ( G ν × G ) k (cid:74) h (cid:75) , the invertible transformation ρ H of O ( U/U ν ) (cid:74) h (cid:75) characterized by h ρ H ( f ) , T i = h f, T ( (cid:15) ⊗ id)( H ) i satisfies ρ H ( f ∗ Ψ h f ) = ρ H ( f ) ∗ Ψ h ρ H ( f ), i.e., we get isomorphic deformation quantizations from twistequivalent associators. From this we obtain { f , f } Ψ = { f , f } Ψ for such Ψ and Ψ .Combined with the identification of H ( B G,G ν ), we obtain a decomposition { f , f } Ψ = { f , f } α + x { f , f } β , (3.15)for some complex number x (which is real for unitary Ψ), with { f , f } α = mr ( l,l ) ( f ⊗ f ) , { f , f } β = im ( t m ν + − t m ν − ) ( r,r ) ( f ⊗ f ) . Note that { f , f } β is invariant for the left translation action of U , while { f , f } α is equivariantfor the Sklyanin bracket on U . The left invariance of { f , f } β implies that the associated Poissonbivectors commute with respect to the Schouten–Nijenhuis bracket. Note also that we can write { f , f } β = mr ( r,r ) ( f ⊗ f ), and in fact it is the Kostant–Kirillov–Souriau bracket if we identify U/U ν with a coadjoint orbit as in Remark 2.15, see [DG95]. The Poisson bracket associated with Ψ KZ ,s ; µ isgiven by x = tanh( πs ).Turning to the side of coisotropic subgroups, by Corollary 3.13, U/K φ admits a Poisson bracket whichis just the restriction of the Sklyanin bracket: { f , f } φ = { f , f } Sk for f i ∈ O ( U/K φ ) ⊂ O ( U ). Thisgives the structure of a Poisson homogeneous space on U/K φ ∼ = U/U ν over ( U, r ).In fact, any Poisson homogeneous structure of U/U ν over ( U, r ) is of the form (3.15) (this seems tobe folklore, but the idea can be traced back to [She91, Appendix]). We thus obtain a correspondencebetween the parameters φ and s through the comparison of the factor x . Remark . The bracket (3.15) defines a Poisson action of ( U, r ) on U/U ν for any x , but there is adistinguished range which naturally shows up in our considerations: − < x < 1. Starting from theKZ-equations the value of tanh( πs ) is confined in this range when s ∈ R and we have a unitary associator.On the side of the Cayley transform, this corresponds to the case that K φ is coisotropic but not aPoisson–Lie subgroup of ( U, r ). In this case the Poisson bivector vanishes on a nondiscrete subset of /U ν , while in the limit case x = ± | x | > Regularity of ribbon braids. We finish this section with a technical result, which we will needlater, showing that in the non-Poisson subgroup case a ribbon braid in the algebra of formal Laurentseries must lie in the algebra of formal power series. More precisely, we will prove the following. Theorem 3.21. Let u σ < u be a Hermitian symmetric pair, and r be an r -matrix of the form (3.1) forsome Cartan subalgebra t < u and a choice of positive roots. Assume we are given a coaction α : U ( G σ ) (cid:74) h (cid:75) → U ( G σ × G ) (cid:74) h (cid:75) of a multiplier Hopf algebra ( U ( G ) (cid:74) h (cid:75) , ∆ h ) such that there exist F ∈ U ( G × G ) (cid:74) h (cid:75) and G ∈ U ( G σ × G ) (cid:74) h (cid:75) satisfying conditions (3.8) – (3.11) . Assume also that there exists a ribbon braid E ∈ Y ρ ∈ Irr G σ ,π ∈ Irr G (cid:0) End( V ρ ) ⊗ End( V π )[ h − , h (cid:75) (cid:1) for this coaction with respect to the R -matrix R = F exp( − ht u ) F − of ( U ( G ) (cid:74) h (cid:75) , ∆ h ) . Then, unless U σ is a Poisson–Lie subgroup of ( U, r ) , we must have E ∈ U ( G σ × G ) (cid:74) h (cid:75) . Consider the K -matrix K = ( (cid:15) ⊗ id)( E ) and put u = ( (cid:15) ⊗ id)( G ). Then from identity (1.5) for ourribbon braid we get ( u ⊗ E ( u − ⊗ 1) = R (1 ⊗ K ) R . Therefore in order to prove the theorem it suffices to show that K ∈ U ( G ) (cid:74) h (cid:75) unless U σ is a Poisson–Liesubgroup of ( U, r ).Identities (1.4) and (1.6) imply K (Ad u )( T ) = (Ad u )( T ) K for all T ∈ U ( G σ ) (cid:74) h (cid:75) , (3.16)∆ h ( K ) = R (1 ⊗ K ) R ( K ⊗ . (3.17)A key step now is to prove the following. Proposition 3.22. With σ , ∆ h and R as in Theorem 3.21, assume we are given an invertible element K ∈ Y π ∈ Irr G (cid:0) End( V π )[ h − , h (cid:75) (cid:1) satisfying conditions (3.16) and (3.17) for some u ∈ U ( G ) (cid:74) h (cid:75) , u = 1 modulo h . Assume also that K ad = ad( K ) ∈ End( g )[ h − , h (cid:75) has a negative order term. Then U σ is a Poisson–Lie subgroup of ( U, r ) . We will prove this by analyzing certain Poisson structures on U/U σ .Take a Cartan subalgebra ˜ t and a system ˜Π of simple roots as in Section 3.1, but for the Hermitiansymmetric pair u σ < u . Then there exists g ∈ U such that Ad g maps t onto ˜ t , while the dual map maps ˜Πonto Π. Now, the Cartan subalgebra t is as in Section 3.1 for the automorphism ν = (Ad g ) − ◦ σ ◦ (Ad g )of u and some Z ν ∈ z ( u ν ).As we remarked in Section 3.4, on the compact symmetric space U/U ν , both the left and the rightactions of the r -matrix r define (real) Poisson bivector fields, denoted by r ( l,l ) and r ( r,r ) . (As part ofthe claim, the bivector field r ( r,r ) on U preserves the right U ν -invariant functions.) Thus, the linearcombinations of these commuting bivector fields define Poisson brackets on U/U ν as in (3.15). Lemma 3.23. The bivector field r ( l,l ) − r ( r,r ) on U/U ν vanishes only at [ e ] . Similarly, the bivector field r ( l,l ) + r ( r,r ) vanishes only at [ ˜ w ] , where ˜ w ∈ U is any lift of the longest element w of the Weyl group.Proof. As U ν is a Poisson–Lie subgroup of ( U, r ), the first statement follows from the well-knowndescription of the Poisson leaves of the reduction r ( l,l ) − r ( r,r ) of the Sklyanin bracket [LW90].As for the bivector field r ( l,l ) + r ( r,r ) , first note that (Ad ˜ w ) ⊗ ( r ) = − r . This means that the Poissonbivector r ( l,l ) + r ( r,r ) on U/U ν vanishes at the point [ ˜ w − ] = [ ˜ w ]. Thus the U -equivariant diffeomorphism U/U ν → U/ (Ad ˜ w )( U ν ) , [ g ] [ g ˜ w − ] , transforms the Poisson bivector r ( l,l ) + r ( r,r ) on U/U ν into a Poisson bivector Π on U/ (Ad ˜ w )( U ν ) whichvanishes at the basepoint.On the other hand, (Ad ˜ w )( U ν ) is again a Poisson–Lie subgroup of U . Hence Π has to agree with thereduction of the Sklyanin bracket, which vanishes only at [ e ] ∈ U/ (Ad ˜ w )( U ν ). As a result r ( l,l ) + r ( r,r ) vanishes only at [ ˜ w ] ∈ U/U ν . (cid:3) dentity (3.17) implies that K satisfies the reflection equation( K ⊗ R (1 ⊗ K ) R = R (1 ⊗ K ) R ( K ⊗ . Using this we are going to introduce another Poisson structure on U/U ν following [DM03a, DM03b].Let us write t for t u . Consider a finite dimensional representation π of G . Put t π = ( π ⊗ π )( t ) andconsider the set M π = { A ∈ End( V π ) | ( A ⊗ A ) t π = t π ( A ⊗ A ) } . We have three actions of g ∈ U on End( V π ) given by multiplication by π ( g ) on the left, on the right, andby conjugation by π ( g ). For X ∈ g , we will denote by X ( l ) , X ( r ) and X (ad) the corresponding vectorfields on End( V π ). Thus, X (ad) = X ( l ) − X ( r ) . The RE bracket on M π is defined by { f , f } RE = m (cid:0) r (ad , ad) + i ( t ( l,r ) − t ( r,l ) ) (cid:1) ( f ⊗ f ) . More precisely, since M π is not a smooth manifold in general, this defines a Poisson bracket on thealgebra of polynomial functions on M π . But in any case what is going to matter to us is only that r (ad , ad) + i ( t ( l,r ) − t ( r,l ) ) is a well-defined bivector field on End( V π ).The following observation is from [DM03b]. Lemma 3.24. Put R π = ( π ⊗ π )( R ) , and assume that A h ∈ End( V π ) (cid:74) h (cid:75) has constant term A = A (0) h ∈M π and satisfies the reflection equation ( A h ⊗ R π (1 ⊗ A h ) R π = R π (1 ⊗ A h ) R π ( A h ⊗ . Then the RE bracket vanishes at A .Proof. Letting r π = ( π ⊗ π )( r ), the bivector field r (ad , ad) + i ( t ( l,r ) − t ( r,l ) ) at the point A ∈ End( V π ) is r π ( A ⊗ A ) + ( A ⊗ A ) r π − (1 ⊗ A ) r π ( A ⊗ − ( A ⊗ r π (1 ⊗ A )+ (1 ⊗ A ) it π ( A ⊗ − ( A ⊗ it π (1 ⊗ A ) . By looking at the order one terms in the reflection equation and using that R = 1 − h ( t + ir ) + O ( h ), R = 1 − h ( t − ir ) + O ( h ), and ( A ⊗ A ) t π = t π ( A ⊗ A ), we see that this bivector is zero. (cid:3) We want to apply this to the element h k K ad for k = − ord( K ad ), so that h k K ad starts with an orderzero term. Denote the orthogonal (with respect to the U -invariant Hermitian form) projection g → m ν ± by P ± , and put P g ± = (Ad g )( P ± ), so P g ± is the projection onto m σ ± . Lemma 3.25. Under the assumptions of Proposition 3.22, the lowest order nonzero coefficient of K ad is,up to a scalar factor, either P g + or P g − .Proof. Let us more generally consider the elements K π = π ( K ) for finite dimensional representations π of G . Denote by k π the order of the lowest nonzero term of K π .Applying the counit to (3.17) we get (cid:15) ( K ) = 1. Consider the contragredient representation ¯ π of G . Theantipode S h for ( U ( G ) (cid:74) h (cid:75) , ∆ h ) has the form vS ( · ) v − for some v ∈ U ( G ) (cid:74) h (cid:75) , v = 1 modulo h . Applying m (id ⊗ S h ) to (3.17) we then get1 = h k π + k ¯ π K π, ( k π ) S ( K ¯ π, ( k ¯ π ) ) + O ( h k π + k ¯ π +1 ) in End( V π )[ h − , h (cid:75) . Hence k π + k ¯ π ≤ 0, and if k π + k ¯ π = 0 then K π, ( k π ) S ( K ¯ π, ( k ¯ π ) ) = 1, while if k π + k ¯ π < K π, ( k π ) S ( K ¯ π, ( k ¯ π ) ) = 0.Consider now the adjoint representation ad. Since it is self-conjugate and by assumption k ad < K ad , ( k ad ) S ( K ad , ( k ad ) ) = 0. By (3.16) we know also that K ad , ( k ad ) is an intertwiner for U σ . As arepresentation of g σ , we have the decomposition g = z ( g σ ) ⊕ [ g σ , g σ ] ⊕ m σ + ⊕ m σ − , where the derived Lie algebra [ g σ , g σ ] is either zero, simple or the sum of two simple ideals. As thesecomponents are mutually nonequivalent, K ad , ( k ad ) is a linear combination of up to 5 projections.The antipode S restricted to the block End( g ) of U ( G ) is the adjoint map with respect to the invariantform ( · , · ) g , that is, ( T X, Y ) g = ( X, S ( T ) Y ) g for T ∈ End( g ) ( X, Y ∈ g ) . Since the invariant form is nondegenerate on the irreducible components of g σ = z ( g σ ) ⊕ [ g σ , g σ ], weconclude that the corresponding projections are S -invariant. We can also conclude that S ( P g + ) = P g − .Therefore the identity K ad , ( k ad ) S ( K ad , ( k ad ) ) = 0 can be true only if K ad , ( k ad ) is a scalar multiple ofeither P g + or P g − . (cid:3) emma 3.26. We have P ± ∈ M ad .Proof. Since t ∗ = t , in order to prove that t ad commutes with P + ⊗ P + it suffices to check that t ad ( m ν + ⊗ m ν + ) ⊂ m ν + ⊗ m ν + . (3.18)Recall that t = t k + t m ν + + t m ν − , where k = u ν , and t m ν ± ∈ m ν ± ⊗ m ν ∓ . As [ m ν + , m ν + ] = 0, we see that t ad = (ad ⊗ ad)( t k ) on m ν + ⊗ m ν + , which obviously implies (3.18). The proof for P − is similar. (cid:3) Consider now the (Ad U )-orbit O ± of P ± . By the previous lemma it is contained in M ad . Since U ν stabilizes P ± and u ν is a maximal proper Lie subalgebra of u , it follows that U ν is the connectedcomponent of the stabilizer of P ± . Hence p ± : U/U ν → O ± , [ g ] (Ad g )( P ± ), is a covering map. Lemma 3.27. The RE bracket on M ad restricts to O ± . Being lifted to U/U ν via p ± , this restrictioncoincides with the bracket defined by r ( l,l ) ∓ r ( r,r ) .Proof. The covering map p ± : U/U ν → O ± intertwines the action by left translations with the adjointaction. From this it is clear that the bivector field r (ad , ad) at the points of O ± ⊂ M ad indeed defines abivector field on O ± , and, being lifted to U/U ν , this gives the bivector field r ( l,l ) .We next want to compare r ( r,r ) with − i ( t ( l,r ) − t ( r,l ) ). We claim that( d [ e ] p + ⊗ d [ e ] p + ) (cid:0) r ( r,r ) ([ e ]) (cid:1) = − i ( t ( l,r ) − t ( r,l ) )( P + ) . Since the bivector fields r ( l,l ) and r ( r,r ) on U/U ν coincide at [ e ] and the pushforward of r ( l,l ) is r (ad , ad) ,this is equivalent to r (ad , ad) ( P + ) = − i ( t ( l,r ) − t ( r,l ) )( P + ) . Using again that [ m ν + , m ν + ] = 0, we get (ad X α ) P + = 0 and P + (ad X − α ) = 0 for all α ∈ Φ +nc . Asimple computation using these properties, together with the fact that U ν stabilizes P + , gives r (ad , ad) ( P + ) = i X α ∈ Φ +nc ( α, α )2 (cid:0) P + (ad X α ) ⊗ (ad X − α ) P + − (ad X − α ) P + ⊗ P + (ad X α ) (cid:1) , and a similar computation for − i ( t ( l,r ) − t ( r,l ) )( P + ) gives the same answer. Thus our claim is proved.Since r ( r,r ) is left U -invariant and − i ( t ( l,r ) − t ( r,l ) ) is (Ad U )-invariant, we then get( d [ g ] p + ⊗ d [ g ] p + ) (cid:0) r ( r,r ) ([ g ]) (cid:1) = − i ( t ( l,r ) − t ( r,l ) )( p + ([ g ])) ( g ∈ U ) . This finishes the proof of the lemma for O + . The proof for O − is similar. (cid:3) Proof of Proposition 3.22. By Lemmas 3.24, 3.25 and 3.26, the RE bracket vanishes either at P g + ∈ O + orat P g − ∈ O − . By Lemma 3.27 this means that either r ( l,l ) − r ( r,r ) or r ( l,l ) + r ( r,r ) vanishes at [ g ] ∈ U/U ν .But then by Lemma 3.23 we have either g ∈ U ν or g ∈ ˜ w U ν , and therefore U σ = (Ad g )( U ν ) is either U ν or (Ad ˜ w )( U ν ). (cid:3) Remark . As r ( l,l ) − r ( r,r ) has to vanish on p − ( P + ), we can also conclude that p + is injective, thatis, U ν is exactly the stabilizer of P + . Similarly, or by symmetry, U ν is the stabilizer of P − . Proof of Theorem 3.21. Assume G σ is not a Poisson–Lie subgroup. To prove the theorem it suffices toshow that K π ∈ End( V π ) (cid:74) h (cid:75) for all finite dimensional representations π of G . By Proposition 3.22 andidentity (3.17) this is already the case if π belongs to the tensor subcategory generated by ad, that is, π factors through the adjoint group G ad = Ad G . For arbitrary π , if K π contains a term of negative degree,then, using that R = 1 modulo h , we see from (3.17) that K π ⊗ n contains a term of negative degree for all n ≥ 1. But when π is irreducible, we have π ⊗ n ∈ Rep G ad for some n ≥ 1, so this cannot happen. (cid:3) Letzter–Kolb coideals Our next goal is to put the Letzter–Kolb coideals in the framework of multiplier algebras. This isagain easy in the non-Hermitian case. In the Hermitian case, we will relate the classical limits of thecoideals to the coisotropic subgroups of the previous section. Combined with a rigidity result for thefusion rules, this will allow us to define multiplier algebra models of the coideals. We will also cast theBalagović–Kolb universal K -matrix in our setting.Throughout this section we fix a nontrivial involutive automorphism θ of u , a Cartan subalgebra t of u and a system Φ + of positive roots such that θ is in Satake form with respect to ( h , b + ), where h = t C . .1. Quantized universal enveloping algebra and Letzter–Kolb coideal subalgebras. Let I bethe label set for the simple roots, so Π = { α i } i ∈ I . As in Section 3.1, for every positive root α we fix anelement X α ∈ g α normalized so that [ X α , X ∗ α ] = H α , and put X − α = X ∗ α .The quantized universal enveloping algebra U h ( g ) is topologically generated over C (cid:74) h (cid:75) by a copyof U ( h ) and elements E i , F i ( i ∈ I ) satisfying the following standard relations:[ H, E i ] = α i ( H ) E i , [ H, F i ] = − α i ( H ) F i , [ E i , F j ] = δ ij e hd i H i − e − hd i H i e hd i − e − hd i , − a ij X n =0 ( − n (cid:20) − a ij n (cid:21) q i E − a ij − ni E j E ni = 0 = − a ij X n =0 ( − n (cid:20) − a ij n (cid:21) q i F − a ij − ni F j F ni , ( i = j ) , where H ∈ h , H i = H α i , q i = q d i = e hd i , d i = ( α i , α i ), ( a ij ) i,j is the Cartan matrix, so a ij = ( α ∨ i , α j ),and (cid:20) mn (cid:21) q i = [ m ] q i ![ n ] q i ![ m − n ] q i ! , [ m ] q i ! = m Y k =1 [ k ] q i , [ k ] q i = q ki − q − ki q i − q − i . We will write U h ( h ) for the copy of U ( h ) (cid:74) h (cid:75) inside U h ( g ).Put K i = e hd i H i . More generally, for ω ∈ h ∗ , let h ω ∈ h be such that α ( h ω ) = ( α, ω ). Define K ω = e hh ω . Then K i = K α i .The coproduct ∆ : U h ( g ) → U h ( g ) ˆ ⊗ U h ( g ) is defined by∆( H ) = H ⊗ ⊗ H ( H ∈ h ) , ∆( E i ) = E i ⊗ K i ⊗ E i , ∆( F i ) = F i ⊗ K − i + 1 ⊗ F i . Finally, the ∗ -structure is defined by H ∗ i = H i , E ∗ i = F i K i , F ∗ i = K − i E i . By assumption, the involutive automorphism θ of g is in Satake form with respect to ( h , b + ) (recallDefinition 3.1). Then θ has the following form: θ = (Ad zm X m ) ◦ τ θ ◦ τ = (Ad zm X ) ◦ τ θ ◦ ω, (4.1)where m and m X are the canonical lifts of w ∈ W and w X ∈ W X to U , ω is the Chevalley automorphism, τ is the diagram automorphism satisfying ω = (Ad m ) ◦ τ , and z is an element of the maximal torusexp( t ). The element z is determined up to a factor in Z ( U ). It automatically satisfies the followingconditions: z i = 1 ( i ∈ X ) , z i ¯ z τ θ ( i ) = ( − α i ,ρ ∨ X ) ( i ∈ I \ X ) , (4.2)where z i = z ( α i ) and ρ ∨ X is half the sum of the positive coroots of the root system generated by X . See,for example, [DCNTY19, Section 2.1] for details.Consider the following parameter sets: C = { c = ( c i ) i ∈ I \ X | c i ∈ C (cid:74) h (cid:75) ∗ , c i = c τ θ ( i ) for i ∈ I C } , S = { s = ( s i ) i ∈ I \ X | s i ∈ C (cid:74) h (cid:75) , s i = 0 for i / ∈ I S } , where C (cid:74) h (cid:75) ∗ denotes the units of C (cid:74) h (cid:75) , that is, the series with nonzero constant terms. We write t = ( c , s )for an element of T = C × S .Fix t ∈ T . For each i ∈ I \ X , we define B i = F i − c i z τ θ ( i ) T w X ( E τ θ ( i ) ) K − i + s i κ i K − i − e − d i h − , (4.3)where κ i = exp( π √− ψ i ) is the square root of z i with 0 ≤ ψ i < 1, and T w X is the Lusztig automorphismassociated to the longest element w X ∈ W X . It will also be convenient to put B i = F i ( i ∈ X ) . Denote by U h ( g X ) ⊂ U h ( g ) the closure in the h -adic topology of the C (cid:74) h (cid:75) -subalgebra generated by theelements H i , E i and F i for i ∈ X . Definition 4.1. We define U t h ( g θ ) ⊂ U h ( g ) as the closure in the h -adic topology of the C (cid:74) h (cid:75) -subalgebragenerated by U h ( h θ ), U h ( g X ) and the elements B i for i ∈ I \ X . emark . In [BK19, DCNTY19], the element z in (4.1) is assumed to have the property z i = 1 for all i ∈ I \ X such that τ θ ( i ) = i , which imposes extra conditions on θ . Although the relevant proofs work inthe generality presented here, it is also possible to reduce the situation to this normalized form as follows.Starting from our convention, choose z ∈ exp( t ) such that z i = κ i for i as above, and z i = 1 for all other i ∈ I . Then θ = (Ad z ) − ◦ θ ◦ (Ad z ) satisfies the normalization condition. Moreover, Ad z lifts to aHopf ∗ -algebra automorphism of U h ( g ), and we have (Ad z )( U t h ( g θ )) = U t h ( g θ ). Remark . The above choice of κ i is, of course, a matter of convention. If we replace κ i by − κ i in (4.3),the corresponding subalgebra will be conjugate to U t h ( g θ ) by an inner automorphism of U h ( g ) defined byan element of the torus.It follows from [Let99, Kol14] that U t h ( g θ ) is a right coideal of U h ( g ):∆( U t h ( g θ )) ⊂ U t h ( g θ ) ˆ ⊗ U h ( g ) . We will only be interested in the coideals U t h ( g θ ) defined by smaller parameter sets T ∗ = C ∗ × S ∗ ⊂ T ∗ C = C ∗ C × S ∗ C ⊂ T . In the non-Hermitian case both T ∗ and T ∗ C consist of just one point defined by c i = e − h ( α − i ,α − i ) , s i = 0 (4.4)for all i ∈ I \ X , where α − i = ( α i − Θ( α i )).In the Hermitian case, recall from our discussion in Section 3.2 that there are one or two distinguishedroots in I \ X . Then we define T ∗ (resp. T ∗ C ), by allowing the following exceptions from (4.4): • S-type: if α o is the unique distinguished root, then we require s o ∈ i R (cid:74) h (cid:75) (resp. s o ∈ C (cid:74) h (cid:75) and s (0) o = ± • C-type: if α o is a distinguished root, then we require c o ∈ R (cid:74) h (cid:75) and c (0) o > c o ∈ C (cid:74) h (cid:75) and c (0) o = ± i ), and, for both T ∗ and T ∗ C , c o c τ θ ( o ) = e − h ( α − o ,α − o ) . By the proof of [DCNTY19, Theorem 3.11] (see also Remark 4.2), the coideals U t h ( g θ ) are ∗ -invariant for t ∈ T ∗ .We will mainly work with T ∗ and then explain how our results extend to generic points of T ∗ C . Thepoint of T ∗ defined by (4.4) for all i ∈ I \ X is denoted by 0, and the corresponding coideal subalgebrais denoted by U θh ( g ). We will also refer to this as the standard or no-parameter case. Thus, in thenon-Hermitian case, U θh ( g ) is the only coideal subalgebra we will be working with.The classical limit of U t h ( g θ ) is given by the following: Definition 4.4. For t ∈ T , we define g θ t to be the Lie subalgebra of g generated by h θ , g X and theelements X − α i + c (0) i θ ( X − α i ) + s (0) i κ i H i ( i ∈ I \ X ) . The image of U t h ( g θ ) under the isomorphism U h ( g ) /hU h ( g ) ∼ = U ( g ) (mapping E i into X α i and F i into X − α i ) is U ( g θ t ). In the standard case, we have c (0) i = 1 and s (0) i = 0 for all i ∈ I \ X , and thecorresponding Lie subalgebra g θ is exactly g θ by [Kol14, Lemma 2.8], see also Proposition 3.15.4.2. Untwisting by Drinfeld twist. Let us quickly review how to relate the quantized universalenveloping algebra to the classical one. Recall the first and the second Whitehead Lemmas: H ( g , V ) =H ( g , V ) = 0 for any finite dimensional g -module V . As is well-known, this implies that there is a C (cid:74) h (cid:75) -algebra isomorphism π : U h ( g ) → U ( g ) (cid:74) h (cid:75) such that π ( H i ) = H i , π ( E i ) = X α i , π ( F i ) = X − α i (mod h ) , (4.5)and if ˜ π is another such isomorphism, then ˜ π = (Ad u ) π for an element u ∈ hU ( g ) (cid:74) h (cid:75) . In asimilar way, for any two C (cid:74) h (cid:75) -algebra homomorphisms ˜ π, π : U h ( g ) → U ( G ) (cid:74) h (cid:75) satisfying (4.5) there is u ∈ h U ( G ) (cid:74) h (cid:75) such that ˜ π = (Ad u ) π . In what follows we fix such a homomorphism π .While a particular choice of π is not going to matter for our results, in some arguments it is convenientto have extra properties. Lemma 4.5. There is a ∗ -preserving C (cid:74) h (cid:75) -algebra isomorphism π : U h ( g ) → U ( g ) (cid:74) h (cid:75) such that π ( H i ) = H i , π ( K − / i E i ) = X α i , π ( F i K / i ) = X − α i (mod h ) . (4.6) roof. We have a homomorphism ρ : U ( g ) → U h ( g ) /h U h ( g ) such that ρ ( H i ) = H i , ρ ( X α i ) = K − / i E i , ρ ( X − α i ) = F i K / i , the key point being that since [ n ] e h = n + O ( h ), the coefficients of the quantum Serre relations reduceto the classical ones modulo h . Taking now an arbitrary C (cid:74) h (cid:75) -algebra isomorphism π : U h ( g ) → U ( g ) (cid:74) h (cid:75) satisfying (4.5), we must have that the homomorphism π ◦ ρ : U ( g ) → U ( g ) (cid:74) h (cid:75) /h U ( g ) (cid:74) h (cid:75) is of the form( π ◦ ρ )( T ) = T + hδ ( T ) ( T ∈ U ( g ))for a derivation δ : U ( g ) → U ( g ). Replacing π by e − hδ ◦ π we get an isomorphism U h ( g ) → U ( g ) (cid:74) h (cid:75) satisfying (4.6).Next, the homomorphism U h ( g ) → U ( g ) (cid:74) h (cid:75) , T π ( T ∗ ) ∗ , also satisfies (4.6). It follows that thereexists u ∈ h U ( g ) (cid:74) h (cid:75) such that π ( T ∗ ) ∗ = uπ ( T ) u − . By taking the adjoints and replacing T by T ∗ we also get π ( T ∗ ) ∗ = u ∗ π ( T )( u ∗ ) − . Hence u ∗ = uv for a central element v ∈ U ( g ) (cid:74) h (cid:75) such that v = 1modulo h . This element must be unitary, hence v / ∈ h U ( g ) (cid:74) h (cid:75) is unitary as well. Replacing u by uv / we can therefore assume that u ∗ = u . But then replacing π by (Ad u / ) π we get a ∗ -preservingisomorphism satisfying (4.6). (cid:3) With π : U h ( g ) → U ( G ) (cid:74) h (cid:75) fixed, there is a unique coproduct ∆ h : U ( G ) (cid:74) h (cid:75) → U ( G × G ) (cid:74) h (cid:75) such that( π ⊗ π )∆ = ∆ h π. By a Drinfeld twist we will mean any element F ∈ h U ( G × G ) (cid:74) h (cid:75) such that( (cid:15) ⊗ id)( F ) = (id ⊗ (cid:15) )( F ) = 1 , ∆ h = F ∆( · ) F − , (id ⊗ ∆)( F − )(1 ⊗ F − )( F ⊗ ⊗ id)( F ) = Φ KZ . (4.7)Consider the r -matrix r defined by (3.1) and the corresponding cobracket δ r ( X ) = [ r, ∆( X )] on g . Lemma 4.6. If π is as in Lemma 4.5, then there is a unitary Drinfeld twist F ∈ U ( g ) ⊗ (cid:74) h (cid:75) such that F = 1 + h ir O ( h ) . (4.8) Proof. We start with an arbitrary Drinfeld twist F ∈ hU ( g ) ⊗ (cid:74) h (cid:75) , which exists by [Dri90]. By ourchoice of π we have ∆ h ( X ) = ∆( X ) + h iδ r ( X )2 + O ( h ) for X ∈ g . It follows that the element S = F (1) − ir ∈ U ( g ) ⊗ commutes with the image of ∆. Since Φ KZ = 1modulo h , identity (4.7) implies (similarly to the proof of Lemma 3.18) that S satisfies the cocycleidentity (id ⊗ ∆)( S ) + 1 ⊗ S − S ⊗ − (∆ ⊗ id)( S ) = 0 . Hence S = T ⊗ ⊗ T − ∆( T ) for a central element T ∈ U ( g ). Replacing F by F (cid:0) (1 + hT ) − ⊗ (1 + hT ) − (cid:1) ∆(1 + hT )we get a Drinfeld twist satisfying (4.8). Replacing further F by F ( F ∗ F ) − / we also get unitarity,see [NT11, Proposition 2.3]. Note that this does not destroy (4.8), since r ∗ = r . (cid:3) Denote the universal R -matrix of U h ( g ) (or, the one for U q ( g ) in the conventions of [DCNTY19]) by R .Then any Drinfeld twist F satisfies( π ⊗ π )( R ) = F exp( − ht u ) F − . (4.9)Indeed, this identity holds for a particular Drinfeld twist by [Dri90], but then it must hold for any Drinfeldtwist by Lemma 3.18. We put R h = ( π ⊗ π )( R ), which is a universal R -matrix for ( U ( G ) (cid:74) h (cid:75) , ∆ h ).4.3. Parameter case and Cayley transform. Suppose that u θ < u is a Hermitian symmetric pair.Let us relate the Lie algebras g θ t to the Cayley transform we considered in Section 3.2.Choose Z θ ∈ z ( u θ ) normalized as ( Z θ , Z θ ) g = − a − θ . Let us choose a Cartan subalgebra ˜ t of u containing z ( u θ ), and choose positive roots as in Section 3.1, but now for the pair ( θ, Z θ ) instead of ( ν, Z ν ). Wedenote the corresponding Borel subalgebra by ˜ b + .Take g ∈ U such that (Ad g )(˜ t ) = t and (Ad g )(˜ b + ) = b + . Put ν = (Ad g ) ◦ θ ◦ (Ad g ) − and Z ν = (Ad g )( Z θ ). Then t and our fixed positive roots are defined as in Section 3.1 for our new pair ( ν, Z ν ).Let g be the Cayley transform for ν with respect to ( X α ) α . emma 4.7. There exist an element z θ ∈ exp( t ) such that (Ad z θ ) ◦ θ ◦ (Ad z θ ) − coincides with theautomorphism θ = (Ad g ) − ◦ ν ◦ (Ad g ) and Z θ = (Ad g z θ ) − ( Z ν ) .Proof. First note that θ is in Satake form with respect to ( h , b + ).We have θ = (Ad g − g ) ◦ θ ◦ (Ad g − g ) − . It follows that θ is in Satake form both with respect to( h , b + ) and ((Ad g − g )( h ) , (Ad g − g )( b + )). By [KW92, Corollary 5.32], we can find g ∈ G θ such that(Ad g g − g )( h ) = h and (Ad g g − g )( b + ) = b + . Then g g − g ∈ exp( h ). Moreover, we still have θ = (Ad g g − g ) ◦ θ ◦ (Ad g g − g ) − . Consider the Cartan decomposition g g − g = z − θ a , so z θ ∈ exp( t ) and a ∈ exp( i t ). As θ, θ are ∗ -preserving and θ ◦ (Ad z − θ a ) = (Ad z − θ a ) ◦ θ , we also have θ ◦ (Ad az θ ) = (Ad az θ ) ◦ θ . It followsthat Ad a commutes with θ . This means that θ ( a ) ∈ Z ( G ) a = Z ( U ) a , hence θ ( a ) = a , and then θ ( a ) = a . Therefore θ = (Ad z θ ) − ◦ θ ◦ (Ad z θ ) = (Ad g z θ ) − ◦ ν ◦ (Ad g z θ ) . We also have (Ad g z θ ) − ( Z ν ) = (Ad z − θ ag − )( Z ν ) = (Ad g g − )( Z ν ) = (Ad g )( Z θ ) = Z θ , where we used that (Ad g − )( Z ν ) ∈ z ( u θ ) is invariant under a ∈ G θ . (cid:3) We can now talk about compact/noncompact positive roots for ( h , b + ) with respect to ν , as in Section 3.Recall that by Proposition 3.5, in the S-type case the unique noncompact simple root α o is exactly thedistinguished root. We have the following characterization for the C-type case. Lemma 4.8. In the C-type case, the unique noncompact simple root α o is determined among thedistinguished roots { α o , α o } by the inequality − iα o ( ˜ Z θ ) > − iα o ( ˜ Z θ ) , where ˜ Z θ is the component of Z θ ∈ g = h ⊕ L α ∈ Φ g α lying in h .Proof. By the definition of the order structure in Section 3.1, we have α o ( − iZ ν ) > α o ( − iZ ν ). ByProposition 3.5, α o and α o have the same restriction to h − = { H | θ ( H ) = − H } . It follows that if Z ν = Z + ν + Z − ν is the decomposition of Z ν with respect to h = h θ ⊕ h − , then − iα o ( Z + ν ) > − iα o ( Z + ν ) . On the other hand, the inverse (Ad g ) − of the Cayley transform acts trivially on h θ and maps h − onto the linear span of the vectors X γ i − X − γ i (Lemma 3.9). As Z θ = (Ad g z θ ) − ( Z ν ), it follows that˜ Z θ = Z + ν , proving the lemma. (cid:3) Thus, in both S-type and C-type cases, once Z θ is fixed (between the two possibilities), ν is uniquelyand explicitly determined: ν acts trivially on h and the root vectors X ± α i for i ∈ I \ { o } , while ν ( X ± α o ) = − X ± α o . Since α o is a noncompact positive root, α o ( − iZ ν ) is a positive number. This,together with the normalization ( Z ν , Z ν ) g = − a − ν , determines Z ν . In the S-type case the pair ( ν, Z ν ) istherefore independent of the choice of Z θ . In the C-type case, changing the sign of Z θ swaps the notionsof compactness/noncompactness for the distinguished roots. Note also that by looking at the basis of therestricted root system obtained by restricting Π \ Π X to h − , we can recover the roots γ , . . . , γ s and thenthe element g .The element z θ is not easily determined, but the following lemma will be enough for our purposes.Recall the factorization of θ given by (4.1). Lemma 4.9. In the S-type case, z θ ( α o ) is a square root of z − o .Proof. Since o is fixed by τ θ , we have θ ( X − α o ) = − z o X α o . As we already observed in the proof ofProposition 3.15, Lemma 3.9 implies that (cid:0) (Ad g ) − ◦ ν ◦ (Ad g ) (cid:1) ( X α o ) = − X − α o . It follows that θ ( X α o ) = − z θ ( α o ) X − α o , and comparing this with the above formula we get z θ ( α o ) = z − o . (cid:3) Consider now the subgroups G φ ( φ ∈ R ) from Definition 3.16. (Note that we have to use θ fromLemma 4.7 as θ in Section 3.3.) It is convenient now to allow also φ ∈ C . Then G φ = (Ad g φ )( G θ ) arestill well-defined subgroups of G . emma 4.10. If u θ < u is Hermitian and t ∈ T ∗ C , then g θ t = (Ad z θ ) − ( g φ ) , where φ ∈ C is any numbersatisfying the following identity: z θ ( α o ) s (0) o κ o = i tan (cid:16) πφ (cid:17) (S-type) or c (0) o = − cot (cid:16) π φ − (cid:17) (C-type) . In particular, the Lie algebras g θ t are all conjugate to g θ in g .Proof. This follows from Proposition 3.15 (and its obvious extension to complex φ ) and the definitionof g θ t , combined with Lemma 4.7. (cid:3) For t ∈ T ∗ C , let G θ t = (Ad z θ ) − ( G φ ) ⊂ G be the (connected) algebraic subgroup integrating g θ t , with φ as in the lemma above. Then K t = (Ad z − θ g φ − )( U ν ) is its compact form. Note that if t ∈ T ∗ , then wecan take φ ∈ R , so that g φ ∈ U and hence k t = g θ t ∩ u . Remark . In the C-type case we get an element Z t θ = (Ad z − θ g φ − )( Z ν ) = (Ad z − θ g φ z θ )( Z θ ) ∈ z ( g θ t ),which by Lemma 3.9 does not depend on the choice of φ (such that c (0) o = − cot( π ( φ − ν from Z t θ . Namely, let ˜ Z t θ be the component of Z t θ in h . Then,again by Lemma 3.9, ˜ Z t θ and ˜ Z θ differ only by an element of h − . Hence α o is determined among thedistinguished roots { α o , α o } by the inequality − iα o ( ˜ Z t θ ) + iα o ( ˜ Z t θ ) > . In the S-type case the element (Ad z − θ g φ − )( Z ν ) ∈ z ( g θ t ) does depend on the choice of φ . Here we cantake any Z t θ ∈ z ( g θ t ) such that ( Z t θ , Z t θ ) g = − a − θ and then make the identity Z t θ = (Ad z − θ g φ − )( Z ν ) anextra condition on φ . This works, because by Corollary 3.6 and Lemma 3.9 we have (Ad g )( Z ν ) = − Z ν .We can formulate this in a more intrinsic way with respect to g θ t as follows. Recall that we have Z ν = i P j H γ j by Corollary 3.6. Then Lemma 3.9 implies(Ad z − θ g φ − )( Z ν ) = 12 cos (cid:16) πφ (cid:17) X j (Ad z − θ )( X − γ j − X γ j ) + i (cid:16) πφ (cid:17) X j H γ j . for all φ . It follows that if Z t θ = (Ad z − θ g φ − )( Z ν ), then( Z t θ , X α o ) g z θ ( α o ) cos( πφ ) = ( X − α o , X α o ) g α, α ) > . Multiplier algebra model of Letzter–Kolb coideals. Back to the general θ , let us next explainhow to cast the Letzter–Kolb coideals in the setting of multiplier algebras. Let P be the weight lattice.Denote by V λ an irreducible g -module with highest weight λ ∈ P + . We denote by π λ : U ( g ) → End( V λ )the corresponding homomorphism and use the same symbol for the extension of π λ to a homomorphism U ( g ) (cid:74) h (cid:75) → End( V λ ) (cid:74) h (cid:75) . We also put π λ,h = π λ π : U h ( g ) → End( V λ ) (cid:74) h (cid:75) . Lemma 4.12. For every t ∈ T ∗ , there exist elements u λ ∈ End( V λ ) (cid:74) h (cid:75) , λ ∈ P + , such that u (0) λ = 1 , (cid:18)M λ ∈ F (Ad u λ ) π λ,h (cid:19)(cid:0) U t h ( g θ ) (cid:1) = (cid:18)M λ ∈ F π λ (cid:19)(cid:0) U ( g θ t ) (cid:74) h (cid:75) (cid:1) (4.10) for any finite subset F ⊂ P + . If π is ∗ -preserving, then u λ can in addition be chosen to be unitary.Proof. Let us first fix a finite subset F ⊂ P + and show that there exist elements u λ , λ ∈ F , satisfying (4.10).Denote by V F the g -module L λ ∈ F V λ and by π F the representation L λ ∈ F π λ . Write π F,h for π F π .Let A h be the commutant of π F,h (cid:0) U t h ( g θ ) (cid:1) in End( V F ) (cid:74) h (cid:75) . It is clear that A h is a closed C (cid:74) h (cid:75) -subalgebraof End( V F ) (cid:74) h (cid:75) and A h ∩ h End( V F ) (cid:74) h (cid:75) = hA h . It follows that A h is a free C (cid:74) h (cid:75) -module and A h /hA h canbe considered as a subalgebra of End( V F ), so that A h is a deformation of this subalgebra. We claim that A h /hA h = End g θ t ( V F ) . The inclusion ⊂ is clear, since the image of U t h ( g θ ) in U h ( g ) /hU h ( g ) ∼ = U ( g ) is U ( g θ t ). For the oppositeinclusion, using the Frobenius isomorphismEnd g θ t ( V F ) ∼ = Hom g θ t ( V , V F ⊗ ¯ V F ) , defined by duality morphisms for g -modules, and a decomposition of V F ⊗ ¯ V F into simple g -modules V µ ,we see that the problem reduces to the question whether every g θ t -invariant vector in V µ can be lifted to a U t h ( g θ )-invariant vector in V µ (cid:74) h (cid:75) . This is indeed possible by a result of Letzter [Let00], see Appendix Bfor more details. ince the algebra End g θ t ( V F ) is semisimple, it has no nontrivial deformations, so there is a C (cid:74) h (cid:75) -algebraisomorphism A h ∼ = End g θ t ( V F ) (cid:74) h (cid:75) that is the identity modulo h . Furthermore, there are no nontrivialdeformations of the identity homomorphism End g θ t ( V F ) → End( V F ), that is, all such deformations aregiven by conjugating by elements of 1 + h End( V F ) (cid:74) h (cid:75) . It follows that there is w ∈ h End( V F ) (cid:74) h (cid:75) suchthat wA h w − = End g θ t ( V F ) (cid:74) h (cid:75) .Next, consider the subalgebra B ⊂ End g θ t ( V F ) spanned by the projections e λ : V F → V λ . Since B ⊂ A h ,we have (Ad w )( B ) ⊂ End g θ t ( V F ) (cid:74) h (cid:75) . As B is also semisimple, the inclusion map B → End g θ t ( V F ) cannotbe nontrivially deformed, that is, there is v ∈ h End g θ t ( V F ) (cid:74) h (cid:75) such that Ad w = Ad v on B . It followsthat the element u = v − w still has the property uA h u − = End g θ t ( V F ) (cid:74) h (cid:75) , (4.11)but in addition it commutes with the projections e λ , λ ∈ P . Hence u = ( u λ ) λ ∈ F for some u λ ∈ h End g θ t ( V λ ) (cid:74) h (cid:75) .By taking the commutants we get from (4.11) that uπ F,h (cid:0) U t h ( g θ ) (cid:1) u − ⊂ π F (cid:0) U ( g θ t ) (cid:1) (cid:74) h (cid:75) , where we used that the g θ t -module V F is completely reducible and hence π F (cid:0) U ( g θ t ) (cid:1) is the commutant ofEnd g θ t ( V F ). The above inclusion becomes an equality modulo h . Since U t h ( g θ ) is complete in the h -adictopology, we then easily deduce that the inclusion is in fact an equality. This finishes the proof of thelemma for a fixed finite set F , apart from the last statement about unitarity.Now, consider an increasing sequence of finite subsets F n ⊂ P + with union P + . For every n , chooseelements u ( n ) = ( u ( n ) λ ) λ ∈ F n , satisfying (4.10) for F = F n . To finish the proof it suffices to show that wecan inductively modify u ( n +1) in such a way that we get u ( n +1) λ = u ( n ) λ for λ ∈ F n .For this, consider the element w = ( u ( n ) λ ( u ( n +1) λ ) − ) λ ∈ F n ∈ End( V F n ) (cid:74) h (cid:75) . We have w (0) = 1 , (Ad w ) (cid:0) π F n (cid:0) U ( g θ t ) (cid:1) (cid:74) h (cid:75) (cid:1) = π F n (cid:0) U ( g θ t ) (cid:1) (cid:74) h (cid:75) . Since π F n (cid:0) U ( g θ t ) (cid:1) is semisimple, it follows that there is an element v ∈ hπ F n (cid:0) U ( g θ t ) (cid:1) (cid:74) h (cid:75) such thatAd w = Ad v on π F n (cid:0) U ( g θ t ) (cid:1) . Lift v to an element u ∈ hU ( g θ t ) (cid:74) h (cid:75) . We then modify u ( n +1) by replacing u ( n +1) λ by u ( n ) λ for λ ∈ F n and by π λ ( u ) u ( n +1) λ for λ ∈ F n +1 \ F n .Finally, assume in addition that π is ∗ -preserving. In this case it suffices to show that at every stageof the above construction of u λ we can get unitary elements with the required properties. Specifically,we claim that if uπ F,h (cid:0) U t h ( g θ ) (cid:1) u − = π F (cid:0) U ( g θ t ) (cid:1) (cid:74) h (cid:75) for a finite set F and an element u , u (0) = 1, thenthe same identity holds for the unitary ( uu ∗ ) − / u . Indeed, taking the adjoints we get π F,h (cid:0) U t h ( g θ ) (cid:1) = u ∗ π F (cid:0) U ( g θ t ) (cid:1) (cid:74) h (cid:75) ( u ∗ ) − . It follows that Ad( uu ∗ ) defines an automorphism β of π F (cid:0) U ( g θ t ) (cid:1) (cid:74) h (cid:75) . As β = idmodulo h , this automorphism has a unique square root β / such that β / = id modulo h . ThenAd( uu ∗ ) − / = β − / on π F (cid:0) U ( g θ t ) (cid:1) (cid:74) h (cid:75) , and our claim is proved. (cid:3) We continue to assume that t ∈ T ∗ . In the Hermitian case, recall the subgroups G θ t < G from theprevious subsection. In the non-Hermitian case, let us put G θ t = G θ . The collection ( u λ ) λ ∈ P + defines anelement u = u t ∈ U ( G ) (cid:74) h (cid:75) such that u (0) = 1 , uπ ( U t h ( g θ )) u − ⊂ U ( G θ t ) (cid:74) h (cid:75) . (4.12)Furthermore, the last inclusion is dense in the sense that the images of both algebras in End( V ) (cid:74) h (cid:75) coincide for any finite dimensional g -module V .Consider the homomorphism α h : U ( G θ t ) (cid:74) h (cid:75) → U ( G × G ) (cid:74) h (cid:75) defined by α h ( x ) = ( u ⊗ h ( u − xu )( u − ⊗ . If x = uπ ( y ) u − for some y ∈ U t h ( g θ ), we have α h ( x ) = α h ( uπ ( y ) u − ) = (cid:0) (Ad u ) π ⊗ π (cid:1) ∆( y ) . By the density of uπ ( U t h ( g θ )) u − in U ( G θ t ) (cid:74) h (cid:75) we conclude that α h ( U ( G θ t ) (cid:74) h (cid:75) ) ⊂ U ( G θ t × G ) (cid:74) h (cid:75) , and thestrict coassociativity ( α h ⊗ id) α h = (id ⊗ ∆ h ) α h holds. Thus, we get a coaction of ( U ( G ) (cid:74) h (cid:75) , ∆ h ) on ( G θ t ) (cid:74) h (cid:75) making the following diagram commutative: U t h ( g θ ) U t h ( g θ ) ˆ ⊗ U h ( g ) U ( G θ t ) (cid:74) h (cid:75) U ( G θ t × G ) (cid:74) h (cid:75) ∆(Ad u t ) π (Ad u t ) π ⊗ πα h Definition 4.13. For t ∈ T ∗ , we call the coaction ( U ( G θ t ) (cid:74) h (cid:75) , α h ) of ( U ( G ) (cid:74) h (cid:75) , ∆ h ) the multiplier algebramodel of the Letzter–Kolb coideal U t h ( g θ ).It is not difficult to see that up to twisting this model does not depend on the choice of π and u .Let us record an immediate consequence of the construction of α h , which we will use later. Proposition 4.14. For every t ∈ T ∗ , there is an element G ∈ U ( G θ t × G ) (cid:74) h (cid:75) such that G (0) = 1 , (id ⊗ (cid:15) )( G ) = 1 , α h = G ∆( · ) G − . If α h is ∗ -preserving, then G can in addition be chosen to be unitary.Proof. Since α h = ∆ mod h , Lemma 1.1 implies the existence of such an element G . If α h is in addition ∗ -preserving, then we can replace G by the unitary G ( G ∗ G ) − / . (cid:3) Remark . By the above arguments and Remark B.8, the multiplier algebra model can also be definedfor all t ∈ T ∗ C excluding a countable set of values of s (0) o (S-type) or c (0) o (C-type). Remark . By Proposition C.1, for every t ∈ T ∗ C , the algebra U t h ( g θ ) is a deformation of U ( g θ t ).In the non-Hermitian case, g θ t = g θ is semisimple and standard arguments show that if π has image U ( g ) (cid:74) h (cid:75) , then there exists u ∈ hU ( g ) (cid:74) h (cid:75) such that uπ ( U θh ( g )) u − = U ( g θ ) (cid:74) h (cid:75) . There also exists G ∈ hU ( g θ ) ⊗ U ( g ) (cid:74) h (cid:75) satisfying the conditions in Proposition 4.14, analogously to Remark 2.7.(Moreover, by the remark following Proposition C.3 we can go beyond the standard case and considerany t = ( c , s ) ∈ T such that c (0) i = 1 for all i ∈ I \ X .) In other words, in the non-Hermitian case themultiplier algebra model does not have any particular advantages over the coideal picture.In the Hermitian case it is still true that U t h ( g θ ) is a trivial algebra deformation of U ( g θ t ), seeProposition C.3. But since in this case the first cohomology of g θ t with coefficients in a finite dimensionalmodule is not always zero, it is not clear whether u and G exist at the level of the universal envelopingalgebras. Remark . Type II symmetric pairs can be dealt with analogously to the non-Hermitian case. Therelevant involution on u ⊕ u in the Satake form is given by θ ( X, Y ) = ( ω ( Y ) , ω ( X )) for the Chevalleyinvolution ω . The corresponding Satake diagram is the disjoint union of two copies of the Dynkin diagramof g , with the corresponding vertices joined by arrows. Cohomological considerations as above, both formultiplier algebras and universal enveloping algebras, carry over.4.5. K -matrix of Balagović–Kolb. Next let us recall the construction of universal K -matrices for thecoideals U t h ( g θ ) according to [BK19, Kol20, DCNTY19]. (Strictly speaking, these papers have an extranormalization condition on θ as in Remark 4.2. We can either adapt their construction to our setting, orwe can first put this extra condition and then use Ad z as in Remark 4.2 to remove it later.)Denote by U q ( g ) the C ( q /d )-subalgebra of U h ( g ) ⊗ C (cid:74) h (cid:75) C [ h − , h (cid:75) generated by K ω ( ω ∈ P ), E i and F i ,where q = e h and d = 4 det(( a ij ) i,j ). (We use the same notation in Appendix B for the algebra definedover K = C [ h − , h (cid:75) , but since we are not going to use that algebra here, this should not lead to confusion.)As usual we denote by x ¯ x the bar involution , the C -linear automorphism of U q ( g ) characterized by q /d = q − /d , K ω = K − ω , E i = E i , F i = F i . In a similar way as before we define coideals U t q ( g θ ) ⊂ U q ( g ) for t = ( c , s ) such that c i , s i ∈ C ( q /d ).We will first construct, following [BK19], the K -matrix for a particular parameter t ∈ T defined by c i = q ( α i , Θ( α i ) − ρ X ) , s i = 0 , where ρ X is half the sum of the positive roots of the root system generated by X . The parameter t = ( c , s ) satisfies the assumptions in [BK19, Section 5.4]. key ingredient of the construction in [BK19] is a quasi- K -matrix X . Denote by U + ⊂ U q ( g ) the C ( q /d )-subalgebra generated by the elements E i , and by U + µ ⊂ U + the subspace of vectors of weight µ ∈ Q + , where Q is the root lattice. Then X = X µ ∈ Q + X µ ( X µ ∈ U + µ ) , where the sum is considered in a completion of U q ( g ) defined similarly to our multiplier algebra U ( G ), butover the field C ( q /d ). The elements X µ are uniquely determined by X = 1 and the following recursiverelations: [ F i , X µ ] = X µ − α i +Θ( α i ) c i X i K i − q − ( α i , Θ( α i )) K − i c i X i X µ − α i +Θ( α i ) ( i ∈ I ) , (4.13)with the convention that X µ − α i +Θ( α i ) = 0 if µ − α i + Θ( α i ) Q + . Here we put X i = 0 ( i ∈ X ) , X i = − z τ θ ( i ) T w X ( E τ θ ( i ) ) ( i ∈ I \ X ) . To use X in our setting, we need the following integrality property. Let R ⊂ C ( q /d ) be the localizationof the ring C [ q /d ] at q /d = 1. Denote by U + , int the R -subalgebra of U + generated by the elements E i ,and put U + , int µ = U + µ ∩ U + , int .Next, let I ∗ ⊂ I \ X be a set of representatives of the τ θ -orbits in I \ X . As we already used inSection 3.2 (although only in the Hermitian setting), the elements α − i = ( α i − Θ( α i )) for i ∈ I ∗ form abasis of the restricted root system, and we have α − τ θ ( i ) = α − i for all i . Proposition 4.18. Take µ ∈ Q + , µ = 0 . If µ has the form µ = X i ∈ I ∗ k i α − i for some k i ∈ Z + , then X µ ∈ ( q /d − U + , int µ . Otherwise X µ = 0 .Proof. Let us start with the second statement, that is, X µ = 0 if either Θ( µ ) = − µ , or Θ( µ ) = − µ but inthe decomposition µ = P i ∈ I ∗ k i α − i some integers k i ≥ µ defined by ht( µ ) = P i ∈ I m i if µ = P i ∈ I m i α i . We verify thecondition by induction on ht( µ ). Since µ − α i + Θ( α i ) = µ − α − i for i ∈ I \ X is either not in Q + or itsatisfies the same assumptions as µ , by the inductive hypothesis we get from (4.13) that [ F i , X µ ] = 0 forall i ∈ I . This means that Lusztig’s skew-derivatives i r ( X µ ) and r i ( X µ ) are zero, which is possible only if X µ = 0, see [Lus10, Proposition 3.1.6 and Lemma 1.2.15].Turning to the first statement, assume µ = P i ∈ I ∗ k i α − i with k i ∈ Z + . Put ht ∗ ( µ ) = P i ∈ I ∗ k i . Wewill prove the statement by induction on ht ∗ ( µ ).Consider the case ht ∗ ( µ ) = 1. Then µ = 2 α − j = α j − Θ( α j ) for some j ∈ I ∗ . From (4.13) we then get[ F i , X µ ] = 0 ( i ∈ I \ { j, τ θ ( j ) } ) , [ F i , X µ ] = c i X i K i − q − ( α i , Θ( α i )) K − i c i X i ( i = j, τ θ ( j )) . (4.14)Denote by U int the R -subalgebra of U q ( g ) generated by the elements K ± i , K i − q − , E i and F i . Then wehave an isomorphism U int / ( q /d − U int → U ( g ) such that K ± i , K i − q − d i H i , E i X α i , F i X − α i . Since X i ∈ U int , by (4.14) we conclude that [ F i , X µ ] ∈ ( q /d − U int for all i ∈ I . We claim that thisimplies that X µ ∈ ( q /d − U int , hence X µ ∈ ( q /d − U + , int µ , as ( q /d − U int ∩ U + = ( q /d − U int , + by the triangular decomposition of U int .Indeed, assuming X µ = 0, let k ∈ Z be the smallest number such that ( q /d − k X µ ∈ U int . If k ≥ q /d − k X µ in U ( g ) is a nonzero element of U ( n + ) µ , and on theother hand this image commutes with X − α i for all i . But this is impossible, hence k ≤ − F i , X µ ] ∈ ( q /d − U int for all i ∈ I . Hence X µ ∈ ( q /d − U + , int µ . (cid:3) Recall that π : U h ( g ) → U ( G ) (cid:74) h (cid:75) denotes a fixed homomorphism satisfying (4.5). When it is convenient,we extend it to U h ( g ) ⊗ C (cid:74) h (cid:75) C [ h − , h (cid:75) and the completion of U q ( g ) from [BK19], but then the targetalgebra should be U ( G )[ h − , h (cid:75) and Q π ∈ Irr G (cid:0) End( V π )[ h − , h (cid:75) (cid:1) , respectively. Corollary 4.19. We have π ( X ) ∈ h U ( G ) (cid:74) h (cid:75) . ollowing [BK19], consider a homomorphism γ : P → C ( q /d ) × such that γ ( α i ) = c i z τ θ ( i ) ( i ∈ I \ X ) , γ ( α i ) = 1 ( i ∈ X ) , (4.15)and put ξ ( ω ) = γ ( ω ) q − ( ω + ,ω + )+ P i ∈ I ( α − i ,α − i ) ω ( $ ∨ i ) , where ω + = ( ω +Θ( ω )) and ( $ ∨ i ) i ∈ I is the dual basis (fundamental coweights) of ( α i ) i ∈ I , see [BK19, (8.1)].It satisfies the relation ξ ( ω + α i ) = γ ( α i ) q − ( α i , Θ( α i )) − ( ω,α i +Θ( α i )) ξ ( ω )for ω ∈ P and i ∈ I , which is enough for most of the purposes. We can view ξ as an element of acompletion of U q ( h ) ⊂ U q ( g ). Then one takes K = X ξT − w X T − w , where T w X and T w now denote the canonical elements implementing the Lusztig automorphisms. Thisgives a universal K -matrix for U t q ( g θ ) in the conventions of [BK19].To pass to our setting, consider the element ω of h ∗ characterized by( ω , α i ) = 0 ( i ∈ X ) , ( ω , α i ) = 14 (Θ( α τ θ ( i ) ) − α τ θ ( i ) − Θ( α i ) + 2 ρ X , α i ) ( i ∈ I \ X ) , and use the isomorphism Ad K ω of U t q ( g θ ) onto U θq ( g ). Namely, define K = τ θ τ (cid:0) (Ad K ω )( K ) (cid:1) = (Ad K ω ) (cid:0) τ θ τ ( K ) (cid:1) , (4.16)where τ θ τ is the automorphism of the Hopf algebra U q ( g ) induced by the automorphism τ θ τ of theDynkin diagram. Finally, using the universal R -matrix of U q ( g ), we put E = R (1 ⊗ K )(id ⊗ τ θ τ )( R ) . (4.17)This is a ribbon τ θ τ -braid for U θq ( g ), hence also for U θh ( g ), see [DCNTY19, Section 3.3]. Then E h = ((Ad u ) π ⊗ π )( E ) ∈ U ( G θ × G ) (cid:74) h (cid:75) is a well-defined ribbon ( τ θ τ ) h -braid for the multiplier algebra model of U θh ( g ), where u is the element (4.12)(for t = 0) and ( τ θ τ ) h denotes the unique automorphism of U ( G ) (cid:74) h (cid:75) such that π ◦ τ θ τ = ( τ θ τ ) h ◦ π. (4.18)We call E (and also E h ) a Balagović–Kolb ribbon ( τ θ τ ) h -braid . Note that this element depends on thechoice of γ , and the set of these twist-braids forms a torsor over Z ( U ). Remark . It is not difficult to see that Corollary 4.19 and identities (4.2) imply that E h = 1 ⊗ gzm X m modulo h for some g ∈ Z ( U ). This is consistent with Theorems 5.5 and 5.10 below.This finishes our discussion of the ribbon twist-braids in the standard case. Assume now that u θ < u is Hermitian and take t ∈ T ∗ C . Note that τ θ τ = id now, since θ is an inner automorphism. The coideal U t h ( g θ ) can be obtained from U θq ( g ) by twisting and h -adic completion similarly to [DCNTY19, TheoremC.7]. Namely, define a character χ t : U θq ( g ) → C [ h − , h (cid:75) as follows: • S-type: χ t ( K ω ) = 1 for ω ∈ P Θ , χ t = (cid:15) on U q ( g X ), χ t ( B i ) = 0 for the nondistinguished vertices i , χ t ( B o ) = s o κ o e − d o h − • C-type: χ t = (cid:15) on U q ( g X ), χ t ( B i ) = 0 for all i , χ t ( K ω ) = λ ( ω ) for ω ∈ P Θ , where λ : P → C [ h − , h (cid:75) × is any homomorphism such that λ ( α i ) = 1 for all i ∈ I \ { o } and λ ( α o ) = c − o e − h ( α − o ,α − o ) . Then ( χ t ⊗ id)∆ maps the generators of U θq ( g ) into those of U t h ( g θ ), except that in the type S case B o ismapped into F o − c o z τ θ ( o ) T w X ( E τ θ ( o ) ) K − o + s o κ o K − o e − d o h − , but this differs only by an additive constant (which may, however, lie in h − C (cid:74) h (cid:75) rather than in C (cid:74) h (cid:75) )from the corresponding generator of U t h ( g θ ). By applying this map to the first leg of E , and using thefactorization of E given in [Kol20], we get a ribbon braid E t for U t h ( g θ ). Then E t h = ((Ad u t ) π ⊗ π )( E t ) s a ribbon braid for the multiplier algebra model of U t h ( g θ ), whenever this model is well-defined. Wecall E t (and E t h ) again a Balagović–Kolb ribbon braid .One problem, however, is that in the S-type case the construction of E t guarantees only that E t h ∈ Y ρ ∈ Irr G θ t ,π ∈ Irr G (cid:0) End( V ρ ) ⊗ End( V π )[ h − , h (cid:75) (cid:1) . Proposition 4.21. For all t ∈ T ∗ C , we have ( π ⊗ π )( E t ) ∈ U ( G × G ) (cid:74) h (cid:75) .Proof. We need only to consider the Hermitian S-type case. Assume first that t ∈ T ∗ , that is, s o ∈ i R (cid:74) h (cid:75) .Then E t h is a ribbon braid for the coaction ( U ( G θ t ) (cid:74) h (cid:75) , α h ) of ( U ( G ) (cid:74) h (cid:75) , ∆ h , R h ). Hence the assertionfollows from Theorem 3.21, which is applicable by the results of Section 4.4 and Corollary 3.13. Forthe general case, observe that by construction the coefficient of h k of the component of ( π ⊗ π )( E t ) inEnd( V ) ⊗ End( W )[ h − , h (cid:75) is a rational function in finitely many parameters s ( n ) o . Since for k < s ( n ) o , it must be zero. (cid:3) In particular, if the multiplier algebra model of U t h ( g θ ) is well-defined for some t ∈ T ∗ C , then we have E t h ∈ U ( G θ t × G ) (cid:74) h (cid:75) . It would still be interesting to find a more explicit construction of E t similar to thatfor E , and provide a more direct proof of the above proposition.5. Comparison theorems We will combine the results of the previous sections to compare the Letzter–Kolb coideals with thequasi-coactions defined by the KZ-equations.5.1. Twisting of ribbon twist-braids. Let us start by refining the twisting procedure from Sec-tion 1.5. Assume H is a reductive algebraic subgroup of G , and ( U ( H ) (cid:74) h (cid:75) , α, Ψ) is a quasi-coaction of( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ). Then, given F ∈ U ( G ) (cid:74) h (cid:75) and G ∈ U ( H × G ) (cid:74) h (cid:75) such that F (0) = 1, G (0) = 1 and( (cid:15) ⊗ id)( F ) = (id ⊗ (cid:15) )( F ) = 1 , (id ⊗ (cid:15) )( G ) = 1 , we get a quasi-coaction ( U ( H ) (cid:74) h (cid:75) , α, Ψ F , G ) of ( U ( G ) (cid:74) h (cid:75) , ∆ h, F , Φ F ).Now, assume in addition that β is an involutive automorphism of ( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ) and v ∈ U ( G ) (cid:74) h (cid:75) is an element such that v (0) = 1, vβ ( v ) = 1 , F = ( v ⊗ v )( β ⊗ β )( F )∆ h ( v ) − . (5.1) Proposition 5.1. Under the above assumptions, β v = vβ ( · ) v − is an involutive automorphism of ( U ( G ) (cid:74) h (cid:75) , ∆ h, F , Φ F ) . Furthermore, suppose that R ∈ U ( G ) (cid:74) h (cid:75) is an R -matrix for ( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ) fixedby β , and that E ∈ U ( H × G ) (cid:74) h (cid:75) is a ribbon β -braid for R . Then R F = F RF − is an R -matrix for ( U ( G ) (cid:74) h (cid:75) , ∆ h, F , Φ F ) fixed by β v , and E G ,v = GE (id ⊗ β )( G ) − (1 ⊗ v − ) = GE (1 ⊗ v − )(id ⊗ β v )( G ) − (5.2) is a ribbon β v -braid for the quasi-coaction ( U ( H ) (cid:74) h (cid:75) , α G , Ψ F , G ) of ( U ( G ) (cid:74) h (cid:75) , ∆ h, F , Φ F , R F ) . We call ( U ( G ) (cid:74) h (cid:75) , ∆ h, F , Φ F , β v ) the twisting of ( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ , β ) by ( F , v ). Proof. The claims are not difficult to check by a direct computation, but let us explain a more conceptualproof using crossed products (or smashed products), cf. [DCNTY19, Remark 1.13]. Namely, consider thealgebra U ( G ) (cid:74) h (cid:75) (cid:111) β Z / Z = { a + a λ β | a, a ∈ U ( G ) (cid:74) h (cid:75) , λ β = 1 , λ β a = β ( a ) λ β } . We can extend in the usual way the coproduct ∆ h on U ( G ) (cid:74) h (cid:75) to a coproduct ˜∆ h on U ( G ) (cid:74) h (cid:75) (cid:111) β Z / Z by letting ˜∆ h ( λ β ) = λ β ⊗ λ β . Then ( U ( G ) (cid:74) h (cid:75) (cid:111) β Z / Z , ˜∆ h , Φ) is a multiplier quasi-bialgebra.Now, given ( F , v ) as above, we can twist ( U ( G ) (cid:74) h (cid:75) (cid:111) β Z / Z , ˜∆ h , Φ) by F to get a new multiplierquasi-bialgebra ( U ( G ) (cid:74) h (cid:75) (cid:111) β Z / Z , ( ˜∆ h ) F , Φ F ). On the other hand, we can first twist ( U ( G ) (cid:74) h (cid:75) , ∆ h , Φ)by F and then consider the crossed product by β v to get ( U ( G ) (cid:74) h (cid:75) (cid:111) β v Z / Z , (∆ h, F ) ∼ , Φ F ). The map f : U ( G ) (cid:74) h (cid:75) (cid:111) β Z / Z → U ( G ) (cid:74) h (cid:75) (cid:111) β v Z / Z , a a, λ β v − λ β v , is an isomorphism of these two multiplier quasi-bialgebras. In particular,( U ( G ) (cid:74) h (cid:75) (cid:111) β v Z / Z , (∆ h, F ) ∼ , Φ F )is indeed a multiplier quasi-bialgebra, and hence β v is an automorphism of ( U ( G ) (cid:74) h (cid:75) , ∆ h, F , Φ F ). et us turn to ribbon twist-braids. First note that R is still an R -matrix for ( U ( G ) (cid:74) h (cid:75) (cid:111) β Z / Z , ˜∆ h , Φ)by its β -invariance. Moreover, we can view ( U ( H ) (cid:74) h (cid:75) , α, Ψ) as a quasi-coaction of this multiplier quasi-bialgebra. Then an element E ∈ U ( H × G ) (cid:74) h (cid:75) is a ribbon β -braid for the original quasi-coaction and R ifand only if E (1 ⊗ λ β ) is a ribbon braid for the new one and R again.Finally, the map f satisfies (id ⊗ f )( GE (1 ⊗ λ β ) G − ) = E G ,v (1 ⊗ λ β v ) , showing that formula (5.2) is a consequence of (1.7) for the crossed products and trivial automorphisms. (cid:3) Remark . Let us also mention a categorical perspective on conditions (5.1), which does not relyon crossed products. The automorphism β defines an autoequivalence F β of ((Rep G ) (cid:74) h (cid:75) , ⊗ h , Φ). Thetwisting by F produces an equivalent category ((Rep G ) (cid:74) h (cid:75) , ⊗ h, F , Φ F ). The functor F β gives rise to anautoequivalence of this new category, which, however, is not defined by any automorphism in general.Conditions (5.1) ensure that this autoequivalence is naturally monoidally isomorphic to an autoequivalencedefined by an automorphism, namely, to F β v .We now return to the setup of Section 4.2. Let π : U h ( g ) → U ( G ) (cid:74) h (cid:75) be a homomorphism satisfying (4.5).Assume β is an involutive automorphism of the Dynkin diagram of g . We denote by the same symbol thecorresponding automorphisms of ( U h ( g ) , ∆) and ( U ( G ) (cid:74) h (cid:75) , ∆); it will always be clear from the contextwhich one we are using. These are automorphisms of the quasi-triangular (multiplier quasi-)bialgebras( U h ( g ) , ∆ , R ) and ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ , R KZ ). We also note that, similarly to (4.18), there is a uniqueautomorphism β h of ( U ( G ) (cid:74) h (cid:75) , ∆ h ) such that π ◦ β = β h ◦ π. Lemma 5.3. Let F be a Drinfeld twist for π (in the sense of Section 4.2). Then there exists a uniqueelement v ∈ h U ( G ) (cid:74) h (cid:75) such that β h = vβ ( · ) v − , F = ( v ⊗ v )( β ⊗ β )( F )∆( v ) − . (5.3) We also have vβ ( v ) = 1 . If, in addition, π is ∗ -preserving and F is unitary, then v is unitary. In other words, once F is fixed, β h is a twisting of the automorphism β of ( U ( G ) (cid:74) h (cid:75) , ∆) in a uniqueway. Proof. Since the homomorphisms π ◦ β and β ◦ π are equal modulo h , there exists w ∈ h U ( G ) (cid:74) h (cid:75) suchthat π ◦ β = (Ad w ) ◦ β ◦ π . Then β h = (Ad w ) ◦ β .We claim that ( w ⊗ w )( β ⊗ β )( F )∆( w ) − is again a Drinfeld twist (for the same π ). Since Φ KZ isinvariant under β , condition (4.7) is satisfied for ( β ⊗ β )( F ), hence also for ( w ⊗ w )( β ⊗ β )( F )∆( w ) − .It remains to check that∆ h = ( w ⊗ w )( β ⊗ β )( F )∆( w ) − ∆( · )∆( w )( β ⊗ β )( F − )( w − ⊗ w − ) , or equivalently,∆ h (cid:0) β h ( · ) (cid:1) = ( w ⊗ w )( β ⊗ β )( F )∆( w ) − ∆ (cid:0) β h ( · ) (cid:1) ∆( w )( β ⊗ β )( F − )( w − ⊗ w − ) . But this is true, as the right hand side of the above identity is easily seen to be equal to ( β h ⊗ β h )∆ h .By Lemma 3.18 it follows that by multiplying w by a central element we get an element v ∈ h U ( G ) (cid:74) h (cid:75) satisfying (5.3). Assume v is another element with the same properties. Then v − v is a central element,hence it also equals v v − and F = ( v v − ⊗ v v − ) F ∆( v v − ) − . By the uniqueness part of Lemma 3.18 we conclude that v v − = 1.Next, since β h and β are both involutive, the element β ( v ) − has the same properties as v , hence β ( v ) − = v . Similarly, if π is ∗ -preserving and F is unitary, then β h is ∗ -preserving as well, and theelement ( v ∗ ) − has the same properties as v , hence ( v ∗ ) − = v . (cid:3) Comparison theorem: non-Hermitian case. We are now ready to prove our main resultsrelating the multiplier algebra models of the Letzter–Kolb coideals to cyclotomic KZ-equations. Let usfirst consider the non-Hermitian case. heorem 5.4. Assume k = u θ < u is a non-Hermitian symmetric pair, with θ in Satake form (4.1) .Then the multiplier algebra model of the Letzter–Kolb coideal U θh ( g ) , which is a coaction ( U ( G θ ) (cid:74) h (cid:75) , α h ) of ( U ( G ) (cid:74) h (cid:75) , ∆ h ) , is obtained by twisting from the quasi-coaction ( U ( G θ ) (cid:74) h (cid:75) , ∆ , Ψ KZ ) of the multiplierquasi-bialgebra ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ) . Any such twisting extends to a twisting between the automorphism τ θ τ of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ) and the automorphism ( τ θ τ ) h of ( U ( G ) (cid:74) h (cid:75) , ∆ h ) .Proof. Using a Drinfeld twist F and an element G provided by Proposition 4.14 we can twist the coaction( U ( G θ ) (cid:74) h (cid:75) , α h ) of ( U ( G ) (cid:74) h (cid:75) , ∆ h ) to a quasi-coaction ( U ( G θ ) (cid:74) h (cid:75) , ∆ , Ψ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ) for some Ψ.The first statement of the theorem follows then from Theorem 2.6. The second statement, on twisting τ θ τ to ( τ θ τ ) h , follows from Lemma 5.3. (cid:3) Theorem 5.5. The twisting provided by Theorem 5.4 establishes a one-to-one correspondence betweenthe following data: • the ribbon τ θ τ -braids for the quasi-coaction ( U ( G θ ) (cid:74) h (cid:75) , ∆ , Ψ KZ ) of the quasi-triangular multiplierquasi-bialgebra ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ , R KZ ) , given by E KZ g = exp( − h (2 t k + C k ))( zm X m g ) ( g ∈ Z ( U )); (5.4) • the Balagović–Kolb ribbon ( τ θ τ ) h -braids E (or their images E h ) for the coideal U θh ( g ) of the quasi-triangular bialgebra ( U h ( g ) , ∆ h , R ) , for different choices of γ satisfying (4.15) .Under this correspondence, we have E (0) h = 1 ⊗ zm X m g .Proof. By Theorem 2.18 we have a complete classification of ribbon θ -braids for the quasi-coaction( U ( G θ ) (cid:74) h (cid:75) , ∆ , Ψ KZ ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ). Since θ = (Ad zm X m ) ◦ τ θ ◦ τ , the multiplication by 1 ⊗ zm X m on the right gives a one-to-one correspondence between the ribbon θ -braids and the ribbon τ θ τ -braids,so the latter ones are given by (5.4). As any Drinfeld twist F satisfies (4.9), formula (5.2) provides acorrespondence between the ribbon τ θ τ -braids and the ( τ θ τ ) h -braids. Since both the ribbon τ θ τ -braidsand the Balagović–Kolb ribbon ( τ θ τ ) h -braids are torsors over the finite group Z ( U ), this gives a bijectivecorrespondence stated in the theorem.Finally, since by definition the elements F , G , and v used in the twisting have constant terms 1, weget the claim about E (0) h . (cid:3) Remark . As we pointed out throughout the paper (see Remarks 2.7 and 4.16), in the non-Hermitiancase there is no real need to consider the multiplier algebra model, so a similar result holds at the level ofthe universal enveloping algebras, also beyond the standard case.Let us formulate this more precisely. Let t = ( c , s ) ∈ T be such that c (0) i = 1 for all i ∈ I \ X (recallalso that, by definition, we have s i = 0 for all i ∈ I \ X ). If we fix algebra isomorphisms U h ( g ) ∼ = U ( g ) (cid:74) h (cid:75) and U t h ( g θ ) ∼ = U ( g θ ) (cid:74) h (cid:75) that are identity modulo h (that is, they are given by (4.5) and Proposition C.3),then the coproduct ∆ : U h ( g ) → U h ( g ) ˆ ⊗ U h ( g ) defines a coproduct ∆ h on U ( g ) (cid:74) h (cid:75) and a coaction α h : U ( g θ ) (cid:74) h (cid:75) → U ( g θ ) ⊗ U ( g ) (cid:74) h (cid:75) of ( U ( g ) (cid:74) h (cid:75) , ∆ h ).The claim then is that this coaction is obtained by twisting from the quasi-coaction ( U ( g θ ) (cid:74) h (cid:75) , ∆ , Ψ KZ )of ( U ( g ) (cid:74) h (cid:75) , ∆ , Φ KZ ). Any such twisting extends to a twisting between the automorphisms τ θ τ of( U ( g ) (cid:74) h (cid:75) , ∆ , Φ KZ ) and ( U ( g ) (cid:74) h (cid:75) , ∆ h ) and, in the standard case t = 0, provides a one-to-one correspondencebetween the ribbon τ θ τ -braids as in Theorem 5.5. Remark . The type II symmetric pairs admit analogues of Theorems 5.4 and 5.5 and Remark 5.6, withessentially the same proofs. Indeed, as we have explained along the way, the intermediate results used inthe proofs, such as Theorems 2.6 and 2.18 and Proposition 4.14, all have analogues for the type II case.5.3. Comparison theorem: Hermitian case. In the Hermitian case we do need to consider themultiplier algebra model in our approach. Theorem 5.8. Assume u θ < u is a Hermitian symmetric pair, with θ in Satake form (4.1) . Take t ∈ T ∗ and choose Z t θ ∈ z ( g θ t ) such that ( Z t θ , Z t θ ) g = − a − θ . Then the coaction ( U ( G θ t ) (cid:74) h (cid:75) , α h ) of ( U ( G ) (cid:74) h (cid:75) , ∆ h ) is obtained by twisting from the quasi-coaction ( U ( G θ t ) (cid:74) h (cid:75) , ∆ , Ψ KZ ,s ; µ ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ) for uniquelydefined s ∈ R and µ ∈ h R (cid:74) h (cid:75) , where Ψ KZ ,s ; µ is defined using Z t θ .The parameter s ∈ R is determined as follows: • S-type: if α o is the unique distinguished root and c = − is (0) o , then s = ± π log (cid:0) (1 + c ) / + c (cid:1) , where ± is the sign of κ o ( Z t θ , X α o ) g ; C-type: if α o is the unique distinguished root such that − iα o ( ˜ Z t θ ) + iα τ θ ( o ) ( ˜ Z t θ ) > , where ˜ Z t θ is thecomponent of Z t θ in h , and c = c (0) o , then s = 2 π log c. Proof. Choose Z θ ∈ z ( u θ ) such that ( Z θ , Z θ ) g = − a − θ . In the C-type case we require also that if o isthe distinguished root as in the formulation of the theorem, then − iα o ( Z θ ) + iα τ θ ( o ) ( Z θ ) > 0, whichdetermines Z θ uniquely. By Lemma 4.7 and the discussion following it we then get a pair ( ν, Z ν ) as inSection 3.By twisting the coaction we may assume that π : U h ( g ) → U ( G ) (cid:74) h (cid:75) defining the multiplier algebramodel is as in Lemma 4.5. Then by Lemma 4.6 there is a unitary Drinfeld twist such that F = 1 + h ir O ( h ) . By Lemma 4.12 we could also choose u satisfying (4.12) to be unitary, which means that by twisting α h we may assume α h to be ∗ -preserving. Hence, by Proposition 4.14, α h is a twisting of ∆ by a unitaryelement G .By Lemma 4.10 we have g θ t = (Ad z θ ) − ( g φ ) and Z t θ = (Ad z − θ g φ − )( Z ν ), where φ is determined asfollows (see also Remark 4.11): • S-type: z θ ( α o ) s (0) o κ o = i tan( πφ ) and ( Z t θ , X α o ) g z θ ( α o ) cos( πφ ) > • C-type: c (0) o = − cot( π ( φ − z θ defines isomorphisms U ( G θ t ) → U ( G φ ) and U ( G ) → U ( G ) and transforms the coaction( U ( G θ t ) (cid:74) h (cid:75) , α h ) of ( U ( G ) (cid:74) h (cid:75) , ∆ h ) into a coaction ( U ( G φ ) (cid:74) h (cid:75) , ˜ α h ) of ( U ( G ) (cid:74) h (cid:75) , ˜∆ h ). As [ z θ ⊗ z θ , r ] = 0,the latter coaction satisfies the assumptions of Theorem 3.17. Hence this coaction is a twisting of thequasi-coaction ( U ( G φ ) (cid:74) h (cid:75) , ∆ , Ψ KZ ,s ; µ ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ) for uniquely determined s ∈ R and µ ∈ h R (cid:74) h (cid:75) ,with s determined by sin (cid:16) πφ (cid:17) = tanh (cid:16) πs (cid:17) . Applying (Ad z θ ) − , we conclude that our original coaction is obtained by twisting from the quasi-coaction ( U ( G θ t ) (cid:74) h (cid:75) , ∆ , Ψ KZ ,s ; µ ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ), and the pair ( s, µ ) is the only one with thisproperty.It remains to verify the formulas for s in the formulation of the theorem.In the S-type case, using s (0) o = ic we can write z θ ( α o ) κ o c = tan( πφ ). We also have z θ ( α o ) κ o = ± πφ ) = ± c (1 + c ) − / , or equivalently πs ± 12 log (cid:18) c (1 + c ) − / − c (1 + c ) − / (cid:19) = ± log (cid:0) (1 + c ) / + c (cid:1) , with the ± being equal to the sign of c − sin( πφ ) = z θ ( α o ) κ o cos( πφ ). This is equal to the sign of κ o ( Z t θ , X α o ) g because ( z θ ( α o ) cos( πφ )) − ( Z t θ , X α o ) g > c (0) o = c > 0, we havesin (cid:16) πφ (cid:17) = c − c + 1 = c − c − c + c − , hence πs = log c . (cid:3) Remark . Throughout the paper we made a number of statements about unitarity. We used themin the proof of Theorem 5.8 to make sure that for t ∈ T ∗ we get s ∈ R and µ ∈ h R (cid:74) h (cid:75) . It follows thatΨ KZ ,s ; µ is unitary, and once we know this, a standard argument based on polar decomposition shows thatif π : U h ( g ) → U ( G ) (cid:74) h (cid:75) and α h are ∗ -preserving, then the twisting can be done by unitary elements. Thesame is true for Theorem 5.4.The parameter µ can in principle be determined by comparing the K -matrices using the next theorem.We will do this in detail in the type AIII case in Section 5.5. Theorem 5.10. The twisting provided by Theorem 5.4 establishes a one-to-one correspondence betweenthe following data: the ribbon braids for the quasi-coaction ( U ( G θ t ) (cid:74) h (cid:75) , ∆ , Ψ KZ ,s ; µ ) of the quasi-triangular multiplierquasi-bialgebra ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ , R KZ ) , given by E KZ ,s ; µ g = exp (cid:16) − h (2 t k t + C k t ) + π (1 − is − iµ )( Z t θ ) (cid:17) g ( g ∈ Z ( U )) , (5.5) • the Balagović–Kolb ribbon braids E t (or their images E t h ) for the coideal U t h ( g θ ) of the quasi-triangularbialgebra ( U h ( g ) , ∆ , R ) , for different choices of γ satisfying (4.15) .Under this correspondence, we have E t h = 1 ⊗ exp( π (1 − is ) Z t θ ) g mod h .Proof. This is proved similarly to Theorem 5.5, but now using the classification result from Theorem 2.19,applied to σ = exp( π ad Z t θ ), and the fact that the multiplication by 1 ⊗ exp( πZ t θ ) on the right gives aone-to-one correspondence between the ribbon σ -braids and the ribbon braids. (cid:3) Analogous results hold for generic t ∈ T ∗ C . More precisely, we have to exclude a countable set of valuesof s (0) o (S-type) and c (0) o (C-type) for the distinguished roots to be sure that a multiplier algebra modelfor U t h ( g θ ) exists, see Remark 4.15. We also have to make sure that Ψ KZ ,s ; µ is well-defined, which meansthat s should be outside a set A satisfying i (1 + 2 Z ) ⊂ A ⊂ i Q × . Proposition 5.11. In the S-type case, for generic t ∈ T ∗ C , the coaction ( U ( G θ t ) (cid:74) h (cid:75) , α h ) of ( U ( G ) (cid:74) h (cid:75) , ∆ h ) is obtained by twisting from the quasi-coaction ( U ( G θ t ) (cid:74) h (cid:75) , ∆ , Ψ KZ ,s ; µ ) of ( U ( G ) (cid:74) h (cid:75) , ∆ , Φ KZ ) for s ∈ C satisfying e πs = (cid:0) (1 + c ) / + c (cid:1) , with c = − is (0) o , and a uniquely determined µ ∈ h C (cid:74) h (cid:75) , wherethe square root (1 + c ) / is chosen such that (1 + c ) / κ o ( Z t θ , X α o ) g > . In the C-type case, thesame holds for s satisfying e πs = c , with c = c (0) o , where o is the unique distinguished root such that − iα o ( ˜ Z t θ ) + iα τ θ ( o ) ( ˜ Z t θ ) > . Such a twisting establishes a one-to-one correspondence between the ribbonbraids (5.5) and the Balagović–Kolb ribbon braids.Proof. The proof is essentially identical to that of Theorems 5.8 and 5.10. Let us only explain where thecondition (1 + c ) / κ o ( Z t θ , X α o ) g > c ) − = cos ( πφ ). We want to choose the square root (1 + c ) / so that sin( πφ ) = c (1 + c ) − / holds. Then we obtain e πs − e πs + 1 = tanh (cid:16) πs (cid:17) = c (1 + c ) − / = (cid:0) (1 + c ) / + c (cid:1) − (cid:0) (1 + c ) / + c (cid:1) + 1 , which gives the asserted formula for e πs . From the proof of Theorem 5.8, we see that the desired choice isgiven by (1 + c ) / = 1 z θ ( α o ) κ o cos( πφ ) . Then we have (1 + c ) / κ o ( Z t θ , X α o ) g = ( Z t θ , X α o ) g z θ ( α o ) cos( πφ ) > , hence the condition in the statement of the theorem. (cid:3) A Kohno–Drinfeld type theorem. The above results allow us to compare certain representationsof type B braid groups.Recall that the braid group Γ n of type B n is generated by elements ρ , σ , . . . , σ n − subject to thefollowing relations: σ i σ j = σ j σ i ( | i − j | > , σ i σ j σ i = σ j σ i σ j ( | i − j | = 1) ,ρ σ i = σ i ρ ( i > , ρ σ ρ σ = σ ρ σ ρ . This is the subgroup of the usual (type A n ) braid group on n + 1 strands consisting of the braids withthe first strand fixed. (a) ρ (b) σ (c) σ Figure 1. Generators of Γ ssume we have a quasi-coaction ( B, α, Ψ) of a quasi-bialgebra ( A, ∆ , Φ), and a ribbon braid E ∈ B ⊗ A with respect to an R -matrix R ∈ A ⊗ A . (As before, we will actually use the corresponding variants formultiplier algebras.) Consider a (left) B -module V and an A -module W . Then we get a representation ofΓ n on V ⊗ W ⊗ n , with ρ acting by E on the zeroth and first factors and σ i acting by the braiding Σ R onthe i -th and ( i + 1)-st factors, where Σ denotes the flip. More precisely, we have to fix a parenthesizationon V ⊗ W ⊗ n and take into account the associativity morphisms, but different choices lead to equivalentrepresentations by the standard coherence argument. For example, for n = 3 we can take(( V ⊗ W ) ⊗ W ) ⊗ W, and then the representation is defined by ρ 7→ E , , σ Ψ − , , (Σ R ) , Ψ , , , σ Ψ − , , (Σ R ) , Ψ , , . Let us first consider the non-Hermitian case. Since the involutive automorphisms of quasi-bialgebrasare nontrivial, we first need to pass to crossed products, as in the proof of Proposition 5.1.On the side of q -deformations, we take the Hopf algebra U h ( g ) (cid:111) τ θ τ Z / Z and consider the ribbonbraid E (1 ⊗ λ τ θ τ ) for a Balagović–Kolb ribbon twist-braid E (defined by γ satisfying (4.15)).Take any U θh ( g )-module V and ( U h ( g ) (cid:111) τ θ τ Z / Z )-module W that are finitely generated and free as C (cid:74) h (cid:75) -modules. Note that a ( U h ( g ) (cid:111) τ θ τ Z / Z )-module is the same as a U h ( g )-module plus a C (cid:74) h (cid:75) -linearisomorphism u : W → W such that u = 1 and ua = ( τ θ τ )( a ) u for all a ∈ U h ( g ). We then get arepresentation of Γ n as described above from E and R .On the side of cyclotomic KZ-equations, we can start with finite dimensional representations of U ( g θ )and U ( g ) (cid:111) τ θ τ Z / Z on ˜ V and ˜ W . The quasi-coaction ( U ( g θ ) (cid:74) h (cid:75) , ∆ , Ψ KZ ) of ( U ( g ) (cid:74) h (cid:75) (cid:111) τ θ τ Z / Z , ∆ , Φ KZ )together with R KZ = e − ht and a ribbon τ θ τ -braid E KZ g (5.4) define a representation of Γ n on ( ˜ V ⊗ ˜ W ⊗ n ) (cid:74) h (cid:75) .Then Theorems 5.4 and 5.5 and Remark 5.6 imply the following. Theorem 5.12. Let u θ < u be a non-Hermitian symmetric pair, with θ in Satake form. Let V be a U θh ( g ) -module, and W be a ( U h ( g ) (cid:111) τ θ τ Z / Z ) -module, that are finitely generated and free as C (cid:74) h (cid:75) -modules. Then the representation of Γ n on V ⊗ C (cid:74) h (cid:75) W ⊗ n defined by E and Σ R is equivalent to theone on (( V /hV ) ⊗ ( W/hW ) ⊗ n ) (cid:74) h (cid:75) defined by ( E KZ g , Ψ KZ , Σ R KZ , Φ KZ ) for the choice of g satisfying ⊗ ( zm X m g ) = E mod h . In the Hermitian case, we can do similar constructions with the following modifications. First, as τ θ τ = id, we don’t have to take crossed products. Thus, given t ∈ T ∗ , a Balagović–Kolb ribbon braid E t for U t h ( g θ ) and Σ R defines a representation of Γ n on V ⊗ C (cid:74) h (cid:75) W ⊗ n . On the side of KZ-equations, by ourconstruction of Ψ KZ ,s ; µ , we can only consider U ( g θ t )-modules ˜ V that can be integrated to representationsof G θ t , or equivalently, that are direct summands of finite dimensional U ( g )-modules. We use the ribbonbraid E KZ ,s ; µ from (5.5).As a consequence of Theorems 5.8 and 5.10, we get the following result. Theorem 5.13. Let u θ < u be a Hermitian symmetric pair, with θ in Satake form, and t ∈ T ∗ . Let V be a U t h ( g θ ) -module and W be a U h ( g ) -module that are finitely generated and free as C (cid:74) h (cid:75) -modules,and assume also that V is a direct summand of a U h ( g ) -module with the same property. Then therepresentation of Γ n on V ⊗ C (cid:74) h (cid:75) W ⊗ n defined by E t and R is equivalent to the representation on (( V /hV ) ⊗ ( W/hW ) ⊗ n ) (cid:74) h (cid:75) defined by ( E KZ ,s ; µ g , Ψ KZ ,s ; µ , Σ R KZ , Φ KZ ) , for the subgroup G θ t < G andparameters ( s, µ ) from Theorem 5.8, for the choice of g ∈ Z ( U ) satisfying ⊗ (exp( πi (1 − is ) Z t θ ) g ) = E t mod h .Remark . Since the subgroups G θ t are conjugate to G θ , we could equally well consider the KZ-equationsonly for G θ < G . We do not do this as the extra choice of conjugator will affect the correspondence E t = E KZ mod h .As a corollary we can also get a version of Theorem 5.12 in the analytic setting. We will prove onesuch result and then discuss how it can be generalized.We can define the algebras U q ( g ) and U θq ( g ) for q ∈ C × not a nontrivial root of unity. Furthermore,as has been noted in [DCNTY19, DCM20], the constructions of a Balagović–Kolb ribbon twist-braid E ,the associators Ψ KZ ,s and so on make sense in this setting. We can therefore consider two types offinite dimensional representations of Γ n , defined by E and monodromy of KZ-equations. To comparesuch representations we need a way to associate a representation of U θq ( g ) to a representation of U ( g θ ).To simplify matters let us consider only representations obtained by restriction from representations f U q ( g ) and U ( g ). The representation theories of U q ( g ) and U ( g ) are well-understood, so for any finitedimensional U ( g )-module V we have its quantum analogue V q . This correspondence extends also torepresentations of the crossed products U ( g ) (cid:111) τ θ τ Z / Z and U q ( g ) (cid:111) τ θ τ Z / Z . Corollary 5.15. Take q > , and assume that u θ < u is a non-Hermitian symmetric pair. Consider afinite dimensional U ( g ) -module V and a finite dimensional ( U ( g ) (cid:111) τ θ τ Z / Z ) -module W , and view V as a U ( g θ ) -module. Then the representation of Γ n on V q ⊗ W ⊗ nq defined by a Balagović–Kolb ribbon τ θ τ -braidfor U θq ( g ) and the universal R -matrix R for U q ( g ) , corresponding to h = log q , is equivalent to therepresentation on V ⊗ W ⊗ n defined by ( E KZ g , Ψ KZ , Σ R KZ , Φ KZ ) given by monodromy of the cyclotomicKZ-equations for the subgroup G θ < G and some choice of g ∈ Z ( U θ ) .Proof. We may assume that V and W are equipped with Hermitian scalar products such that they giverise to unitary representations of U . In a similar way, the assumption q > U q ( g ) is a ∗ -algebra and its representations on V q and W q can be turned into ∗ -representations.Theorem 5.5 gives us a bijection between the Balagović–Kolb ribbon τ θ τ -braids for U θh ( g ) and theribbon τ θ τ -braids (5.4). By specialization this gives us a bijection also in the analytic setting, but apriori it is not given by any formula similar to (5.2), as it is not clear when G can be specialized.Now, using this bijection, it is convenient to extend the representations of Γ n to Γ n × Z ( U ), with Z ( U )acting on the first factors W q and W of V q ⊗ W ⊗ nq and V ⊗ W ⊗ n , and prove a formally stronger statementthat these representations of Γ n × Z ( U ) are equivalent. The representations have the same character,since they are obtained by specialization from the formal case and in that case the representations areequivalent. Therefore it suffices to prove that the representations are completely reducible. For this, inturn, it suffices to show that in both cases the operators of the representations span ∗ -algebras.Observe in general that in the presence of a ∗ -involution, if we have a quasi-coaction ( B, α, Ψ) of aquasi-bialgebra ( A, ∆ , Φ) and a ribbon braid E with respect to an R -matrix R , with unitary Ψ and Φ andthe R -matrix satisfying R ∗ = R , then E ∗ is also a ribbon braid. Indeed, analogues of identities (1.4)and (1.5) for E ∗ are obtained immediately by taking the adjoints. For (1.6), we in addition have toconjugate by R and then flip the last two tensor factors.Since in the formal setting we have a complete classification of ribbon twist-braids, we conclude thatevery Balagović–Kolb ribbon τ θ τ -braid for U θh ( g ) and every ribbon τ θ τ -braid (5.4), being multiplied by1 ⊗ λ τ θ τ on the right, has the property that it coincides with its adjoint up to a factor 1 ⊗ g ( g ∈ Z ( U )).(For the twist-braids (5.4) this is also not difficult to see by definition, and for the Balagović–Kolb’s onesthis can be checked by an explicit computation as well [DCM20].) Hence the same is true in the analyticsetting, which implies the desired property of the representations. (cid:3) Remark . Corollary 5.15 remains true for generic q ∈ C . Briefly, this can be proved by viewing bothrepresentations as defined over a field of meromorphic functions in q . These representations have thesame character by Theorem 5.12. They can also be shown to be completely reducible, essentially becauseeverything is determined by restriction to q > 0, and for every such q the representations are completelyreducible. Hence they are equivalent, and then by specialization we get an equivalence for generic valuesof q . Remark . In the Hermitian case, for q > 0, we can define an analogue of the parameter set T ∗ forwhich U t q ( g θ ) are ∗ -coideals, see [DCNTY19]. Then the proof of the above corollary still works for such t ,but with a caveat. Assume t is obtained by specialization from a parameter in our set T ∗ . Then to beable to use Theorem 5.13, or even formulate the result in the analytic setting, we need µ provided byTheorem 5.8 to be specializable. Assuming we have an explicit formula for µ as a function of t that canbe specialized, this can be further generalized to generic q ∈ C . In the type AIII case analyzed below wesee that this is indeed the case, and it is natural to expect that the same it true in all other cases.5.5. Example: AIII case. In this section we look in detail at the AIII symmetric pairs, that is, thepairs s ( u p ⊕ u N − p ) < su N for 0 < p ≤ N/ N ≥ u = su N , g = sl N ( C ). The normalized invariant form is ( X, Y ) g = Tr( XY ). As the Cartansubalgebra h we take the diagonal matrices with trace zero. Let e ij be the matrix units of M N ( C ). Define L i ∈ h ∗ by L i ( P j a j e jj ) = a i . As a system of simple roots and generators of g we takeΠ = { α i = L i − L i +1 } ≤ i ≤ N − , H i = e ii − e i +1 ,i +1 , X α i = e i,i +1 , X − α i = e i +1 ,i . Note that ( L i , L i ) = 1 − N , ( L i , L j ) = − N ( i = j ) . (5.6) efine Z ν = i diag (cid:16) − pN , . . . , − pN | {z } p , − pN , . . . , − pN | {z } N − p (cid:17) , ν = Ad exp( πZ ν ) . Then u ν = s ( u p ⊕ u N − p ) and the pair ( ν, Z ν ) is as in Section 3.1. We will write k for u ν . The uniquenoncompact simple root is α p . N − p (a) S-type 1 pN − pN − (b) C-type Figure 2. Satake diagrams for AIII symmetric pairsThe S-type case corresponds to N = 2 p . Then the distinguished simple root is α p , X = ∅ , and as aninvolution θ in Satake form we take θ = Ad m , m = A N = − . . . ( − N − , so that z = 1 in (4.1). It is clear that ( ν, Z ν ) is associated with θ as described after Lemma 4.8. For every t ∈ T ∗ we fix a normalized element Z t θ ∈ z ( g θ t ) by requiring ( Z t θ , X α p ) g > 0. For the standard case t = 0we write Z θ = Z θ ∈ z ( g θ ).The C-type case corresponds to 0 < p < N/ 2. In this case the distinguished simple roots are α p and α N − p , X = { α p +1 , . . . , α N − p − } . As an involution θ in Satake form we take θ = Ad zm m X , z = e πip/N diag(( − p , . . . , ( − p | {z } p , − ( − N − p , . . . , − ( − N − p ) . Note that m m X = A p − ( − N − p I N − p ( − N − p A p , zm m X = e πip/N − A tp I N − p − A p . Again, ( ν, Z ν ) is the unique pair associated with θ for which α p is a noncompact root. For every t ∈ T ∗ we fix a normalized element Z t θ ∈ z ( g θ t ) by requiring − iα p ( ˜ Z t θ ) + iα N − p ( ˜ Z t θ ) > 0. Again for the standardcase we write Z θ = Z θ ∈ z ( g θ ). Theorem 5.18. With the above choices, the parameters s ∈ R and µ ∈ h R (cid:74) h (cid:75) associated with t ∈ T ∗ according to Theorem 5.8 are determined as follows (with q = e h ): • S-type: s + µ = 2 π log (cid:16)(cid:16) − q ( q + 1) s p (cid:17) / − q / ( q + 1)2 is p (cid:17) ; • C-type: s + µ = 2 π log c p + hπ .In particular, the standard case t = 0 corresponds to s + µ = 0 . We will prove the theorem by comparing the eigenvalues of K -matrices. In the AIII case this isfacilitated by the knowledge of solutions of the reflection equation [Mud02].As in Section 4.5, we will work over the field C ( q /d ) and then pass to C [ h − , h (cid:75) . The fundamentalrepresentation of U q ( g ) on V = C ( q /d ) N is given by π V ( E i ) = q / e i,i +1 , π V ( F i ) = q − / e i +1 ,i , π V ( K i ) = qe ii + q − e i +1 ,i +1 . riting R = q − /N ( π V ⊗ π V )( R ) for the universal R -matrix R , we have R = X i,j q − δ ij e ii ⊗ e jj + ( q − − q ) X i We have π V ( T w ) = q N − A N . Proof. By definition (see, e.g., [Jan96, Section 8.6]), we have T w = T [1] T [2] · · · T [ N − , T [ k ] = T k T k − · · · T , and the operators π V ( T i ) are given by π V ( T i ) e j = e j ( j = i, i + 1) , π V ( T i ) e i = − q / e i +1 , π V ( T i ) e i +1 = q / e i . This gives the result. (cid:3) In the nonstandard case we have K t = ( χ t ⊗ id)( R (1 ⊗ K ) R )for an appropriate character χ t : U θq ( g ) → C [ h − , h (cid:75) , as described in Section 4.5. rite T = (id ⊗ π V )( R ) ∈ U q ( g ) ⊗ M N ( C ( q /d )). Then T is an upper triangular matrix withcoefficients in U q ( b − ), and K t = ( χ t ⊗ id)( T (1 ⊗ K ) T ∗ ) , where we remind that h ∗ = h . Lemma 5.20. We have K t N = χ t (cid:16) K − NN K − NN · · · K N − N N − (cid:17) K N . Proof. By definition we have K t N = χ t (cid:16) X i,j T Ni T ∗ j (cid:17) K ij . Since T is upper triangular, the contribution of i = N is zero. The form of (5.7), and the fact that K isnot scalar, further tell us that K Nj = 0 for j = 1. Therefore K t N = χ t ( T NN T ∗ ) K N . Since T NN T ∗ = T NN T = K − NN K − NN · · · K N − N N − by the usual factorization of R (see, e.g., [Kas95,Section XVII.2]), this proves the lemma. (cid:3) To get further information on the K -matrices we will use that K t must commute with π V ( U t h ( g )). Wewill treat the S-type and C-type cases separately.In the S-type case, Θ( L i ) = L N − i +1 . The coideal U t h ( g θ ) is generated by U h ( h θ ), the elements B i = F i − q − E N − i K − i ( i < p ) , B p = F p − q − E p K − p + s p ( K − p − q − − , and their adjoints. Proposition 5.21. In the S-type case, for every t ∈ T ∗ , we have K t = ( − p − q p − p (cid:18) q / ( q + 1) s p I p − A tp A p (cid:19) . (5.11) Proof. It is easily seen that the nonconstant matrices (5.7) commuting with the generators of U t h ( g θ ) areprecisely of the form y (cid:18) q / ( q + 1) s p I p − A tp A p (cid:19) for y ∈ C [ h − , h (cid:75) × . (Note that these solutions were also described in [KS09, Section 5].)To determine y , we look at the matrix coefficient K t N . By Lemma 5.20 it is independent of t , since χ t istrivial on the Cartan part in the S-type case. In the standard case we compute K N using definition (5.9).By (5.10) and (5.6) we have π V ( ξ ) = q p − , while Lemma 5.19 describes the action of T w . Since e X lies in a completion of U q ( n + ) and has the weight zero component 1, we conclude that K t N = K N = q p − p . Hence y = ( − p − q p − p . (cid:3) Proof of Theorem 5.18: S-type case. Note that the matrix K t given by (5.11) leaves the two-dimensionalspaces spanned by e i and e p − i +1 invariant. From this we see that it has eigenvalues x ± = ( − p − q p − p (cid:16) q / ( q + 1)2 s p ± i (cid:16) − q ( q + 1) s p (cid:17) / (cid:17) , each of multiplicity p . (Recall that we are assuming s p ∈ ih R (cid:74) h (cid:75) , so (cid:0) − q ( q +1) s p (cid:1) / is well-defined asan element of R (cid:74) h (cid:75) .)On the other hand, consider a K -matrix as in (5.8) but using ( k , Z ν ) instead of ( u θ t , Z t θ ), and denoteits image under the fundamental representation of su N by M . (For the moment we leave the choice of g ∈ Z ( U ) free.) Then K t is conjugate to M by a formal matrix. Let us compute the eigenvalues of M .The Casimir operator C k equals C su p ⊕ su p − Tr( Z ν ) − Z ν . Since the Casimir operator of su p acts asthe scalar p − p in the fundamental representation of su p , it follows that C k acts as p − p + p = p − p .Hence the eigenvalues of M are y ± = ± q p − p ie ± π ( s + µ ) e πik p , each of multiplicity p , for some 0 ≤ k ≤ p − t follows that y + coincides with x + or x − . Since s is real, by looking at the order zero terms we canconclude that this is possible only if e πikp = ± e π ( s + µ ) = (cid:16) − q ( q + 1) s p (cid:17) / ± q / ( q + 1)2 is p . Furthermore, since we already know the formula for s by Theorem 5.8, we see that for s (0) p = 0 the signmust be − and g ∈ Z ( SU ( N )) is the scalar matrix e πikp = ( − p − .It remains to handle the case s (0) p = 0, so that g θ t = g θ . Then Theorem 5.8 implies that s = 0. We firstclaim that Z θ = ( − p A N . Since θ = Ad A N , we have A N ∈ z ( u θ ), hence this formula must be true up to a sign. Then the requirement( Z θ , X p ) g > M for the image of (5.8) under the fundamental representation of su N . From the aboveformula for Z θ , we see that M preserves the span of e i and e p − i +1 for each i , analogously to the situationfor K t observed above. Restricting to the span of e and e N , we find M ( e + ie N ) = ( − p iq p − p e ( − p πµ e πikp ( e + ie N ) , where e πikp is the effect of g ∈ Z ( U ), and a similar formula for e − ie N (which we do not use).Now, we also know that M and π V ( K t ) are conjugate by a formal matrix with constant term I N . Inparticular K t has eigenvectors which are deformations of e ± ie N , with the same eigenvalues.From (5.11), the restriction of K t to the span of e and e N gives K t η = − q p − p (cid:18) i (cid:16) − t (cid:17) + t (cid:19) η, with η = (cid:18)(cid:18) − t − i (cid:16) − t (cid:17)(cid:19) e + e N (cid:19) , where t = ( − p q ( q + 1) s p ∈ ih R (cid:74) h (cid:75) . This eigenvector η is a deformation of − i ( e + ie N ), hence weobtain the equality of eigenvalues( − p iq p − p e ( − p πµ e πikp = − q p − p (cid:18) i (cid:16) − t (cid:17) + t (cid:19) , or equivalently, e ( − p πµ e πikp = ( − p − (cid:18)(cid:16) − t (cid:17) − i t (cid:19) . When p is even, this implies e πikp = − µ follows by taking the logarithm. When p is odd, we first obtain e πikp = 1, and then the formula for µ follows by taking the logarithm of inverses(note that ( √ − x + x )( √ − x − x ) = 1). (cid:3) Next let us consider the C-type case. Then Θ( L i ) = L i for p + 1 ≤ i ≤ N − p , and Θ( L i ) = L N − i +1 for all other i . The coideal U t h ( g θ ) is generated by U h ( h θ ), U q ( g X ), the elements B i = F i − q − T w X ( E N − i ) K − i ( i < p ) , B p = F p − c p T w X ( E N − p ) K − p and their adjoints. Similarly to Lemma 5.19 we have π V ( T w X ) = I p q N − − p A N − p I p . (5.12)It follows that π V ( B i ) = q − / e i +1 ,i − q − / e N − i,N − i +1 ( i < p ) , π V ( B p ) = q − / e p +1 ,p − q N − p c p e p +1 ,N − p +1 . In the next proposition we assume that the character χ t is defined using the unique homomorphism P → R (cid:74) h (cid:75) ∗ with values in the power series with positive constant terms such that α p c − p q − and α i i = p . Proposition 5.22. In the C-type case, for every t ∈ T ∗ , we have K t = ( λ + µ ) I p − q N +12 − p c p λA tp λI N − p q − N +12 + p c − p µA p , (5.13) here λ = e − πi pN q N − ( N − p ) − pN c − pN p and µ = e πi N − pN q N − p + N − pN c N − p ) N p .Proof. Again, some elementary computations show that a nonconstant solution (5.7) commutes with thegenerators of π V ( U t h ( g θ )) if and only if it has the form (5.13), with no restrictions on λ and µ .Consider first the standard case. Then c p = q − ( α − p ,α − p ) = q − / . By (5.10) and (5.6) we have π V ( ξ ) e N = ( − p e − πi pN q N − e N , from which we deduce, similarly to the proof of Proposition 5.21, that K N = ( − p e − πi pN q N − N . It follows that µ = − e − πi pN q N − p . In a similar way, using (5.12), we compute λ = K p +1 ,p +1 = e − πi pN q N − ( N − p ) . For general t ∈ T ∗ , by Lemma 5.20 we have K t N = c − pN p q − pN K N = ( − p e − πi pN q − pN + N − N c − pN p , which gives the asserted formula for µ . Similarly to the proof of Lemma 5.20 we also have λ = K t p +1 ,p +1 = χ t (cid:16) N − p X i = p +1 T p +1 ,i T ∗ p +1 ,i (cid:17) K ii = χ t (cid:16) N − p X i = p +1 T p +1 ,i T ∗ p +1 ,i (cid:17) K p +1 ,p +1 . Using again the usual factorization of the R -matrix we see that T p +1 ,i for p + 1 < i ≤ N − p is an elementof U q ( h ) U q ( n − X ) of weight L i − L p +1 , while T p +1 ,p +1 = K N · · · K pN p K p +1 N − p +1 · · · K N − N − N − . By definition of χ t we conclude that λ differs from the standard case by the factor χ t ( T p +1 ,p +1 ) = q − pN c − pN p . This gives the required formula for λ . (cid:3) Proof of Theorem 5.18: C-type case. Similarly to the S-type case, the matrix K t given by (5.13) haseigenvalues − e − πi pN q N − p + N − pN c N − p ) N p , e − πi pN q N − ( N − p ) − pN c − pN p of multiplicities p and N − p , resp., while the image of (5.8) in the fundamental representation of su N has eigenvalues − e πikN e − πi pN q N − p e N − pN π ( s + µ ) , e πikN e − πi pN q N − ( N − p ) e − pN π ( s + µ ) of multiplicities p and N − p , resp., for some 0 ≤ k ≤ N − 1. By looking at the order zero terms weconclude that k = 0 and q − pN c − pN p = e − pN π ( s + µ ) , giving the formula for s + µ in the statement of the theorem. (cid:3) Appendix A. Co-Hochschild cohomology Our goal is to prove an analogue of Corollary 2.4 for universal enveloping algebras. The result actuallyfollows from [Cal06], but since the proof in [Cal06] relies on a deep theorem of Dolgushev [Dol05], we willgive a more elementary proof.Let V be a finite dimensional vector space over C (we could consider any field of characteristic zero),and W ⊂ V be a subspace. Viewing V as an abelian Lie algebra, consider its universal enveloping algebra( U ( V ) , ∆). As an algebra it is the symmetric algebra Sym( V ). But we will mostly need only the coalgebrastructure, in which case we write Sym c ( V ). Consider the tensor algebra T (Sym c ( V )) of the vector spaceSym c ( V ). We then make Sym c ( W ) ⊗ T (Sym c ( V )) into a cochain complex by defining d : Sym c ( W ) ⊗ Sym c ( V ) ⊗ n → Sym c ( W ) ⊗ Sym c ( V ) ⊗ ( n +1) by formula (2.1), so dT = T , ,...,n +1 − T , ,...,n +1 + · · · + ( − n T , ,...,n ( n +1) + ( − n +1 T , ,...,n . onsider the linear map p V/W : Sym c ( V ) → V /W obtained by composing the projection Sym c ( V ) → Sym ( V ) = V with the quotient map V → V /W . It extends to an algebra homomorphism T (Sym c ( V )) → V V /W, which we continue to denote by p V/W . Consider also the counit (cid:15) : Sym c ( W ) → C . Proposition A.1. The map (cid:15) ⊗ p V/W defines a quasi-isomorphism of (Sym c ( W ) ⊗ T (Sym c ( V )) , d ) onto ( V V /W, . The result is well-known for W = 0, see, e.g., [Kas95, Theorem XVIII.7.1]. We will deduce theproposition from this particular case using the formalism of twisting morphisms .Let ( A, m A , d A ) be a differential graded algebra with product m A and cohomological differential d A : A n → A n +1 . Let ( C, ∆ C ) be a coalgebra (concentrated in degree 0). A twisting morphism is a linearmap α : C → A satisfying d A α + α ? α = 0 , where α ? α = m A ( α ⊗ α )∆ C . Then the degree 1 map id ⊗ d A + d α , with d α = (id ⊗ m A )(id ⊗ α ⊗ id)(∆ C ⊗ id) , defines the structure of a cochain complex on the graded vector space C ⊗ A . We denote this complex by C ⊗ α A .We further assume that • A n and C have auxiliary gradings, called the weight gradings ; we write A w : m,ch : n for the weight m part of A n and C w : m for the weight m part of C ; • m A , d A , ∆ C and α have degree 0 for the weight grading; • C and A are nonnegatively graded with respect to both the weight degree and the cohomologicaldegree.In our application A = T (Sym c ( V )), C = Sym c ( W ) and the weight gradings are defined by declaringthe elements of Sym n ( V ) and Sym n ( W ) to be of weight n . Lemma A.2 (cf. [LV12, Lemma 2.1.5]) . Let A and B be weight graded differential graded algebras asabove and f : A → B be a quasi-isomorphism (i.e., a weight degree preserving homomorphism of dg algebrasinducing an isomorphism in cohomology). Then for any weight graded coalgebra C and any twistingmorphism α : C → A such that α ( C w :0 ) = 0 , the map id ⊗ f : C ⊗ α A → C ⊗ β B is a quasi-isomorphism,where β = f α .Proof. The assumption α ( C w :0 ) = 0 implies that d α “carries” weight from C to A . This can be used tobuild a spectral sequence.Specifically, define a decreasing filtration F s of C ⊗ α A by setting F s = M m ≥ s C ⊗ A w : m . We clearly have (id ⊗ d A )( F s ) ⊂ F s . Since (id ⊗ α )∆ C ( C w : n ) belongs to L k ≥ C w : n − k ⊗ A w : k bythe assumption α ( C w :0 ) = 0, we have d α ( F s ) ⊂ F s +1 . Note also that, for each weight n , we have F w : n = ( C ⊗ α A ) w : n and F w : ns = 0 for s > n . Hence the associated spectral sequence starting with E s,t = F ch : s + ts /F ch : s + ts +1 = C ⊗ A w : s,ch : s + t is convergent at each weight, and the E -differential d : E s,t → E s,t +10 is just id ⊗ d A . Therefore E s,t = C ⊗ H s + t ( A w : s ) . We can do the same construction for B . Then id ⊗ f induces an isomorphism at the E -page. Combinedwith the convergence of the spectral sequences, we obtain the assertion. (cid:3) Proof of Proposition A.1. Denote the complex we get for W = 0 by ( A, d A ), so that A = T (Sym c ( V )).For general W , let C = Sym c ( W ) and define a linear map α : C = Sym c ( W ) → A = Sym c ( V ) as thezero map on C = Sym ( W ) and the inclusion map Sym n ( W ) → Sym n ( V ) for n ≥ 1. Then ( α ? α )(1) = 0and ( α ? α )( x ) = ∆( x ) − ⊗ x − x ⊗ x ∈ Sym n ( W ) , n ≥ . It follows that α is a twisting morphism. We also have d α ( x ⊗ y ) = ∆( x ) ⊗ y − x ⊗ ⊗ y ( x ∈ Sym c ( W ) , y ∈ T m (Sym c ( V ))) . herefore the complex C ⊗ α A is exactly (Sym c ( W ) ⊗ T (Sym c ( V )) , d ).Since the proposition is true for W = 0, by Lemma A.2 we conclude that the map id ⊗ p V defines aquasi-isomorphism of (Sym c ( W ) ⊗ T (Sym c ( V )) , d ) onto (Sym( W ) ⊗ V V, d ), where the new differential d = d p V α is given by d (1 ⊗ y ) = 0 and d ( x · · · x m ⊗ y ) = X i x · · · ˆ x i · · · x m ⊗ x i ∧ y ( x , . . . , x m ∈ W, y ∈ V V ) . It remains to show that the homomorphism of graded algebras Sym( W ) ⊗ V V → V V /W defined by (cid:15) and the quotient map V → V /W gives a quasi-isomorphism of (Sym( W ) ⊗ V V, d ) onto ( V V /W, V = W , that is, Sym( W ) ⊗ V W is quasi-isomorphic to C concentratedin degree 0. The general case follows from this and the standard isomorphism of graded algebras V V ∼ = V W ˆ ⊗ V V /W . (cid:3) Let a be a finite dimensional complex Lie algebra and b < a a Lie subalgebra. For n = 0 , , . . . , put˜ B n a , b = U ( b ) ⊗ U ( a ) ⊗ n . These spaces form a cochain complex by the same formula as in (2.1). The differential d cH is equivariantwith respect to the diagonal adjoint action of b , so we also obtain a subcomplex B a , b = ( ˜ B a , b ) b .It is well-known that the symmetrization map defines an a -equivariant coalgebra isomorphism ofSym c ( a ) onto U ( a ), see [Kas95, Theorem V.2.5]. Since the definition of d cH uses only the coalgebrastructures, it follows that the complexes (Sym c ( b ) ⊗ T (Sym c ( a )) , d ) and ( ˜ B a , b , d cH ) are isomorphic.Therefore Proposition A.1 implies the following. Proposition A.3. For any finite dimensional complex Lie algebra a and a Lie subalgebra b , there is a b -equivariant quasi-isomorphism of ( ˜ B a , b , d cH ) onto ( V a / b , .Remark A.4 . For any X , . . . , X n ∈ a , the element 1 ⊗ X ⊗ · · · ⊗ X n ∈ ˜ B n a , b is a cocycle. By the definitionof (cid:15) ⊗ p a / b and the symmetrization map, the image of this cocycle in V n a / b is ˜ X ∧ · · · ∧ ˜ X n , where ˜ X i is the image of X i in a / b .Consider now a reductive algebraic subgroup H < G as in Section 2.1. Corollary A.5. The cohomology ( ˜ B g , h , d cH ) is isomorphic to V g / h and the cohomology of ( B g , h , d cH ) is isomorphic to ( V g / h ) h . Furthermore, the embedding ˜ B g , h → ˜ B G,H is a quasi-isomorphism, and if H is connected, then B g , h → B G,H is a quasi-isomorphism as well.Proof. The first statement follows immediately from Proposition A.3 and the fact that computing thecohomology commutes with taking the h -invariants in the reductive case.Consider now the embedding ˜ B g , h → ˜ B G,H . This induces a pairing between cochains in ˜ B g , h andchains f ⊗ · · · ⊗ f n in ˜ B nG,H = O ( H ) ⊗ O ( G ) ⊗ n from the proof of Proposition 2.1. Unpacking thedefinitions, for the cocycle 1 ⊗ X ⊗ · · · ⊗ X n ∈ ˜ B n g , h from Remark A.4, we have h f ⊗ · · · ⊗ f n , ⊗ X ⊗ · · · ⊗ X n i = f ( e ) n Y i =1 d e f i ( X i ) . Restricting to cycles of ˜ B G,H , this reduces to the canonical duality pairing between V g / h ∼ = H( ˜ B g , h ) and( V g / h ) ∼ = H( ˜ B G,H ), see the identification of [BGI71, Proposition VII.2.5].Let us be more concrete. Choose a basis X , . . . , X m in a complement of h in g . By Remark A.4,the cohomology classes of c i ...i n = 1 ⊗ X i ⊗ · · · ⊗ X i n ∈ ˜ B n g , h , i < · · · < i n , form a basis in H n ( ˜ B g , h ).Choose right H -invariant functions f , . . . , f m ∈ O ( G ) such that d e f i ( X j ) = δ ij . Define functions a i ...i n on H × G n by a i ...i n ( g , g , . . . , g n ) = X σ ∈ S n sgn( σ ) f i ( g σ (1) ) . . . f i n ( g σ ( n ) ) , so that we have h a i ...i n , c j ...j n i = δ i j . . . δ i n j n for all i < · · · < i n and j < · · · < j n . Moreover, usingthat f i ( g ) = f i ( e ) for g ∈ H , one can check that the a i ...i n are cycles in ˜ B G,H . We thus obtain anondegenerate pairing between H n ( ˜ B g , h ) and H n ( ˜ B G,H ) which factors through H n ( ˜ B g , h ) → H n ( ˜ B G,H ).Hence the last map is an isomorphism.Finally, by considering the h -invariants we conclude that if H is connected, then B g , h → B G,H is aquasi-isomorphism as well. (cid:3) ppendix B. Spherical vectors The goal of this appendix is to prove the following result essentially due to Letzter [Let00]. Sinceher setting and assumptions are slightly different, we will give a complete argument for the reader’sconvenience. Theorem B.1. In the notation of Section 4.4, assume t ∈ T ∗ and λ ∈ P + are such that the highestweight g -module V λ has a nonzero g θ t -invariant vector. Then this vector can be lifted to a U t h ( g θ ) -invariantvector in V λ (cid:74) h (cid:75) . In the non-Hermitian case we have g θ t = g θ = g θ . In the Hermitian case, by Lemma 4.10, the Liesubalgebra g θ t < g is conjugate to g θ . Hence in both cases, by [Kna02, Theorem 8.49] and its proof,necessary (and, as we will see shortly, sufficient) conditions for the existence of a nonzero g θ t -invariantvector in V λ are the following: the weight λ ∈ P + vanishes on h Θ , (B.1)( λ, α ∨ i ) ∈ Z for all i ∈ I such that Θ( α i ) = − α i . (B.2) Lemma B.2. Given i ∈ I , we have Θ( α i ) = − α j for some j if and only if i ∈ I \ X and α i is orthogonalto α k for all k ∈ X , in which case we also have j = τ θ ( i ) .Proof. Assume Θ( α i ) = − α j . Since Θ( α k ) = α k for all k ∈ X , we must have i ∈ I \ X , and sinceΘ( α i ) + α τ θ ( i ) ∈ Z X and the set I \ X is τ θ -invariant, it follows also that j = τ θ ( i ). As Θ( α i ) = − w X α τ θ ( i ) ,we have 0 ≥ ( α j , α k ) = − (Θ( α i ) , α k ) = ( w X α τ θ ( i ) , α k ) = ( α τ θ ( i ) , w X α k ) ≥ , for all k ∈ X , where the last inequality holds as w X α k ∈ Φ − X . Therefore α j = α τ θ ( i ) is orthogonal to α k for all k ∈ X , hence the same is true for α i as well. This proves the lemma in one direction, the otherdirection is obvious. (cid:3) It is well-known that ( U, U θ ), or equivalently ( U, K t ), is a Gelfand pair. Since every U t h ( g θ )-invariantvector reduces modulo h to a K t -invariant vector, to prove Theorem B.1 it therefore suffices to show thatthere is a nonzero U t h ( g θ )-invariant vector whenever conditions (B.1) and (B.2) are satisfied.Denote by K the field C [ h − , h (cid:75) . Instead of working with U h ( g ) and U t h ( g θ ), we extend the scalarsto K . Recall that we denote e h by q . Consider the K -subalgebra U q ( g ) of U h ( g ) ⊗ C (cid:74) h (cid:75) K generated by E i , F i and K ω ( ω ∈ P ). Consider also the K -subalgebra U t q ( g θ ) generated by K ω ( ω ∈ P Θ ), K ± i , E i , F i ( i ∈ X ) and B i ( i ∈ I \ X ). Then V qλ = V λ (cid:74) h (cid:75) ⊗ C (cid:74) h (cid:75) K = V λ ⊗ C K is the irreducible U q ( g )-modulewith highest weight λ . If we can show that it contains a nonzero U t q ( g θ )-invariant vector v , then h n v becomes a U t h ( g θ )-invariant vector in V λ (cid:74) h (cid:75) for n ∈ N large enough. To prove that such a vector v existsit is enough, in turn, to show that if ξ λ ∈ V qλ is the highest weight vector, then ξ λ U t q ( g θ ) + ξ λ , where U t q ( g θ ) + denotes the augmentation ideal of U t q ( g θ ). Indeed, since U t q ( g θ ) is ∗ -invariant, the U t q ( g θ )-module V qλ is completely reducible by [Let00, Theorem 3.3]. Then the projection of ξ λ onto a complementarysubmodule to U t q ( g θ ) + ξ λ is a nonzero invariant vector. Therefore it suffices to establish the followingresult. Theorem B.3 (cf. [Let00, Theorem 4.3]) . Assume t ∈ T and λ ∈ P + is a weight satisfying conditions (B.1) and (B.2) . Then for the highest weight vector ξ λ ∈ V qλ we have ξ λ / ∈ U t q ( g θ ) + ξ λ . Note that for this result we no longer need ∗ -invariance, so we can take any parameter t ∈ T . Theproof also works for any field extension of Q ( q /d ) in place of K (with parameters c i and s i taken fromthis field), where d is the determinant of the Cartan matrix.We start by analyzing the rank one case. Lemma B.4. Consider g = sl ( C ) and the element B = F − cEK − + s ( K − − ∈ U q ( sl ) , with c ∈ K × and s ∈ K . Then, for every n ∈ Z + , the highest weight module V qn contains a nonzero vectorkilled by B .Proof. For n = 0 the lemma is obvious. For n = 1, the U q ( sl )-module V q has the basis ξ , F ξ , F ξ over K , and the actions of E and K on this basis are given by KF k ξ = q − k F k ξ , EF k ξ = [2 − ( k − q [ k ] q F k − ξ . One can then easily check that the vector v = ξ + s (1 − q ) cq [2] q F ξ + 1 cq [2] q F ξ ies in the kernel of B .For n ≥ 2, the vector ξ ⊗ n ∈ ( V q ) ⊗ n has weight n and generates a U q ( sl )-submodule isomorphic to V qn .As ∆ q ( B ) = B ⊗ K − + 1 ⊗ B , the vector v ⊗ n ∈ ( V q ) ⊗ n is killed by B . Since its weight n component isnonzero, the projection of this vector onto V qn ⊂ ( V q ) ⊗ n is a nonzero vector killed by B . (cid:3) To deal with the general case, let us introduce the following notation. For a multi-index J = ( j , . . . , j n ),define F J = F j · · · F j n and B J = B j · · · B j n . We also let wt( J ) = α j + · · · + α j n . Denote by U − theunital K -subalgebra of U q ( g ) generated by the elements F j , j ∈ I , by M X, + the unital K -subalgebra of U t q ( g θ ) generated by the elements E j , j ∈ X , and by U Θ the K -algebra generated by the elements K ω , ω ∈ P Θ . Fix λ ∈ P + . Lemma B.5. Choose a finite collection J of multi-indices such that ξ λ and F J ξ λ , J ∈ J , form a basisof V qλ over K . Then the vectors ξ λ and B J ξ λ , J ∈ J , also form a basis of V qλ .Proof. By the definition of B i , we have that B J ξ λ equals F J ξ λ plus a linear combination of vectors ofhigher weights. A simple induction argument using this property gives the result. (cid:3) For each i ∈ X , put m i = ( λ, α ∨ i ) ∈ Z + . Lemma B.6. There are collections J i , i ∈ I , of multi-indices such that the elements , F J ( J ∈ J ,with J as in the previous lemma) and F J F m i +1 i ( J ∈ J i , i ∈ I ) form a basis of U − over K . Moreover theelements , B J ( J ∈ J ) and B J B m i +1 i ( J ∈ J i , i ∈ I ) form a basis of the right U Θ M X, + -module U t q ( g θ ) .More generally, for any choice of degree m i + 1 polynomials p i ∈ K [ x ] , the elements , B J ( J ∈ J ) and B J p i ( B i ) ( J ∈ J i , i ∈ I ) form a basis of the right U Θ M X, + -module U t q ( g θ ) .Proof. Consider the Verma module L λ with highest weight vector v λ of weight λ . The first part of thelemma follows from the well-known facts that the map U − → L λ , a av λ , is a linear isomorphism andthe kernel of the quotient map L λ → V λ is P i U − F m i +1 i v λ .The second part of the lemma follows then from [Kol14, Proposition 6.2]. To be more precise, someof our generators B i differ from the ones used by Kolb by scalar summands. Let us denote Kolb’sgenerators by ˜ B i . Then every element B J equals ˜ B J plus a linear combination of the elements ˜ B J withwt( J ) < wt( J ). Similarly to the previous lemma, we see that whenever { ˜ B J } J ∈ ˜ J is a basis of the right U Θ M X, + -module U t q ( g θ ), then { B J } J ∈ ˜ J also forms a basis. For the same reason we can add to every B J any linear combination of the elements B J with wt( J ) < wt( J ) and still get a basis. In particular, wecan replace every element of the form B J B m i +1 i by B J p i ( B i ). (cid:3) Lemma B.7. If λ ∈ P + satisfies (B.2) , we can find for all i ∈ I , degree m i + 1 polynomials p i ∈ K [ x ] such that p i ( B i ) ξ λ = 0 and p i (0) = 0 .Proof. Consider three cases. If i ∈ X , then B i = F i and we can take p i ( x ) = x m i +1 .Next, assume i ∈ I \ X but Θ( α i ) = − α i . Then i I ns , hence s i = 0 and B i = F i − c i z τ θ ( i ) T w X ( E τ θ ( i ) ) K − i . As − Θ( α i ) = w X α τ θ ( i ) ∈ ∆ + is different from α i , we have − k Θ( α i ) − ( n − k ) α i n ≥ k ≥ k elements T w X ( E τ θ ( i ) ) and n − k elements F i has weight − k Θ( α i ) − ( n − k ) α i , itmust therefore kill ξ λ . It follows that B ni ξ λ = F ni ξ λ for any n ≥ 1. In particular, we have B m i +1 i ξ λ = 0,so in this case we can again take p i ( x ) = x m i +1 .Finally, assume i is such that Θ( α i ) = − α i . Then, by Lemma B.2, we have i ∈ I ns , hence B i = F i − c i z i E i K − i + s i κ i K − i − q − d i − . The elements E i , F i , K ± i generate a copy of U q di ( sl ) in U q ( g ). By acting on ξ λ we get a spin m i U q di ( sl )-module with basis ξ λ , F i ξ λ , . . . , F m i i ξ λ over K . By Lemma B.5, the elements ξ λ , B i ξ λ , . . . , B m i i ξ λ also form a basis. As m i ∈ Z + by assumption, we can apply Lemma B.4 and conclude that there isa nonzero polynomial f ∈ K [ x ] of degree m ≤ m i such that B i f ( B i ) ξ λ = 0. Hence we can take p i ( x ) = x m i +1 − m f ( x ). (cid:3) Proof of Theorem B.3. Take a ∈ U t q ( g θ ) + . We want to show that aξ λ = ξ λ . By Lemma B.6 we canwrite a as a + X J ∈J B J a J + X i ∈ I X J ∈J i B J p i ( B i ) b i ; J , here a , a J and b i ; J are in U Θ M X, + and p i are the polynomials from Lemma B.7. Note that since (cid:15) ( B i ) = 0 and the polynomials p i have zero constant terms, we must have (cid:15) ( a ) = 0. By assumption (B.1)we have yξ λ = (cid:15) ( y ) ξ λ for every y ∈ U Θ M X, + . Hence aξ λ = X J ∈J (cid:15) ( a J ) B J ξ λ , which is different from ξ λ by our choice of J . (cid:3) Remark B.8 . Theorem B.1 remains true for t ∈ T ∗ C if we exclude a finite set (depending on λ ) of valuesof s (0) o (S-type) or c (0) o (C-type). Indeed, let us look for spherical vectors of the form ξ λ + P J ∈J c J F J ξ λ , c J ∈ K , where J is as in Lemma B.5. The sphericity condition gives us a system of linear equations for c J with coefficients that are rational functions (with complex coefficients) in q , s o or c o . A spherical vectorexists if and only if the rank of the matrix A of this system is the same as the rank of the augmentedmatrix B . Take a submatrix of A of maximal size giving a nonzero minor for some t ∈ T ∗ . Then thelowest order nonzero term of the minor’s expansion in h is a rational function of s (0) o or c (0) o , so thecorresponding minor remains nonzero for all t ∈ T ∗ C excluding a finite set of values of s (0) o or c (0) o . On theother hand, if we take any larger minor of B and consider its expansion in h , then the coefficients will berational functions in the parameters s ( n ) o or c ( n ) o . These functions must vanish for all purely imaginary(S-type) or real (C-type) values of the parameters, hence they are identically zero. Appendix C. Coideals as deformations For t ∈ T , recall g θ t < g from Definition 4.4. Proposition C.1. For every t ∈ T , the C (cid:74) h (cid:75) -module U t h ( g θ ) is topologically free and the homomorphism U h ( g ) → U ( g ) induces an isomorphism U t h ( g θ ) /hU t h ( g θ ) ∼ = U ( g θ t ) .Proof. Since U h ( g ) is topologically free and U t h ( g θ ) ⊂ U h ( g ) is closed, to prove both statements it sufficesto show that U t h ( g θ ) ∩ hU h ( g ) = hU t h ( g θ ) . For this, in turn, it is enough to check that for all n ≥ U t h ( g θ ) ∩ hU h ( g ) ⊂ hU t h ( g θ ) + h n U h ( g ) . (C.1)Let J X = { J = ( i , . . . , i k ) | i j ∈ X } ⊂ ∪ ∞ k =0 X k be such that the elements X α i · · · X α ik with ( i , . . . , i k ) ∈ J form a basis of U ( n + X ) ⊂ U ( g X ). Consideralso a larger set J ⊂ ∪ ∞ k =0 I k giving a basis of U ( n + ). For J = ( i , . . . , i k ) ∈ J , put E J = E i · · · E i k ∈ U h ( g ) and B J = B i · · · B i k ∈ U h ( g ) . Let H , . . . , H l be a basis of h . Then by the proof of [Kol14, Proposition 6.1], the image of the set { E J ( H ) k · · · ( H l ) k l B J | J ∈ J , k i ≥ , J ∈ J } in U ( g ) is a basis. This implies that this set is a basis of the free C (cid:74) h (cid:75) / ( h n )-module U h ( g ) /h n U h ( g ). Onthe other hand, the same argument as in the proof of [Kol14, Proposition 6.2] shows that { E J ( H ) k · · · ( H l ) k l B J | J ∈ J X , k i ≥ , J ∈ J } generates U h ( g θ t ) / ( U h ( g θ t ) ∩ h n U h ( g )) as a C (cid:74) h (cid:75) / ( h n )-module. These two facts clearly imply (C.1). (cid:3) The following lemma slightly generalizes the second Whitehead lemma. Lemma C.2. Assume a is a finite dimensional Lie algebra over a field of characteristic zero such thatthe derived Lie subalgebra [ a , a ] is semisimple and has codimension . Then H ( a , V ) = 0 for all finitedimensional a -modules V .Proof. This follows from [Dix55, Proposition 1] and the usual second Whitehead lemma, see also[Zus08]. (cid:3) Assume now that t ∈ T ∗ C . In the non-Hermitian case the set T ∗ C consists of one point and we have g θ = g θ . In the Hermitian case, by Lemma 4.10 we have g θ t ∼ = g θ . Therefore in both cases the abovelemma applies to g θ t . As U t h ( g θ ) is a deformation of U ( g θ t ) by Proposition C.1, this leads to the followingresult. roposition C.3. For all t ∈ T ∗ C , the isomorphism U t h ( g θ ) /hU t h ( g θ ) ∼ = U ( g θ t ) lifts to an isomorphism U t h ( g θ ) ∼ = U ( g θ t ) (cid:74) h (cid:75) of C (cid:74) h (cid:75) -algebras. 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MR2431117 ↑ C Vrije Universiteit Brussel Email address : [email protected] Universitetet i Oslo Email address : [email protected] OsloMet - storbyuniversitetet Email address : [email protected] Universitetet i Oslo Email address : [email protected]@math.uio.no