Jump to ContentJump to Main Navigation
Show Summary Details
More options 


Journal fÃŒr die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie


IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2017: 1.49

Online
ISSN
1435-5345
See all formats and pricing
More options 

Volume 2019, Issue 747

Issues

Universal K-matrix for quantum symmetric pairs

Martina Balagović / Stefan Kolb
Published Online: 2016-07-12 | DOI: https://doi.org/10.1515/crelle-2016-0012

Abstract

Let 𝔀 be a symmetrizable Kac–Moody algebra and let Uq⁢(𝔀) denote the corresponding quantized enveloping algebra. In the present paper we show that quantum symmetric pair coideal subalgebras B𝐜,𝐬 of Uq⁢(𝔀) have a universal K-matrix if 𝔀 is of finite type. By a universal K-matrix for B𝐜,𝐬 we mean an element in a completion of Uq⁢(𝔀) which commutes with B𝐜,𝐬 and provides solutions of the reflection equation in all integrable Uq⁢(𝔀)-modules in category 𝒪. The construction of the universal K-matrix for B𝐜,𝐬 bears significant resemblance to the construction of the universal R-matrix for Uq⁢(𝔀). Most steps in the construction of the universal K-matrix are performed in the general Kac–Moody setting.

In the late nineties T. tom Dieck and R. HÀring-Oldenburg developed a program of representations of categories of ribbons in a cylinder. Our results show that quantum symmetric pairs provide a large class of examples for this program.

1 Introduction

1.1 Background

Let 𝔀 be a symmetrizable Kac–Moody algebra and Ξ:𝔀→𝔀 an involutive Lie algebra automorphism. Let 𝔚={x∈𝔀∣Ξ⁢(x)=x} denote the fixed Lie subalgebra. We call the pair of Lie algebras (𝔀,𝔚) a symmetric pair. Assume that Ξ is of the second kind, which means that the standard Borel subalgebra 𝔟+ of 𝔀 satisfies dim⁡(Ξ⁢(𝔟+)∩𝔟+)<∞. In this case the universal enveloping algebra U⁢(𝔚) has a quantum group analog B𝐜,𝐬=B𝐜,𝐬⁢(Ξ) which is a right coideal subalgebra of the Drinfeld–Jimbo quantized enveloping algebra Uq⁢(𝔀), see [22, 23, 18]. We call (Uq⁢(𝔀),B𝐜,𝐬) a quantum symmetric pair.

The theory of quantum symmetric pairs was first developed by M. Noumi, T. Sugitani, and M. Dijkhuizen for all classical Lie algebras in [27, 29, 28, 8]. The aim of this program was to perform harmonic analysis on quantum group analogs of compact symmetric spaces. This allowed an interpretation of Macdonald polynomials as quantum zonal spherical functions. Independently, G. Letzter developed a comprehensive theory of quantum symmetric pairs for all semisimple 𝔀 in [22, 23]. Her approach uses the Drinfeld–Jimbo presentation of quantized enveloping algebras and hence avoids casework. Letzter’s theory also aimed at applications in harmonic analysis for quantum group analogs of symmetric spaces [24, 25]. The algebraic theory of quantum symmetric pairs was extended to the setting of Kac–Moody algebras in [18].

Over the past two years it has emerged that quantum symmetric pairs play an important role in a much wider representation theoretic context. In a pioneering paper H. Bao and W. Wang proposed a program of canonical bases for quantum symmetric pairs [3]. They performed their program for the symmetric pairs

(𝔰⁢𝔩2⁢N,𝔰⁢(𝔀⁢𝔩N×𝔀⁢𝔩N)) and (𝔰⁢𝔩2⁢N+1,𝔰⁢(𝔀⁢𝔩N×𝔀⁢𝔩N+1))

and applied it to establish Kazhdan–Lusztig theory for the category 𝒪 of the ortho-symplectic Lie superalgebra 𝔬𝔰𝔭(2n+1∣2m). Bao and Wang developed the theory for these two examples in astonishing similarity to Lusztig’s exposition of quantized enveloping algebras in [26]. In a closely related program M. Ehrig and C. Stroppel showed that quantum symmetric pairs for

(𝔀⁢𝔩2⁢N,𝔀⁢𝔩N×𝔀⁢𝔩N) and (𝔀⁢𝔩2⁢N+1,𝔀⁢𝔩N×𝔀⁢𝔩N+1)

appear via categorification using parabolic category 𝒪 of type D (see [11]). The recent developments as well as the previously known results suggest that quantum symmetric pairs allow as deep a theory as quantized enveloping algebras themselves. It is reasonable to expect that most results about quantized enveloping algebras have analogs for quantum symmetric pairs.

One of the fundamental properties of the quantized enveloping algebra Uq⁢(𝔀) is the existence of a universal R-matrix which gives rise to solutions of the quantum Yang–Baxter equation for suitable representations of Uq⁢(𝔀). The universal R-matrix is at the heart of the origins of quantum groups in the theory of quantum integrable systems [10, 14] and of the applications of quantum groups to invariants of knots, braids, and ribbons [31]. Let

Δ:Uq⁢(𝔀)→Uq⁢(𝔀)⊗Uq⁢(𝔀)

denote the coproduct of Uq⁢(𝔀) and let Δop denote the opposite coproduct obtained by flipping tensor factors. The universal R-matrix RU of Uq⁢(𝔀) is an element in a completion 𝒰0(2) of Uq⁢(𝔀)⊗Uq⁢(𝔀), see Section 3.2. It has the following two defining properties:

  • (1)

    In 𝒰0(2) the element RU satisfies the relation Δ⁢(u)⁢RU=RU⁢Δop⁢(u) for all u∈Uq⁢(𝔀).

  • (2)

    The relations

    (Δ⊗id)⁢(RU)=R23U⁢R13U,(id⊗Δ)⁢(RU)=R12U⁢R13U

    hold. Here we use the usual leg notation for threefold tensor products.

The universal R-matrix gives rise to a family R^=(RM,N) of commutativity isomorphisms R^M,N:M⊗N→N⊗M for all category 𝒪 representations M,N of Uq⁢(𝔀). In our conventions one has R^M,N=RU∘flipM,N where flipM,N denotes the flip of tensor factors. The family R^ can be considered as an element in an extension 𝒰(2) of the completion 𝒰0(2) of Uq⁢(𝔀)⊗Uq⁢(𝔀), see Section 3.3 for details. In 𝒰(2) property (1) of RU can be rewritten as follows:

  • (1’)

    In 𝒰(2) the element R^ commutes with Δ⁢(u) for all u∈Uq⁢(𝔀).

By definition the family of commutativity isomorphisms R^=(R^M,N) is natural in M and N. The above relations mean that R^ turns category 𝒪 for Uq⁢(𝔀) into a braided tensor category.

The analog of the quantum Yang–Baxter equation for quantum symmetric pairs is known as the boundary quantum Yang–Baxter equation or (quantum) reflection equation. It first appeared in I. Cherednik’s investigation of factorized scattering on the half line [6] and in E. Sklyanin’s investigation of quantum integrable models with non-periodic boundary conditions [33, 21]. In [21, Section 6.1] an element providing solutions of the reflection equation in all representations was called a ‘universal K-matrix’. Explicit examples of universal K-matrices for Uq⁢(𝔰⁢𝔩2) appeared in [7, (3.31)] and [20, (2.20)].

A categorical framework for solutions of the reflection equation was proposed by T. tom Dieck and R. HÀring-Oldenburg under the name braided tensor categories with a cylinder twist [34, 35, 12]. Their program provides an extension of the graphical calculus for braids and ribbons in ℂ×[0,1] as in [31] to the setting of braids and ribbons in the cylinder ℂ∗×[0,1], see [12]. It hence corresponds to an extension of the theory from the classical braid group of type AN-1 to the braid group of type BN. Tom Dieck and HÀring-Oldenburg called the analog of the universal R-matrix in this setting a universal cylinder twist. They determined a family of universal cylinder twists for Uq⁢(𝔰⁢𝔩2) by direct calculation [35, Theorem 8.4]. This family essentially coincides with the universal K-matrix in [20, (2.20)] where it was called a universal solution of the reflection equation.

1.2 Universal K-matrix for coideal subalgebras

Special solutions of the reflection equation were essential ingredients in the initial construction of quantum symmetric pairs by Noumi, Sugitani, and Dijkhuizen [27, 29, 28, 8]. For this reason it is natural to expect that quantum symmetric pairs give rise to universal K-matrices. The fact that quantum symmetric pairs B𝐜,𝐬 are coideal subalgebras of Uq⁢(𝔀) moreover suggests to base the concept of a universal K-matrix on a coideal subalgebra of a braided (or quasitriangular) Hopf algebra.

Recall that a subalgebra B of Uq⁢(𝔀) is called a right coideal subalgebra if

Δ⁢(B)⊂B⊗Uq⁢(𝔀).

In the present paper we introduce the notion of a universal K-matrix for a right coideal subalgebra B of Uq⁢(𝔀). A universal K-matrix for B is an element 𝒊 in a suitable completion 𝒰 of Uq⁢(𝔀) with the following properties:

  • (1)

    In 𝒰 the universal K-matrix 𝒊 commutes with all b∈B.

  • (2)

    The relation

    Δ⁢(𝒊)=(𝒊⊗1)⋅R^⋅(𝒊⊗1)⋅R^(1.1)

    holds in the completion 𝒰(2) of Uq⁢(𝔀)⊗Uq⁢(𝔀).

See Definition 4.12 for details. By the definition of the completion 𝒰, a universal K-matrix is a family 𝒊=(KM) of linear maps KM:M→M for all integrable Uq⁢(𝔀)-modules in category 𝒪. Moreover, this family is natural in M. The defining properties (1) and (2) of 𝒊 are direct analogs of the defining properties (1’) and (2) of the universal R-matrix RU. The fact that R^ commutes with Δ⁢(𝒊) immediately implies that 𝒊 satisfies the reflection equation

R^⋅(𝒊⊗1)⋅R^⋅(𝒊⊗1)=(𝒊⊗1)⋅R^⋅(𝒊⊗1)⋅R^

in 𝒰(2). By (1.1) and the naturality of 𝒊 a universal K-matrix for B gives rise to the structure of a universal cylinder twist on the braided tensor category of integrable Uq⁢(𝔀)-modules in category 𝒪. Universal K-matrices, if they can be found, hence provide examples for the theory proposed by tom Dieck and HÀring-Oldenburg. The new ingredient in our definition is the coideal subalgebra B. We will see in this paper that B plays a focal role in finding a universal K-matrix.

The notion of a universal K-matrix can be defined for any coideal subalgebra of a braided bialgebra H with universal R-matrix RH∈H⊗H. This works in complete analogy to the above definition for B and Uq⁢(𝔀), and it avoids completions, see Section 4.3 for details. Following the terminology of [34, 35] we call a coideal subalgebra B of H cylinder-braided if it has a universal K-matrix.

A different notion of a universal K-matrix for a braided Hopf algebra H was previously introduced by J. Donin, P. Kulish, and A. Mudrov in [9]. Let R21H∈H⊗H denote the element obtained from RH by flipping the tensor factors. Under some technical assumptions the universal K-matrix in [9] is just the element RH⁢R21H∈H⊗H. Coideal subalgebras only feature indirectly in this setting. We explain this in Section 4.4.

In a dual setting of coquasitriangular Hopf algebras the relations between the constructions in [9], the notion of a universal cylinder twist [34, 35], and the theory of quantum symmetric pairs was already discussed by J. Stokman and the second named author in [19]. In that paper universal K-matrices were found for quantum symmetric pairs corresponding to the symmetric pairs (𝔰⁢𝔩2⁢N,𝔰⁢(𝔀⁢𝔩N×𝔀⁢𝔩N)) and (𝔰⁢𝔩2⁢N+1,𝔰⁢(𝔀⁢𝔩N×𝔀⁢𝔩N+1)). However, a general construction was still outstanding.

1.3 Main results

The main result of the present paper is the construction of a universal K-matrix for every quantum symmetric pair coideal subalgebra B𝐜,𝐬 of Uq⁢(𝔀) for 𝔀 of finite type. This provides an analog of the universal R-matrix for quantum symmetric pairs. Moreover, it shows that important parts of Lusztig’s book [26, Chapters 4 and 32] translate to the setting of quantum symmetric pairs.

The construction in the present paper is significantly inspired by the example classes (𝔰⁢𝔩2⁢N,𝔰⁢(𝔀⁢𝔩N×𝔀⁢𝔩N)) and (𝔰⁢𝔩2⁢N+1,𝔰⁢(𝔀⁢𝔩N×𝔀⁢𝔩N+1)) considered by Bao and Wang in [3]. The papers [3] and [11] both observed the existence of a bar involution for quantum symmetric pair coideal subalgebras B𝐜,𝐬 in this special case. Bao and Wang then constructed an intertwiner Υ∈𝒰 between the new bar involution and Lusztig’s bar involution. The element Î¥ is hence an analog of the quasi R-matrix in Lusztig’s approach to quantum groups, see [26, Theorem 4.1.2]. Similar to the construction of the commutativity isomorphisms in [26, Chapter 32] Bao and Wang construct a B𝐜,𝐬-module homomorphism 𝒯M:M→M for any finite-dimensional representation M of Uq⁢(𝔰⁢𝔩N). If M is the vector representation, they show that 𝒯M satisfies the reflection equation and they establish Schur–Jimbo duality between the coideal subalgebra and a Hecke algebra of type BN acting on V⊗N.

In the present paper we consider quantum symmetric pairs in full generality and formulate results in the Kac–Moody setting whenever possible. The existence of the bar involution

¯B:B𝐜,𝐬→B𝐜,𝐬,x↩x¯B

for the quantum symmetric pair coideal subalgebra B𝐜,𝐬 was already established in [2]. Following [3, Section 2] closely we now prove the existence of an intertwiner between the two bar involutions. More precisely, we show in Theorem 6.10 that there exists a nonzero element 𝔛∈𝒰 which satisfies the relation

x¯B⁢𝔛=𝔛⁢x¯ for all x∈B𝐜,𝐬.(1.2)

We call the element 𝔛 the quasi K-matrix for B𝐜,𝐬. It corresponds to the intertwiner Υ in the setting of [3].

Recall from [18, Theorem 2.7] that the involutive automorphism Ξ:𝔀→𝔀 is determined by a pair (X,τ) up to conjugation. Here X is a subset of the set of nodes of the Dynkin diagram of 𝔀 and τ is a diagram automorphism. The Lie subalgebra 𝔀X⊂𝔀 corresponding to X is required to be of finite type. Hence there exists a longest element wX in the parabolic subgroup WX of the Weyl group W. The Lusztig automorphism TwX may be considered as an element in the completion 𝒰 of Uq⁢(𝔀), see Section 3. We define

𝒊′=𝔛⁢Ο⁢TwX-1∈𝒰(1.3)

where Ο∈𝒰 denotes a suitably chosen element which acts on weight spaces by a scalar. The element 𝒊′ defines a linear isomorphism

𝒊M′:M→M(1.4)

for every integrable Uq⁢(𝔀)-module M in category 𝒪. In Theorem 7.5 we show that 𝒊M′ is a B𝐜,𝐬-module homomorphism if one twists the B𝐜,𝐬-module structure on both sides of (1.4) appropriately. The element 𝒊′ exists in the general Kac–Moody case.

For 𝔀 of finite type there exists a longest element w0∈W and a corresponding family of Lusztig automorphisms Tw0=(Tw0,M)∈𝒰. In this case we define

𝒊=𝔛⁢Ο⁢TwX-1⁢Tw0-1∈𝒰.(1.5)

For the symmetric pairs (𝔰⁢𝔩2⁢N,𝔰⁢(𝔀⁢𝔩N×𝔀⁢𝔩N)) and (𝔰⁢𝔩2⁢N+1,𝔰⁢(𝔀⁢𝔩N×𝔀⁢𝔩N+1)) the construction of 𝒊 coincides with the construction of the B𝐜,𝐬-module homomorphisms 𝒯M in [3] up to conventions. The longest element w0 induces a diagram automorphism τ0 of 𝔀 and of Uq⁢(𝔀). Any Uq⁢(𝔀)-module M can be twisted by an algebra automorphism φ:Uq⁢(𝔀)→Uq⁢(𝔀) if we define ▷⁡um=φ⁢(u)⁢m for all u∈Uq⁢(𝔀), m∈M. We denote the resulting twisted module by Mφ. We show in Corollary 7.7 that the element 𝒊 defines a B𝐜,𝐬-module isomorphism

𝒊M:M→Mτ⁢τ0(1.6)

for all finite-dimensional Uq⁢(𝔀)-modules M. Alternatively, this can be written as

𝒊⁢b=τ0⁢(τ⁢(b))⁢𝒊 for all b∈B𝐜,𝐬.

The construction of the bar involution for B𝐜,𝐬, the intertwiner 𝔛, and the B𝐜,𝐬-module homomorphism 𝒊 are three expected key steps in the wider program of canonical bases for quantum symmetric pairs proposed in [3]. The existence of the bar involution was explicitly stated without proof and reference to the parameters in [3, Section 0.5] and worked out in detail in [2]. Weiqiang Wang has informed us that he and Huanchen Bao have constructed 𝔛 and 𝒊M′ independently in the case X=∅, see [4].

In the final Section 9 we address the crucial problem to determine the coproduct Δ⁢(𝒊) in 𝒰(2). The main step to this end is to determine the coproduct of the quasi K-matrix 𝔛 in Theorem 9.4. Even for the symmetric pairs (𝔰⁢𝔩2⁢N,𝔰⁢(𝔀⁢𝔩N×𝔀⁢𝔩N)) and (𝔰⁢𝔩2⁢N+1,𝔰⁢(𝔀⁢𝔩N×𝔀⁢𝔩N+1)), this calculation goes beyond what is contained in [3]. It turns out that if τ⁢τ0=id, then the coproduct Δ⁢(𝒊) is given by formula (1.1). Hence, in this case 𝒊 is a universal K-matrix as defined above for the coideal subalgebra B𝐜,𝐬. If τ⁢τ0≠id, then we obtain a slight generalization of properties (1) and (2) of a universal K-matrix. Motivated by this observation we introduce the notion of a φ-universal K-matrix for B if φ is an automorphism of a braided bialgebra H and B is a right coideal subalgebra, see Section 4.3. With this terminology it hence turns out in Theorem 9.5 that in general 𝒊 is a τ⁢τ0-universal K-matrix for B𝐜,𝐬. The fact that τ⁢τ0 may or may not be the identity provides another conceptual explanation for the occurrence of two distinct reflection equations in the Noumi–Sugitani–Dijkhuizen approach to quantum symmetric pairs.

1.4 Organization

Sections 2–5 are of preparatory nature. In Section 2 we fix notation for Kac–Moody algebras and quantized enveloping algebras, mostly following [15, 26, 13]. In Section 3 we discuss the completion 𝒰 of Uq⁢(𝔀) and the completion 𝒰0(2) of Uq⁢(𝔀)⊗Uq⁢(𝔀). In particular, we consider Lusztig’s braid group action and the commutativity isomorphisms R^ in this setting.

Section 4.1 is a review of the notion of a braided tensor category with a cylinder twist as introduced by tom Dieck and HÀring-Oldenburg. We extend their original definition by a twist in Section 4.2 to include all the examples obtained from quantum symmetric pairs later in the paper. The categorical definitions lead us in Section 4.3 to introduce the notion of a cylinder-braided coideal subalgebra of a braided bialgebra. By definition this is a coideal subalgebra which has a universal K-matrix. We carefully formulate the analog definition for coideal subalgebras of Uq⁢(𝔀) to take into account the need for completions. Finally, in Section 4.4 we recall the different definition of a universal K-matrix from [9] and indicate how it relates to cylinder braided coideal subalgebras as defined here.

Section 5 is a brief summary of the construction and properties of the quantum symmetric pair coideal subalgebras B𝐜,𝐬 in the conventions of [18]. In Section 5.3 we recall the existence of the bar involution for B𝐜,𝐬 following [2]. The quantum symmetric pair coideal subalgebra B𝐜,𝐬 depends on a choice of parameters, and the existence of the bar involution imposes additional restrictions. In Section 5.4 we summarize our setting, including all restrictions on the parameters 𝐜,𝐬.

The main new results of the paper are contained in Sections 6–9. In Section 6 we prove the existence of the quasi K-matrix 𝔛. The defining condition (1.2) gives rise to an overdetermined recursive formula for the weight components of 𝔛. The main difficulty is to prove the existence of elements satisfying the recursion. To this end, we translate the inductive step into a more easily verifiable condition in Section 6.2. This condition is expressed solely in terms of the constituents of the generators of B𝐜,𝐬, and it is verified in Section 6.4. This allows us to prove the existence of 𝔛 in Section 6.5. A similar argument is contained in [3, Section 2.4] for the special examples (𝔰⁢𝔩2⁢N,𝔰⁢(𝔀⁢𝔩N×𝔀⁢𝔩N)) and (𝔰⁢𝔩2⁢N+1,𝔰⁢(𝔀⁢𝔩N×𝔀⁢𝔩N+1)). However, the explicit formulation of the conditions in Proposition 6.3 seems to be new.

In Section 7 we consider the element 𝒊′∈𝒰 defined by (1.3). In Section 7.1 we define a twist of Uq⁢(𝔀) which reduces to the Lusztig action Tw0 if 𝔀 is of finite type. We also record an additional Assumption (τ0) on the parameters. In Section 7.2 this assumption is used in the proof that 𝒊M′:M→M is a B𝐜,𝐬-module isomorphism of twisted B𝐜,𝐬-modules. In the finite case this immediately implies that the element 𝒊 defined by (1.5) gives rise to an B𝐜,𝐬-module isomorphism (1.6). Up to a twist this verifies the first condition in the definition of a universal K-matrix for B𝐜,𝐬.

The map Ο involved in the definition of 𝒊′ is discussed in more detail in Section 8. So far, the element Ο was only required to satisfy a recursion which guarantees that 𝒊M′ is a B𝐜,𝐬-module homomorphism. In Section 8.1 we choose Ο explicitly and show that our choice satisfies the required recursion. In Section 8.2 we then determine the coproduct of this specific Ο considered as an element in the completion 𝒰. Moreover, in Section 8.3 we discuss the action of Ο on Uq⁢(𝔀) by conjugation. This simplifies later calculations.

In Section 9 we restrict to the finite case. We first perform some preliminary calculations with the quasi R-matrices of Uq⁢(𝔀) and Uq⁢(𝔀X). This allows us in Section 9.2 to determine the coproduct of the quasi K-matrix 𝔛, see Theorem 9.4. Combining the results from Sections 8 and 9 we calculate the coproduct Δ⁢(𝒊) and prove a τ⁢τ0-twisted version of formula (1.1) in Section 9.3. This shows that 𝒊 is a τ⁢τ0-universal K-matrix in the sense of Definition 4.12.

2 Preliminaries on quantum groups

In this section we fix notation and recall some standard results about quantum groups. We mostly follow the conventions in [26] and [13].

2.1 The root datum

Let I be a finite set and let A=(ai⁢j)i,j∈I be a symmetrizable generalized Cartan matrix. By definition there exists a diagonal matrix D=diag(ϵi∣i∈I) with coprime entries ϵi∈ℕ such that the matrix DA is symmetric. Let (𝔥,Π,Π√) be a minimal realization of A as in [15, Section 1.1]. Here Π={αi∣i∈I} and Π√={hi∣i∈I} denote the set of simple roots and the set of simple coroots, respectively. We write 𝔀=𝔀⁢(A) to denote the Kac–Moody Lie algebra corresponding to the realization (𝔥,Π,Π√) of A as defined in [15, Section 1.3].

Let Q=℀⁢Π be the root lattice and define Q+=ℕ0⁢Π. For λ,Ό∈𝔥∗ we write λ>ÎŒ if λ-Ό∈Q+∖{0}. For ÎŒ=∑imi⁢αi∈Q+ let ht⁢(ÎŒ)=∑imi denote the height of ÎŒ. For any i∈I the simple reflection σi∈GL⁢(𝔥∗) is defined by

σi⁢(α)=α-α⁢(hi)⁢αi.

The Weyl group W is the subgroup of GL⁢(𝔥∗) generated by the simple reflections σi for all i∈I. For simplicity set rA=|I|-rank⁢(A). Extend Π√ to a basis

Πext√=Π√∪{ds∣s=1,
,rA}

of 𝔥 and set Qext√=℀⁢Πext√. Assume additionally that αi⁢(ds)∈℀ for all i∈I, s=1,
,rA. By [15, Section 2.1] there exists a nondegenerate, symmetric, bilinear form (⋅,⋅) on 𝔥 such that

(hi,h)=αi⁢(h)ϵi for all ⁢h∈𝔥,i∈I,(dm,dn)=0 for all ⁢n,m∈{1,
,rA}.

Hence, under the resulting identification of 𝔥 and 𝔥∗ we have hi=αi/ϵi. The induced bilinear form on 𝔥∗ is also denoted by the bracket (⋅,⋅). It satisfies (αi,αj)=ϵi⁢ai⁢j for all i,j∈I. Define the weight lattice by

P={λ∈𝔥∗∣λ⁢(Qext√)⊆℀}.

Remark 2.1.

The abelian groups Y=Qext√ and X=P together with the embeddings I→Y, i↩hi and I→X, i↊αi form an X-regular and Y-regular root datum in the sense of [26, Section 2.2].

Define βi∈𝔥∗ by βi⁢(h)=(di,h), set

Πext=Π∪{βi∣i=1,
,rA},

and let Qext=℀⁢Πext. Then

P√={h∈𝔥∣Qext⁢(h)⊆℀}

is the coweight lattice. Let ϖi√ for i∈I denote the basis vector of P√ dual to αi. Let B denote the rA×|I|-matrix with entries αj⁢(di). Define an (rA+|I|)×(rA+|I|) matrix by

Aext=(AD-1⁢BtB0).

By construction, one has det⁡(Aext)≠0. The pairing (⋅,⋅) induces ℚ-valued pairings on P×P and P×P√. The above conventions lead to the following result.

Lemma 2.2.

The pairing (⋅,⋅) takes values in 1det⁡(Aext)⁢Z on P×P and on P×P√.

2.2 Quantized enveloping algebras

With the above notations we are ready to introduce the quantized enveloping algebra Uq⁢(𝔀). Let d∈ℕ be the smallest positive integer such that ddet⁡(Aext)∈℀. Let q1/d be an indeterminate and let 𝕂 a field of characteristic zero. We will work with the field 𝕂⁢(q1/d) of rational functions in q1/d with coefficients in 𝕂.

Remark 2.3.

The choice of ground field is dictated by two reasons. Firstly, by Lemma 2.2 it makes sense to consider q(λ,ÎŒ) as an element of 𝕂⁢(q1/d) for any weights λ,Ό∈P. This will allow us to define the commutativity isomorphism R^, see Example 3.4 and formula (3.8). Secondly, in the construction of the function Ο in Section 8 we will require factors of the form qλ⁢(ϖi√) for λ∈P and i∈I, see formula (8.1). Again, Lemma 2.2 shows that such factors lie in 𝕂⁢(q1/d).

Following [26, Section 3.1.1] the quantized enveloping algebra Uq⁢(𝔀) is the associative 𝕂⁢(q1/d)-algebra generated by elements Ei, Fi, Kh for all i∈I and h∈Qext√ satisfying the following defining relations:

  • (i)

    K0=1 and Kh⁢Kh′=Kh+h′ for all h,h′∈Qext√.

  • (ii)

    Kh⁢Ei=qαi⁢(h)⁢Ei⁢Kh for all i∈I, h∈Qext√.

  • (iii)

    Kh⁢Fi=q-αi⁢(h)⁢Fi⁢Kh for all i∈I, h∈Qext√.

  • (iv)

    Ei⁢Fj-Fj⁢Ei=ÎŽi⁢j⁢Ki-Ki-1qi-qi-1 for all i∈I where qi=qϵi and Ki=Kϵi⁢hi.

  • (v)

    the quantum Serre relations given in [26, Section 3.1.1 (e)].

We will use the notation qi=qϵi and Ki=Kϵi⁢hi all through this text. Moreover, for

ÎŒ=∑i∈Ini⁢αi∈Q

we will use the notation

KÎŒ=∏i∈IKini.

We make the quantum-Serre relations (v) more explicit. Let [1-ai⁢jn]qi denote the qi-binomial coefficient defined in [26, Section 1.3.3]. For any i,j∈I define a non-commutative polynomial Si⁢j in two variables by

Si⁢j⁢(x,y)=∑n=01-ai⁢j(-1)n⁢[1-ai⁢jn]qi⁢x1-ai⁢j-n⁢y⁢xn.

By [26, Section 33.1.5] the quantum Serre relations can be written in the form

Si⁢j⁢(Ei,Ej)=Si⁢j⁢(Fi,Fj)=0 for all i,j∈I.

The algebra Uq⁢(𝔀) is a Hopf algebra with coproduct Δ, counit ε, and antipode S given by

Δ⁢(Ei)=Ei⊗1+Ki⊗Ei,ε⁢(Ei)=0,S⁢(Ei)=-Ki-1⁢Ei,(2.1)Δ⁢(Fi)=Fi⊗Ki-1+1⊗Fi,ε⁢(Fi)=0,S⁢(Fi)=-Fi⁢Ki,Δ⁢(Kh)=Kh⊗Kh,ε⁢(Kh)=1,S⁢(Kh)=K-h

for all i∈I, h∈Qext√. We denote by Uq⁢(𝔀′) the Hopf subalgebra of Uq⁢(𝔀) generated by the elements Ei,Fi, and Ki±1 for all i∈I. Moreover, for any i∈I let Uqi⁢(𝔰⁢𝔩2)i be the subalgebra of Uq⁢(𝔀) generated by Ei,Fi,Ki and Ki-1. The Hopf algebra Uqi⁢(𝔰⁢𝔩2)i is isomorphic to Uqi⁢(𝔰⁢𝔩2) up to the choice of the ground field.

As usual we write U+, U-, and U0 to denote the 𝕂⁢(q1/d)-subalgebras of Uq⁢(𝔀) generated by {Ei∣i∈I}, {Fi∣i∈I}, and {Kh∣h∈Qext√}, respectively. We also use the notation U≥=U+⁢U0 and U≀=U-⁢U0 for the positive and negative Borel part of Uq⁢(𝔀). For any U0-module M and any λ∈P let

Mλ={m∈M|▷⁡Khm=qλ⁢(h)⁢m⁢ for all ⁢h∈Qext√}

denote the corresponding weight space. We can apply this notation in particular to U+, U-, and Uq⁢(𝔀) which are U0-modules with respect to the left adjoint action. We obtain algebra gradings

U+=⊕Ό∈Q+UÎŒ+,U-=⊕Ό∈Q+U-ÎŒ-,Uq⁢(𝔀)=⊕Ό∈QUq⁢(𝔀)ÎŒ.(2.2)

2.3 The bilinear pairing 〈⋅,⋅〉

Let k be any field, let A and B be k-algebras, and let 〈⋅,⋅〉:A×B→k be a bilinear pairing. Then 〈⋅,⋅〉 can be extended to A⊗n×B⊗n by setting

〈⊗iai,⊗ibi〉=∏i=1n〈ai,bi〉.

In the following we will use this convention for k=𝕂⁢(q1/d), A=U≀, B=U≥, and n=2 and 3 without further remark.

There exists a unique 𝕂⁢(q1/d)-bilinear pairing

〈⋅,⋅〉:U≀×U≥→𝕂⁢(q1/d)(2.3)

such that for all x,x′∈U≥, y,y′∈U≀, g,h∈Qext√, and i,j∈I the following relations hold

〈y,x⁢x′〉=〈Δ⁢(y),x′⊗x〉,〈y⁢y′,x〉=〈y⊗y′,Δ⁢(x)〉,(2.4)〈Kg,Kh〉=q-(g,h),〈Fi,Ej〉=ÎŽi⁢j⁢-1qi-qi-1,(2.5)〈Kh,Ei〉=0,〈Fi,Kh〉=0.(2.6)

Here we follow the conventions of [13, Section 6.12] in the finite case. In the Kac–Moody case the existence of the pairing 〈⋅,⋅〉 follows from the results in [26, Chapter 1]. Relations (2.4)–(2.6) imply that for all x∈U+,y∈U-, and g,h∈Qext√ one has

〈y⁢Kg,x⁢Kh〉=q-(g,h)⁢〈y,x〉.(2.7)

The pairing 〈⋅,⋅〉 respects weights in the following sense. For ÎŒ,Μ∈Q+ with Ό≠Μ the restriction of the pairing to U-Îœ-×UÎŒ+ vanishes identically. On the other hand, the restriction of the paring to U-ÎŒ-×UÎŒ+ is nondegenerate for all Ό∈Q+. The nondegeneracy of this restriction implies the following lemma, which we will need in the proof of Theorem 9.4.

Lemma 2.4.

Let X,X′∈∏Ό∈Q+U+⁢KΌ⊗UÎŒ+. If

〈y⊗z,X〉=〈y⊗z,X′〉 for all y,z∈U-,

then X=X′.

Proof.

We may assume that X′=0. Write X=∑Ό∈Q+XÎŒ, with

XÎŒ=∑iXÎŒ,i(1)⁢KΌ⊗XÎŒ,i(2)∈U+⁢KΌ⊗UÎŒ+.

Consider X~ÎŒ=∑iXÎŒ,i(1)⊗XÎŒ,i(2)∈U+⊗UÎŒ+. For any y∈U-, Ό∈Q+, and z∈UÎŒ- we then have

0=〈y⊗z,X〉=〈y⊗z,XΌ〉⁢=(2.7)⁢〈y⊗z,XΌ′〉.

By the nondegeneracy of the pairing on UÎŒ+×UÎŒ- it follows that XΌ′=0. Consequently, XÎŒ=0 for all Ό∈Q+, and hence X=0 as claimed. ∎

2.4 Lusztig’s skew derivations ri and ri

Let 𝐟′ be the free associative 𝕂⁢(q1/d)-algebra generated by elements 𝖿i for all i∈I. The algebra 𝐟′ is a U0-module algebra with

▷⁡Kh𝖿i=qαi⁢(h)⁢𝖿i.

As in (2.2) one obtains a Q+-grading

𝐟′=⊕Ό∈Q+𝐟Ό′.

The natural projection 𝐟′→U+, 𝖿i↩Ei respects the Q+-grading. There exist uniquely determined 𝕂⁢(q1/d)-linear maps ri,ri:𝐟′→𝐟′ such that

ri⁢(𝖿j)=ÎŽi⁢j,ri⁢(x⁢y)=q(αi,Îœ)⁢ri⁢(x)⁢y+x⁢ri⁢(y),(2.8)ri⁢(𝖿j)=ÎŽi⁢j,ri⁢(x⁢y)=ri⁢(x)⁢y+q(αi,ÎŒ)⁢x⁢ri⁢(y)(2.9)

for any x∈𝐟Ό′ and y∈𝐟Μ′. The above equations imply in particular that ri⁢(1)=0=ri⁢(1). By [26, Section 1.2.13] the maps ri and ri factor over U+, that is there exist linear maps ri,ri:U+→U+, denoted by the same symbols, which satisfy relations (2.8) and (2.9) for all x∈UÎŒ+, y∈UÎœ+ and with 𝖿j replaced by Ej. The maps ri and ri on U+ satisfy the following three properties, each of which is equivalent to the definition given above.

  • (1)

    For all x∈U+ and all i∈I one has

    [x,Fi]=1(qi-qi-1)⁢(ri⁢(x)⁢Ki-Ki-1⁢ri⁢(x)),(2.10)

    see [26, Proposition 3.1.6].

  • (2)

    For all x∈UÎŒ+ one has

    Δ⁢(x)=x⊗1+∑iri⁢(x)⁢Ki⊗Ei+(rest)1,(2.11)Δ⁢(x)=KΌ⊗x+∑iEi⁢KÎŒ-αi⊗ri⁢(x)+(rest)2

    where (rest)1∈∑α∉Π∪{0}UÎŒ-α+⁢Kα⊗Uα+ and (rest)2∈∑α∉Π∪{0}Uα+⁢KÎŒ-α⊗UÎŒ-α+, see [13, Section 6.14].

  • (3)

    For all x∈U+, y∈U-, and i∈I one has

    〈Fi⁢y,x〉=〈Fi,Ei〉⁢〈y,ri⁢(x)〉,〈y⁢Fi,x〉=〈Fi,Ei〉⁢〈y,ri⁢(x)〉(2.12)

    see [26, Section 1.2.13]

Property (3) and the original definition of ri and ri as skew derivations are useful in inductive arguments. Properties (1) and (2), on the other hand, carry information about the algebra and the coalgebra structure of Uq⁢(𝔀), respectively.

Property (3) above and the nondegeneracy of the pairing 〈⋅,⋅〉 imply that for any x∈UÎŒ+ with Ό∈Q+∖{0} one has

x=0⇔ri⁢(x)=0⁢ for all ⁢i∈I⇔ri⁢(x)=0⁢ for all ⁢i∈I,(2.13)

see also [26, Lemma 1.2.15]. Moreover, property (2) and the coassociativity of the coproduct imply that for any i,j∈I one has

ri∘rj=rj∘ri,(2.14)

see [13, Lemma 10.1]. Note that this includes the case i=j.

Similarly to the situation for the algebra U+, the maps ri,ri:𝐟′→𝐟′ also factor over the canonical projection 𝐟′→U-, 𝖿i→Fi which maps 𝐟Ό′ to U-ÎŒ- for all Ό∈Q+. The maps ri,ri:U-→U- satisfy (2.8) and (2.9) for all x∈U-ÎŒ-, y∈U-Îœ- with 𝖿j replaced by Fj. Moreover, the maps ri,ri:U-→U- can be equivalently described by analogs of properties (1)–(3) above. For example, in analogy to (3) one has

〈y,Ei⁢x〉=〈Fi,Ei〉⁢〈ri⁢(y),x〉,〈y,x⁢Ei〉=〈Fi,Ei〉⁢〈ri⁢(y),x〉(2.15)

for all x∈U-, y∈U-, and i∈I.

As in [26, Section 3.1.3] let σ:Uq⁢(𝔀)→Uq⁢(𝔀) denote the 𝕂⁢(q1/d)-algebra antiautomorphism determined by

σ⁢(Ei)=Ei,σ⁢(Fi)=Fi,σ⁢(Kh)=K-h for all i∈I, h∈Qext√.(2.16)

The map σ intertwines the skew derivations ri and ri as follows:

σ∘ri=ri∘σ for all i∈I.(2.17)

Recall that the bar involution on Uq⁢(𝔀) is the 𝕂-algebra automorphism

¯:Uq⁢(𝔀)→Uq⁢(𝔀),x↩x¯

defined by

q1/d¯=q-1/d,Ei¯=Ei,Fi¯=Fi,Kh¯=K-h(2.18)

for all i∈I, h∈Qext√. The bar involution on Uq⁢(𝔀) also intertwines the skew derivations ri and ri in the sense that

ri⁢(x¯)=q(αi,ÎŒ-αi)⁢ri⁢(x)¯ for all x∈UÎŒ+, Ό∈Q+,(2.19)

see [26, Lemma 1.2.14].

3 The completion 𝒰 of Uq⁢(𝔀)

It is natural to consider completions of the infinite-dimensional algebra Uq⁢(𝔀) and related algebras. The quasi R-matrix for Uq⁢(𝔀), for example, lies in a completion of U-⊗U+, and the universal R-matrix lies in a completion of U≀⊗U≥, see Section 3.3. Similarly, the universal K-matrix we construct in this paper lies in a completion 𝒰 of Uq⁢(𝔀). This completion is commonly used in the literature, see for example [32, Section 1.3]. Here, for the convenience of the reader, we recall the construction and properties of the completion 𝒰 in quite some detail. This allows us to introduce further concepts, such as the Lusztig automorphisms, as elements of 𝒰. It also provides a more conceptual view on the quasi R-matrix and the commutativity isomorphisms.

3.1 The algebra 𝒰

Let 𝒪int denote the category of integrable Uq⁢(𝔀)-modules in category 𝒪. Recall that category 𝒪 consists of Uq⁢(𝔀)-modules M which decompose into finite-dimensional weight spaces M=⊕λ∈PMλ and on which the action of U+ is locally finite. Moreover, the weights of M are contained in a finite union ⋃i(λi-ℕ0⁢Π) for some λi∈P. Objects in 𝒪int are additionally locally finite with respect to the action of Uqi⁢(𝔰⁢𝔩2)i for all i∈I. Simple objects in 𝒪int are irreducible highest weight modules with dominant integral highest weight [26, Corollary 6.2.3]. If 𝔀 is of finite type, then 𝒪int is the category of finite-dimensional type 1 representations.

Let 𝒱⁢𝑒𝑐𝑡 be the category of vector spaces over 𝕂⁢(q1/d). Both 𝒱⁢𝑒𝑐𝑡 and 𝒪int are tensor categories, and the forgetful functor

ℱ⁢𝑜𝑟:𝒪int→𝒱⁢𝑒𝑐𝑡

is a tensor functor. Let 𝒰=End⁢(ℱ⁢𝑜𝑟) be the set of natural transformations from ℱ⁢𝑜𝑟 to itself. The category 𝒪int is equivalent to a small category and hence 𝒰 is indeed a set. More explicitly, elements of 𝒰 are families (φM)M∈Ob⁢(𝒪int) of vector space endomorphisms

φM:ℱ⁢𝑜𝑟⁢(M)→ℱ⁢𝑜𝑟⁢(M)

such that the diagram

commutes for any Uq⁢(𝔀)-module homomorphism ψ:M→N. Natural transformations of ℱ⁢𝑜𝑟 can be added and multiplied by a scalar, both operations coming from the linear structure on vector spaces. Composition of natural transformations gives a multiplication on 𝒰 which turns 𝒰 into a 𝕂⁢(q1/d)-algebra.

Example 3.1.

The action of Uq⁢(𝔀) on objects of 𝒪int gives an algebra homomorphism Uq⁢(𝔀)→𝒰 which is injective, see [26, Proposition 3.5.4] and [13, Section 5.11]. We always consider Uq⁢(𝔀) a subalgebra of 𝒰.

Example 3.2.

Let U+^=∏Ό∈Q+UÎŒ+ and let (XÎŒ)Ό∈Q+∈U+^. Let M∈Ob⁢(𝒪int) and m∈M. As the action of U+ on M is locally finite there exist only finitely many Ό∈Q+ such that XΌ⁢m≠0. Hence the expression

∑Ό∈Q+XΌ⁢m(3.1)

is well defined. In this way the element (XÎŒ)Ό∈Q+∈U+^ defines an endomorphism of ℱ⁢𝑜𝑟, and we may thus consider U+^ as a subalgebra of 𝒰. We sometimes write elements of U+^ additively as

X=∑Ό∈Q+XÎŒ.

In view of (3.1) this is compatible with the inclusions U+⊆U+^⊆𝒰.

Example 3.3.

Let Ο:P→𝕂⁢(q1/d) be any map. For M∈Ob⁢(𝒪int) define a linear map ΟM:M→M by ΟM⁢(m)=Ο⁢(λ)⁢m for all m∈Mλ. Then the family (ΟM)M∈Ob⁢(𝒪int) is an element in 𝒰. By slight abuse of notation we denote this element by Ο as well.

Lusztig showed that 𝒰 contains a homomorphic image of the braid group corresponding to W. For any M∈Ob⁢(𝒪int) and any i∈I the Lusztig automorphism TiM:M→M is defined on m∈Mλ with λ∈P by

TiM⁢(m)=∑a,b,c≥0a-b+c=λ⁢(hi)(-1)b⁢qib-a⁢c⁢Ei(a)⁢Fi(b)⁢Ei(c)⁢m.(3.2)

The family Ti=(TiM) defines an element in 𝒰. By [26, Proposition 5.2.3] the elements Ti of 𝒰 are invertible with inverse Ti-1=(TiM-1) given by

TiM-1⁢(m)=∑a,b,c≥0a-b+c=λ⁢(hi)(-1)b⁢qia⁢c-b⁢Fi(a)⁢Ei(b)⁢Fi(c)⁢m.

By [26, Section 39.43] the elements Ti for i∈I satisfy the braid relations

Ti⁢Tj⁢Ti⁢ ⏟mi⁢j factors=Tj⁢Ti⁢Tj⁢ ⏟mi⁢j factors

where mi⁢j denotes the order of σi⁢σj∈W. Hence, for any w∈W there is a well-defined element Tw∈𝒰 given by

Tw=Ti1⁢Ti2⁢ ⁢Tik

if w=σi1⁢σi2⁢ ⁢σik is a reduced expression.

We also use the symbol Ti for i∈I to denote the corresponding algebra automorphism of Uq⁢(𝔀) denoted by Ti,1′′ in [26, Section 37.1]. This is consistent with the above notation, in the sense that for any u∈Uq⁢(𝔀), any M∈Ob⁢(𝒪int), and any m∈M we have

Ti⁢M⁢(u⁢m)=Ti⁢(u)⁢Ti⁢M⁢(m).

Hence Ti, as an automorphism of Uq⁢(𝔀), is nothing but conjugation by the invertible element Ti∈𝒰. In this way we obtain automorphisms Tw of Uq⁢(𝔀) for all w∈W.

Furthermore, the bar involution for Uq⁢(𝔀) intertwines Ti and Ti-1. More explicitly, for u∈UÎŒ+ one has

Ti⁢(u)=(-1)Ό⁢(hi)⁢q(Ό,αi)⁢Ti-1⁢(u¯)¯,

see [26, Section 37.2.4]

3.2 The coproduct on 𝒰

To define a coproduct on 𝒰 consider the functor

ℱ⁢𝑜𝑟(2):𝒪int×𝒪int→𝒱⁢𝑒𝑐𝑡,(M,N)↊ℱ⁢𝑜𝑟⁢(M⊗N)=ℱ⁢𝑜𝑟⁢(M)⊗ℱ⁢𝑜𝑟⁢(N),(f,g)↊ℱ⁢𝑜𝑟⁢(f⊗g).

Let 𝒰0(2)=End⁢(ℱ⁢𝑜𝑟(2)) denote the set of natural transformations from ℱ⁢𝑜𝑟(2) to itself. Again, 𝒰0(2) is an algebra for which the multiplication ⋅ is given by composition of natural transformations. The map

i(2):𝒰⊗𝒰→𝒰0(2),(φM)⊗(ψN)↩(φM⊗ψN)

is an injective algebra homomorphism. However, it is not surjective, as the following example shows.

Example 3.4.

For M,N∈Ob⁢(𝒪int) define a linear map

κM,N:M⊗N→M⊗N,m⊗n↩q(ÎŒ,Îœ)⁢m⊗n if m∈MΌ and n∈NÎœ.

The collection κ=(κM,N)M,N∈Ob⁢(𝒪int) lies in 𝒰0(2). However, one can show that κ is not of the form ∑k=1nfi⊗gi for any n∈ℕ and any collection fi,gi∈𝒰. Hence κ does not lie in the image of the map i(2) described above.

The element κ is an important building block of the universal R-matrix for Uq⁢(𝔀), see Section 3.3. For κ to be well defined the ground field needs to contain q(ÎŒ,Îœ) for all ÎŒ,Μ∈P. This gives one of the reasons why we work over the field 𝕂⁢(q1/d).

Any natural transformation φ∈𝒰 can be restricted to all ℱ⁢𝑜𝑟⁢(M⊗N), M,N∈Ob⁢(𝒪int). Moreover, restriction is compatible with composition and linear combinations of natural transformations. Hence we obtain an algebra homomorphism

Δ𝒰:𝒰→𝒰0(2),(φM)M∈Ob⁢(𝒪int)↩(φM⊗N)M,N∈Ob⁢(𝒪int).

We call Δ𝒰 the coproduct of 𝒰. The restriction of Δ𝒰 to Uq⁢(𝔀) coincides with the coproduct of Uq⁢(𝔀) from Section 2.2. For this reason we will drop the subscript 𝒰 and just denote the coproduct on 𝒰 by Δ.

We would also like to consider families of linear maps flipping the two tensor factors by a similar formalism. To that end consider the functor

ℱ⁢𝑜𝑟(2)⁢op:𝒪int×𝒪int→𝒱⁢𝑒𝑐𝑡,(M,N)↊ℱ⁢𝑜𝑟⁢(N⊗M)=ℱ⁢𝑜𝑟⁢(N)⊗ℱ⁢𝑜𝑟⁢(M),(f,g)↊ℱ⁢𝑜𝑟⁢(g⊗f).

Define 𝒰1(2)=Hom⁢(ℱ⁢𝑜𝑟(2),ℱ⁢𝑜𝑟(2)⁢op). For M,N∈Ob⁢(𝒪int) let

flipM,N:M⊗N→N⊗M,m⊗n↩n⊗m

denote the flip of tensor factors. Then flip=(flipM,N)M,N∈Ob⁢(𝒪int) is an element of 𝒰1(2). The direct sum

𝒰(2)=𝒰0(2)⊕𝒰1(2)

is a â„€2-graded algebra where multiplication ⋅ is given by composition of natural transformations. This is the natural algebra for the definition of the commutativity isomorphisms in the next subsection.

3.3 Quasi R-matrix and commutativity isomorphisms

Let Ό∈Q+ and let {bÎŒ,i} be a basis of U-ÎŒ-. Let {bÎŒi} be the dual basis of UÎŒ+ with respect to the pairing (2.3). Define

RÎŒ=∑ibÎŒ,i⊗bÎŒi∈U-⊗U+.

The element RΌ is independent of the chosen basis {bΌ,i}. The quasi R-matrix

R=∑Ό∈Q+RÎŒ

gives a well-defined element 𝒰0(2). Indeed, for M,N∈Ob⁢(𝒪int) only finitely many summands RÎŒ act nontrivially on any element of M⊗N.

Remark 3.5.

The element R∈𝒰0(2) coincides with the quasi-R-matrix defined in [26, Section 4.1.4] and in [13, Section 7.2] in the finite case. Those references use the symbol Θ for the quasi-R-matrix, but we change notation to avoid confusion with the involutive automorphism Θ:𝔥∗→𝔥∗ defined in Section 5.1.

The quasi R-matrix has a second characterization in terms of the bar involution (2.18) of Uq⁢(𝔀). Define a bar involution ¯ on Uq⁢(𝔀)⊗Uq⁢(𝔀) by

u⊗v¯=u¯⊗v¯.

By [26, Theorem 4.1.2] the quasi-R-matrix is the uniquely determined element

R=∑Ό∈Q+RΌ∈∏Ό∈Q+U-ÎŒ-⊗UÎŒ+

with RΌ∈U-ÎŒ-⊗UÎŒ+ and R0=1⊗1 for which

Δ⁢(u¯)⁢R=R⁢Δ⁢(u)¯ for all ⁢u∈Uq⁢(𝔀).(3.3)

Moreover, R is invertible, with

R-1=R¯.(3.4)

Remark 3.6.

If 𝔀 is of finite type, then the quasi-R-matrix R can be factorized into a product of R-matrices for 𝔰⁢𝔩2. Choose a reduced expression w0=σi1⁢ ⁢σit for the longest element w0 of W. For j=1,
,t set γj=σi1⁢σi2⁢ ⁢σij-1⁢(αij) and define

Eγj=Ti1⁢Ti2⁢ ⁢Tij-1⁢(Eij),Fγj=Ti1⁢Ti2⁢ ⁢Tij-1⁢(Fij).(3.5)

Then {γ1,
,γt} is the set of positive roots of 𝔀, and (3.5) are the root vectors used in the construction of the PBW basis corresponding to the chosen reduced expression for w0. For j=1,
,t define

R[j]=∑r≥0(-1)r⁢qij-r⁢(r-1)/2⁢(qij-qij-1)r[r]qij!⁢Fγjr⊗Eγjr(3.6)

and for i∈I set Ri=R[j] if γj=αi. By [13, Remark 8.29] one has

R=R[t]⋅R[t-1]⋅
⋅R[2]⋅R[1].(3.7)

The quasi-R-matrix R and the transformation κ defined in Example 3.4 give rise to a family of commutativity isomorphisms. Define

R^=R⋅κ-1⋅flip(3.8)

in 𝒰(2). By [26, Theorem 32.1.5] the maps

R^M,N:M⊗N→N⊗M(3.9)

are isomorphisms of Uq⁢(𝔀)-modules for all M,N∈Ob⁢(𝒪int). Moreover, the isomorphisms R^M,N satisfy the hexagon property

R^M,N⊗N′=(idN⊗R^M,N′)⁢(R^M,N⊗idN′),R^M⊗M′,N=(R^M,N⊗idM′)⁢(idM⊗R^M′,N′)

for all M,M′,N,N′∈Ob⁢(𝒪int), see [26, Section 32.2]. This implies that 𝒪int is a braided tensor category as defined for example in [17, Section XIII.1.1].

Remark 3.7.

In the construction ([26, Chapter 32]) of the commutativity isomorphisms R^M,N it is assumed that 𝔀 is of finite type. Moreover, Lusztig defines the commutativity isomorphisms on tensor products of integrable weight modules. Lusztig’s arguments extend to the Kac–Moody case if one restricts to category 𝒪. We retain the assumption of integrability so that the Lusztig automorphisms TiM given by (3.2) are well defined. The restrictions imposed by R^M,N and TiM force us to work with the category 𝒪int.

It follows from the definitions of the completion 𝒰 and the coproduct Δ:𝒰→𝒰(2) that in 𝒰(2) one has

R^⋅Δ⁢(u)=Δ⁢(u)⋅R^  for all ⁢u∈𝒰.(3.10)

In the proof of the next lemma we will use this property for u=Ti. Moreover, by [26, Proposition 5.3.4] the Lusztig automorphisms Ti∈𝒰 satisfy

Δ⁢(Ti)=Ti⊗Ti⋅Ri-1(3.11)

where Ri was defined just below (3.6). To generalize the above formula we recall the following well-known lemma, see for example [5, Proposition 8.3.11]. We include a proof to assure that we have the correct formula in our conventions. Recall that for 𝔀 of finite type w0∈W denotes the longest element. Define, moreover, R21=flip⋅R⋅flip∈𝒰0(2).

Lemma 3.8.

Assume that g is of finite type. Then the relations

Δ⁢(Tw0)=Tw0⊗Tw0⋅R-1,(3.12)Δ⁢(Tw0-1)=Tw0-1⊗Tw0-1⋅κ⋅R21⋅κ-1(3.13)

hold in U0(2).

Proof.

First observe that equations (3.12) and (3.13) are equivalent. Indeed, equation (3.10) for u=Tw0 implies that equation (3.12) is equivalent to

R^⋅Δ⁢(Tw0)=κ-1⋅flip⋅Tw0⊗Tw0.

The inverse of the above equation is (3.13). It remains to verify that (3.13) holds. Applying the equivalence between (3.12) and (3.13) to (3.11) one obtains

Δ⁢(Ti-1)=Ti-1⊗Ti-1⋅κ⋅Ri21⋅κ-1

where Ri21=flip⋅Ri⋅flip. Hence for Tw0=Ti1⁢Ti2⁢ ⁢Tit one has

Δ⁢(Tw0-1)=Δ⁢(Tit-1)⁢⋯⁢Δ⁢(Ti2-1)⋅Δ⁢(Ti1-1)=Tit-1⊗Tit-1⋅κ⋅Rit21⋅κ-1⁢⋯⁢Ti1-1⊗Ti1-1⋅κ⋅Ri121⋅κ-1=Tw0-1⊗Tw0-1⋅κ⋅R21[t]⋅R21[t-1]⁢⋯⁢R21[2]⋅R21[1]⋅κ-1

where R21[j]=flip⋅R[j]⋅flip. By (3.7) one obtains relation (3.13). ∎

4 Braided tensor categories with a cylinder twist

As explained in Section 3.3 the commutativity isomorphisms (3.9) turn 𝒪int into a braided tensor category. For any V∈Ob⁢(𝒪int) there exists a graphical calculus for the action of R^ on V⊗n in terms of braids in ℂ×[0,1], see [17, Corollary XIII.3.8]. If 𝔀 is finite dimensional, then 𝒪int has a duality in the sense of [17, Section XIV.2] and there exists a ribbon element which turns 𝒪int into a ribbon category as defined in [17, Section XIV.3.2]. The graphical calculus extends to ribbon categories, see [17, Theorem XIV 5.1] also for original references.

In [34] T. tom Dieck outlined a program to extend the graphical calculus to braids or ribbons in the cylinder ℂ∗×[0,1]. The underlying braid group corresponds to a Coxeter group of type B. In the papers [34, 35, 12] tom Dieck and R. HÀring-Oldenburg elaborated a categorical setting for such a graphical calculus, leading to the notion of tensor categories with a cylinder braiding. In the present section we recall this notion. In Section 4.2 we will also give a slight generalization which captures all the examples which we obtain from quantum symmetric pairs in Section 9.3. These examples are determined by a coideal subalgebra of the braided Hopf algebra Uq⁢(𝔀). Cylinder braiding in this setting naturally leads to the notion of a cylinder-braided coideal subalgebra of a braided bialgebra H which we introduce in Section 4.3. The key point is that a cylinder-braided coideal subalgebra of H has a universal K-matrix which provides solutions of the reflection equation in all representations of H.

4.1 Cylinder twists and the reflection equation

To define cylinder twists, let (𝒜,⊗,I,a,l,r) be a tensor category as defined in [17, Definition XI.2.1]. Let ℬ be another category and assume that there exists a functor ∗:ℬ×𝒜→ℬ which we write as

(M,N)↩M∗N,(f,g)↩f∗g

on objects M∈Ob⁢(ℬ),N∈Ob⁢(𝒜) and morphisms f,g in ℬ and 𝒜, respectively. The functor ∗ is called a right action of 𝒜 on ℬ if there exist natural isomorphisms α and ρ with

αM,N,N′:(M∗N)∗N′→M∗(N⊗N′)for M∈Ob⁢(ℬ), N,N′∈Ob⁢(𝒜),ρM:M∗I→Mfor M∈Ob⁢(ℬ)

which satisfy the pentagon and the triangle axiom given in [34, (2.1), (2.2)]. A category ℬ together with a right action of 𝒜 on ℬ is called a right 𝒜-module category.

Example 4.1.

As seen in Section 3.3, the category 𝒜=𝒪int is a braided tensor category. Let B⊆Uq⁢(𝔀) be a right coideal subalgebra, that is a subalgebra satisfying

Δ⁢(B)⊆B⊗Uq⁢(𝔀).

Let ℬ be the category with Ob⁢(ℬ)=Ob⁢(𝒪int) and Homℬ⁢(M,N)=HomB⁢(M,N) for all M,N∈Ob⁢(ℬ). Then ℬ is a right 𝒜-module category with ∗ given by M∗N=M⊗N.

From now on, following [34], we will consider the following data:

  • (1)

    (𝒜,⊗,I,a,l,r,c) is a braided tensor category with braiding cM,N:M⊗N→N⊗M for all M,N∈Ob⁢(𝒜).

  • (2)

    (ℬ,∗,α,ρ) is a right 𝒜-module category.

  • (3)

    𝒜 is a subcategory of ℬ with Ob⁢(𝒜)=Ob⁢(ℬ). In other words, Hom𝒜⁢(M,N) is a subset of Homℬ⁢(M,N) for all M,N∈Ob⁢(𝒜)=Ob⁢(ℬ).

  • (4)

    ∗, α, ρ restrict to ⊗, a, r on 𝒜×𝒜.

We call (ℬ,𝒜) a tensor pair if the above conditions (1)–(4) are satisfied. By condition (3) there exists a forgetful functor

ℱ⁢𝑜𝑟ℬ𝒜:𝒜→ℬ.

Definition 4.2.

Let (ℬ,𝒜) be a tensor pair. A natural transformation

t=(tM)M∈Ob⁢(𝒜):ℱ⁢𝑜𝑟ℬ𝒜→ℱ⁢𝑜𝑟ℬ𝒜

is called a ℬ-endomorphism of 𝒜. If tM:ℱ⁢𝑜𝑟ℬ𝒜⁢(M)→ℱ⁢𝑜𝑟ℬ𝒜⁢(M) is an automorphism for all M∈Ob⁢(𝒜), then t is called a ℬ-automorphisms of 𝒜.

In other words, a ℬ-endomorphism of 𝒜 is a family t=(tM)M∈Ob⁢(𝒜) of morphisms tM∈Homℬ⁢(M,M) such that

tN∘f=f∘tM(4.1)

for all f∈Hom𝒜⁢(M,N).

Example 4.3.

The pair (ℬ,𝒜) from Example 4.1 is a tensor pair. In this setting a ℬ-endomorphism of 𝒪int is an element t∈𝒰 which commutes with all elements of the coideal subalgebra B⊂𝒰. In other words, the maps tM:M→M are B-module homomorphisms for all M∈Ob⁢(𝒪int).

The following definition provides the main structure investigated by tom Dieck and HÀring-Oldenburg in [34, 35, 12].

Definition 4.4 ([34]).

Let (ℬ,𝒜) be a tensor pair. A cylinder twist for (ℬ,𝒜) consists of a ℬ-automorphism t=(tM)M∈Ob⁢(𝒜) of 𝒜 such that

tM⊗N=(tM∗1N)⁢cN,M⁢(tN∗1M)⁢cM,N(4.2)

for all M,N∈Ob⁢(𝒜)=Ob⁢(ℬ).

The definition of a cylinder twist in [34] involves a second equation. This equation, however, is a consequence of (4.2). This was already observed in [34, Proposition 2.10].

Proposition 4.5.

Let (B,A) be a tensor pair with a cylinder twist (tM)M∈Ob⁢(A). Then the relation

(tM∗1N)⁢cN,M⁢(tN∗1M)⁢cM,N=cN,M⁢(tN∗1M)⁢cM,N⁢(tM∗1N)(4.3)

holds for all M,N∈Ob⁢(A).

Proof.

As cN,M is a morphism in 𝒜, relation (4.1) implies that

tM⊗N∘cN,M=cN,M∘tN⊗M.

If one inserts relation (4.2) into both sides of the above equation, one obtains equation (4.3). ∎

In [35] equation (4.3) is called the four-braid relation. Here we follow the mathematical physics literature [21] and call (4.3) the reflection equation. Equation (4.2) is know as the fusion procedure, see [21, Section 6.1], as it allows us to fuse the two solutions tM and tN of the reflection equation for M and N, respectively, to a new solution tM⊗N for the tensor product M⊗N.

4.2 Twisted cylinder twists

Let (ℬ,𝒜) be a tensor pair. To cover the examples considered in the present paper in full generality, we introduce a slight generalization of tom Dieck’s notion of a cylinder twist for (ℬ,𝒜). This generalization involves a second twist which suggests the slightly repetitive terminology.

Let t⁢w:𝒜→𝒜 be braided tensor equivalence given by

M↩Mt⁢w,f↩ft⁢w∈Hom⁢(Mt⁢w,Nt⁢w) for all M,N∈Ob⁢(𝒜), f∈Hom⁢(M,N).

This means that t⁢w is a braided tensor functor as defined in [17, Definition XIII.3.6] and an equivalence of categories. A family t=(tM)M∈Ob⁢(𝒜) of morphisms tM∈Homℬ⁢(M,Mt⁢w) is called a ℬ-t⁢w-endomorphism of 𝒜 if

tN∘f=ft⁢w∘tM(4.4)

for all f∈Hom𝒜⁢(M,N). In other words, a ℬ-t⁢w-endomorphism of 𝒜 is a natural transformation t:ℱ⁢𝑜𝑟𝒜ℬ→ℱ⁢𝑜𝑟𝒜ℬ∘t⁢w.

Definition 4.6.

Let (ℬ,𝒜) be a tensor pair and t⁢w:𝒜→𝒜 a braided tensor equivalence. A t⁢w-cylinder twist for (ℬ,𝒜) consists of a ℬ-t⁢w-automorphism t=(tM)M∈Ob⁢(𝒜) of 𝒜 such that

tM⊗N=(tM∗1Nt⁢w)⁢cNt⁢w,M⁢(tN∗1M)⁢cM,N(4.5)

for all M,N∈Ob⁢(𝒜)=Ob⁢(ℬ).

Let (ℬ,𝒜) be a tensor pair with a t⁢w-cylinder twist. The relation cN,Mt⁢w=cNt⁢w,Mt⁢w and (4.4) imply that

tM⊗N∘cN,M=cNt⁢w,Mt⁢w∘tN⊗M.

As in the proof of Proposition 4.5 one now obtains

(tM∗1Nt⁢w)⁢cNt⁢w,M⁢(tN∗1M)⁢cM,N=cNt⁢w,Mt⁢w⁢(tN∗1Mt⁢w)⁢cMt⁢w,N⁢(tM∗1N).(4.6)

Example 4.7.

Consider the setting of Example 4.1. Let φ:Uq⁢(𝔀)→Uq⁢(𝔀) be a Hopf algebra automorphism. For any M∈Ob⁢(𝒪int) let Mφ be the integrable representation with left action ∙φ given by u∙φm=φ⁢(u)⁢m for all u∈Uq⁢(𝔀), m∈M. By [36, Theorem 2.1] one has φ⁢(U+)=U+ and φ⁢(U0)=U0 and hence Mφ∈Ob⁢(𝒪int). Moreover, as φ⁢(U0)=U0 the map φ induces a group isomorphism φP:P→P. We assume additionally that φP is an isometry, that is (φP⁢(λ),φP⁢(ÎŒ))=(λ,ÎŒ) for all λ,Ό∈P. Then one obtains an auto-equivalence of braided tensor categories

t⁢w:𝒪int→𝒪int

given by t⁢w⁢(M)=Mφ and t⁢w⁢(f)=f. In this case relations (4.5) and (4.6) become

tM⊗N=(tM⊗1)⁢R^Nφ,M⁢(tN⊗1)⁢R^M,N,(4.7)(tM⊗1)⁢R^Nφ,M⁢(tN⊗1)⁢R^M,N=R^Nφ,Mφ⁢(tN⊗1)⁢R^Mφ,N⁢(tM⊗1),(4.8)

respectively, for any M,N∈Ob⁢(𝒪int).

4.3 Cylinder-braided coideal subalgebras and the universal K-matrix

We can formalize Examples 4.1, 4.3, and 4.7 in the setting of bialgebras and their coideal subalgebras. For the convenience of the reader we recall the relevant notions in the setting of the present paper.

Definition 4.8 ([17, Definition VIII.2.2]).

A bialgebra H with coproduct

ΔH:H→H⊗H

is called braided (or quasitriangular) if there exists an invertible element RH∈H⊗H such that the following two properties hold:

  • (1)

    For all x∈H one has

    ΔHop⁢(x)=(RH)-1⁢ΔH⁢(x)⁢RH(4.9)

    where ΔHop=flip∘ΔH:H→H⊗H denotes the opposite coproduct.

  • (2)

    The element R satisfies the relations

    (ΔH⊗idH)⁢(RH)=R23H⁢R13H,(idH⊗ΔH)⁢(RH)=R12H⁢R13H(4.10)

    where we use the usual leg-notation.

In this case the element RH is called a universal R-matrix for H.

Let H be a braided bialgebra with universal R-matrix RH=∑isi⊗ti∈H⊗H. In this situation the category 𝒜=H-mod of H-modules is a braided tensor category with braiding

cM,NH:M⊗N→N⊗M,m⊗n↩∑isi⁢n⊗ti⁢m(4.11)

for all M,N∈Ob⁢(𝒜), see [17, Section VIII.3].

Remark 4.9.

The conventions in Definition 4.8 slightly differ from the conventions in [17]. The reason for this is that following [26] we use the braiding R⋅κ-1⋅flip for 𝒪int and hence the braiding RH∘flip for H-mod. To match conventions observe that RH in Definition 4.8 coincides with R21 in [17, Definition VIII.2.2].

Let B be a right coideal subalgebra of H. As in Example 4.3 define ℬ to be the category with Ob⁢(ℬ)=Ob⁢(𝒜) and Homℬ⁢(M,N)=HomB⁢(M,N) for all M,N∈Ob⁢(𝒜). Then (ℬ,𝒜) is a tensor pair. For any bialgebra automorphism φ:H→H define

RH,φ=(id⊗φ)⁢(RH).

In analogy to the notion of a universal R-matrix the following definition is natural.

Definition 4.10.

Let H be a braided bialgebra with universal R-matrix RH∈H⊗H and let φ:H→H be an automorphism of braided bialgebras. We say that a right coideal subalgebra B of H is φ-cylinder-braided if there exists an invertible element 𝒊∈H such that

𝒊⁢b=φ⁢(b)⁢𝒊 for all b∈B,(4.12)Δ⁢(𝒊)=(𝒊⊗1)⁢RH,φ⁢(1⊗𝒊)⁢R21H.(4.13)

In this case we call 𝒊 a φ-universal K-matrix for the coideal subalgebra B. If φ=idH, then we simply say that B is cylinder-braided and that 𝒊 is a universal K-matrix for B.

The bialgebra automorphism φ defines a braided tensor equivalence tw:𝒜→𝒜 given by M↩Mφ where as before Mφ denotes the H-module which coincides with M as a vector space and has the left action h⊗m↊φ⁢(h)⁢m. In the above setting a φ-universal K-matrix for the coideal subalgebra B defines a family of maps

tM:M→M,m↊𝒊⁢m,for all M∈Ob⁢(𝒜).(4.14)

By construction the natural transformation t=(tM)M∈Ob⁢(𝒜) is a tw-cylinder twist for the tensor pair (ℬ,𝒜).

Remark 4.11.

Observe the parallel between Definition 4.8 and Definition 4.10 in the case φ=idH. Indeed, condition (4.9) means that the maps cM,NH defined by (4.11) are H-module homomorphisms while condition (4.12) means that the maps tM defined by (4.14) are B-module homomorphisms if φ=idH. Similarly, conditions (4.10) and (4.13) both express compatibility with the tensor product.

Definition 4.10 can be extended to include the quantized universal enveloping algebra Uq⁢(𝔀) which is braided only in the completion. In this case we also need to allow for 𝒊 to lie in the completion 𝒰. We repeat Definition 4.10 in this setting for later reference. Recall the notation from Section 3 and from Example 4.7. For any Hopf algebra automorphisms φ:Uq⁢(𝔀)→Uq⁢(𝔀) define an element R^φ∈𝒰(2) by

(R^φ)M,N=R^Mφ,N for all M,N∈Ob⁢(𝒪int).

In the following definition we reformulate condition (4.13) in terms of R^ and R^φ.

Definition 4.12.

Let φ:Uq⁢(𝔀)→Uq⁢(𝔀) be a Hopf algebra automorphism. A right coideal subalgebra B⊆Uq⁢(𝔀) is called φ-cylinder-braided if there exists an invertible element 𝒊∈𝒰 such that the relation

𝒊⁢b=φ⁢(b)⁢𝒊 for all b∈B(4.15)

holds in 𝒰 and the relation

Δ⁢(𝒊)=(𝒊⊗1)⋅R^φ⋅(𝒊⊗1)⋅R^(4.16)

holds in 𝒰(2). In this case we call 𝒊 a φ-universal K-matrix for the coideal subalgebra B. If φ=idUq⁢(𝔀), then we simply say that B is cylinder-braided and that 𝒊 is a universal K-matrix for B.

Similarly to the discussion for the bialgebra H above, a cylinder-braided coideal subalgebra of Uq⁢(𝔀) naturally gives rise to a cylinder twist. For later reference we summarize the situation in the following remark.

Remark 4.13.

Let B⊆Uq⁢(𝔀) be a right coideal subalgebra and let (ℬ,𝒪int) be the tensor pair from Example 4.1. Moreover, let φ:Uq⁢(𝔀)→Uq⁢(𝔀) be a Hopf-algebra automorphism and let t⁢w:𝒪int→𝒪int be the corresponding braided tensor equivalence as in Example 4.7. An element 𝒊∈𝒰 is a φ-universal K-matrix for B if and only if 𝒊 is a t⁢w-cylinder twist of (ℬ,𝒪int). In this case, in particular, the element t=𝒊∈𝒰 satisfies the fusion procedure (4.7) and the reflection equation (4.8) for all M,N∈𝒪int.

4.4 Cylinder braided coideal subalgebras via characters

In [9] J. Donin, P. Kulish, and A. Mudrov introduced the notion of a universal solution of the reflection equation which they also called a universal K-matrix. In contrast to Definition 4.10, this notion does not refer to a coideal subalgebra of a Hopf algebra. Nevertheless, there is a close relationship between Definition 4.10 and the notion of a universal K-matrix in [9], and it is the purpose of the present section to explain this. This material will not be used in later parts of the present paper.

As in Section 4.3 let (H,RH) be a braided Hopf algebra over a field k. We retain the conventions from Definition 4.8 and hence the symbol ℛ in [9] corresponds to R21H in our conventions. Let H∗=Homk⁢(H,k) denote the linear dual space of H. Recall from [30] that the braided Hopf H algebra is called factorizable if the linear map

H∗→H,f↩(f⊗id)⁢(R12H⁢R21H)

is an isomorphism of vector spaces. This is only possible if H is finite dimensional. If H is factorizable, then Donin, Kulish, and Mudrov call the element

𝒊dkm=R12H⁢R21H∈H⊗H

the universal K-matrix of H. It follows from (4.10) that

(id⊗Δ)⁢(𝒊dkm)=R01H⁢R02H⁢R20H⁢R10H⁢R12H⁢(R12H)-1(4.17)=R01H⁢R10H⁢R12H⁢R02H⁢R20H⁢R21H⁢(R12H⁢R21H)-1=𝒊01dkm⁢R12H⁢𝒊02dkm⁢R21H⁢(R12H⁢R21H)-1

where we label the tensor legs of H⊗3 by 0, 1, 2. The above formula is closely related to formula (4.13) for Δ⁢(𝒊). There are two differences, however, namely the occurrence of the additional factor (R12H⁢R21H)-1 and the fact that (4.17) holds in H⊗3 while formula (4.13) holds in H⊗2. Moreover, the element 𝒊dkm makes no reference to a coideal subalgebra of Uq⁢(𝔀).

To address the first difference, recall from [17, Definition XIV.6.1] that the braided Hopf algebra (H,RH) is called a ribbon algebra if there exists a central element ΞH∈H such that

Δ⁢(ΞH)=(R12H⁢R21H)-1⁢(ΞH⊗ξH),ε⁢(ΞH)=1,S⁢(ΞH)=ΞH.

If such a ribbon element ΞH exists, then the element

𝒊dkm,Ξ=(1⊗ξH-1)⁢𝒊dkm∈H⊗H

satisfies the relation

(id⊗Δ)⁢(𝒊dkm,Ξ)=𝒊01dkm,Ξ⁢R12H⁢𝒊02dkm,Ξ⁢R21H(4.18)

in H⊗3.

To eliminate the additional tensor factor in (4.17) and (4.18) let

f:H→k

be a character, that is a one-dimensional representation. Define

Bf={(f⊗id)⁢Δ⁢(h)∣h∈H}(4.19)

and observe that Bf is a right coideal subalgebra of H. The element

𝒊dkm,Ξ,f=(f⊗ξH-1)⁢𝒊dkm∈H(4.20)

commutes with all elements of Bf because Δ⁢(h) commutes with 𝒊dkm=R12H⁢R21H for all h∈H by (4.9). By (4.18) one has

Δ⁢(𝒊dkm,Ξ,f)=(𝒊dkm,Ξ,f⊗1)⁢RH⁢(1⊗𝒊dkm,Ξ,f)⁢R21H

which coincides with relation (4.13) in Definition 4.10. We summarize the above discussion.

Proposition 4.14.

Let (H,RH,ξH) be a factorizable ribbon Hopf algebra over a field k and let f:H→k be a character. Then the right coideal subalgebra Bf defined by (4.19) is cylinder braided with universal K-matrix

𝒊dkm,Ξ,f=(f⊗ξH-1)⁢(R12H⁢R21H)∈H.

By the above proposition the element 𝒊dkm,Ξ,f satisfies the reflection equation in every tensor product M⊗N of representations of H. As the ribbon element ΞH is central, the element (f⊗1)⁢(R12H⁢R21H) also satisfies the reflection equation.

Remark 4.15.

Assume that 𝔀 is of finite type. If one naively translates the construction of Proposition 4.14 to the setting of Uq⁢(𝔀), then the resulting universal K-matrix is the identity element because Uq⁢(𝔀) does not have any interesting characters. However, in [9] a universal K-matrix is also defined for non-factorizable H. In this case one chooses 𝒊dkm to be the canonical element in H~∗⊗H where H~∗ denotes a twisted version of the dual Hopf algebra H∗. One obtains a universal K-matrix by application of a character f of H~∗. This framework translates to the setting of Uq⁢(𝔀) if one replaces H~∗ by the braided restricted dual of Uq⁢(𝔀). The braided restricted dual of Uq⁢(𝔀) is isomorphic as an algebra to the (right) locally finite part

Fr⁢(Uq⁢(𝔀))={x∈Uq⁢(𝔀)∣dim⁡(adr⁢(Uq⁢(𝔀))⁢(x))<∞}

where adr⁢(u)⁢(x)=S⁢(u(1))⁢x⁢u(2) for u,x∈Uq⁢(𝔀) denotes the right adjoint action. The locally finite part has many nontrivial characters, and a cylinder braiding for 𝒪int can be associated to each of them, see [19, Propositions 2.8, 3.14].

The constructions in [9] and in this subsection, however, do not answer the question how to find characters of H~∗. For H=Uq⁢(𝔀) this amounts to finding numerical solutions of the reflection equation which satisfy additional compatibility conditions. For 𝔀=𝔰⁢𝔩n⁢(ℂ) this is a manageable problem, see [19, Remark 5.11]. It would be interesting to find a conceptual classification of characters of Fr⁢(Uq⁢(𝔀)) for all 𝔀 of finite type.

5 Quantum symmetric pairs

In the remainder of this paper we will show that quantum symmetric pair coideal subalgebras of Uq⁢(𝔀) are φ-cylinder-braided as in Definition 4.12 for a suitable automorphism φ of Uq⁢(𝔀). To set the scene we now recall the construction and properties of quantum symmetric pairs. We will in particular recall the existence of the intrinsic bar involution from [2] in Section 5.3. Quantum symmetric pairs depend on a choice of parameters and the existence of the bar involution imposes further restrictions. In Section 5.4, for later reference, we summarize our setting and assumptions including the restrictions on parameters.

5.1 Involutive automorphisms of the second kind

Let 𝔟+ denote the positive Borel subalgebra of 𝔀. An automorphism Ξ:𝔀→𝔀 is said to be of the second kind if

dim⁡(Ξ⁢(𝔟+)∩𝔟+)<∞.

Involutive automorphisms of the second kind of 𝔀 were essentially classified in [16], see also [18, Theorem 2.7]. In this subsection we recall the combinatorial data underlying this classification.

For any subset X of I let 𝔀X denote the corresponding Lie subalgebra of 𝔀. The sublattice QX of Q generated by {αi∣i∈X} is the root lattice of 𝔀X. If 𝔀X is of finite type, then let ρX and ρX√ denote the half sum of positive roots and positive coroots of 𝔀X, respectively. The Weyl group WX of 𝔀X is the parabolic subgroup of W generated by all σi with i∈X. If 𝔀X is of finite type, then let wX∈WX denote the longest element. Let Aut⁢(A) denote the group of permutations τ:I→I such that the entries of the Cartan matrix A=(ai⁢j) satisfy ai⁢j=aτ⁢(i)⁢τ⁢(j) for all i,j∈I. Let Aut⁢(A,X) denote the subgroup of all τ∈Aut⁢(A) which additionally satisfy τ⁢(X)=X.

Involutive automorphisms of 𝔀 of the second kind are parametrized by combinatorial data attached to the Dynkin diagram of 𝔀. This combinatorial data is a generalization of Satake diagrams from the finite-dimensional setting to the Kac–Moody case, see [1], [18, Definition 2.3].

Definition 5.1.

A pair (X,τ) consisting of a subset X⊆I of finite type and an element τ∈Aut⁢(A,X) is called admissible if the following conditions are satisfied:

  • (1)

    τ2=idI.

  • (2)

    The action of τ on X coincides with the action of -wX.

  • (3)

    If j∈I∖X and τ⁢(j)=j then αj⁢(ρX√)∈℀.

We briefly recall the construction of the involutive automorphisms Ξ=Ξ⁢(X,τ) corresponding to the admissible pair (X,τ), see [18, Section 2] for details. Let ω:𝔀→𝔀 denote the Chevalley involution as in [15, (1.3.4)]. Any τ∈Aut⁢(A,X) can be lifted to a Lie algebra automorphism τ:𝔀→𝔀. Moreover, for X⊂I of finite type let Ad⁢(wX):𝔀→𝔀 denote the corresponding braid group action of the longest element in WX. Finally, let s:I→𝕂× be a function such that

s⁢(i)=1if ⁢i∈X⁢ or ⁢τ⁢(i)=i,(5.1)s⁢(i)s⁢(τ⁢(i))=(-1)αi⁢(2⁢ρX√)if ⁢i∉X⁢ and ⁢τ⁢(i)≠i.(5.2)

Such a function always exists. The map s gives rise to a group homomorphism sQ:Q→𝕂× such that

sQ⁢(αi)=s⁢(i).

This in turn allows us to define a Lie algebra automorphism Ad⁢(s):𝔀→𝔀 such that the restriction of Ad⁢(s) to any root space 𝔀α is given by multiplication by sQ⁢(α).

Remark 5.2.

In [18, (2.7)] and in [2, (3.2)] we chose the values s⁢(i) for i∈I to be certain fourth roots of unity. This had the advantage that Ad⁢(s) commutes with the involutive automorphism corresponding to the admissible pair (X,τ). However, the only properties of s used in [18, 2], and in the present paper are the relations (5.1) and (5.2). It is hence possible to choose s⁢(i)∈{-1,1}. This is more suitable for the categorification program in [11] and for the program of canonical bases for coideal subalgebras in [3].

With the above notations at hand we can now recall the classification of involutive automorphisms of the second kind in terms of admissible pairs.

Theorem 5.3 ([16], [18, Theorem 2.7]).

The map

(X,τ)↊Ξ⁢(X,τ)=Ad⁢(s)∘Ad⁢(wX)∘τ∘ω

gives a bijection between the set of Aut⁢(A)-orbits of admissible pairs for g and the set of Aut⁢(g)-conjugacy classes of involutive automorphisms of the second kind.

Let 𝔚={x∈𝔀∣Ξ⁢(X,τ)⁢(x)=x} denote the fixed Lie subalgebra of 𝔀. We refer to (𝔀,𝔚) as a symmetric pair. The involution Ξ=Ξ⁢(X,τ) leaves 𝔥 invariant. The induced map Θ:𝔥∗→𝔥∗ is given by

Θ=-wX∘τ

where τ⁢(αi)=ατ⁢(i) for all i∈I, see [18, Section 2.2, (2.10)]. Hence Θ restricts to an involution of the root lattice. Let QΘ be the sublattice of Q consisting of all elements fixed by Θ. For later use we note that

Θ⁢(ατ⁢(i))-ατ⁢(i)=Θ⁢(αi)-αi for all i∈I,(5.3)

see [2, Lemma 3.2].

5.2 The construction of quantum symmetric pairs

We now recall the definition of quantum symmetric pair coideal subalgebras following [18]. For the remainder of this paper let (X,τ) be an admissible pair and s:I→𝕂× a function satisfying (5.1) and (5.2). Let ℳX=Uq⁢(𝔀X) denote the subalgebra of Uq⁢(𝔀) generated by the elements Ei, Fi, Ki±1 for all i∈X. Correspondingly, let ℳX+ and ℳX- denote the subalgebras of ℳX generated by the elements in the sets {Ei∣i∈X} and {Fi∣i∈X}, respectively.

Note that the derived Lie subalgebra 𝔀′ is invariant under the involutive automorphism Ξ=Ξ⁢(X,τ). One can define a quantum group analog

Ξq:Uq⁢(𝔀′)→Uq⁢(𝔀′)

of Ξ, see [18, Definition 4.3] for details. The quantum involution Ξq is a 𝕂⁢(q1/d)-algebra automorphism but it is not a coalgebra automorphism and Ξq2≠idUq⁢(𝔀′). However, the map Ξq has the following desirable properties:

Ξq|ℳX=idℳX,Ξq⁢(KÎŒ)=KΘ⁢(ÎŒ)for all Ό∈Q,Ξq⁢(Ki-1⁢Ei)=-s⁢(τ⁢(i))-1⁢TwX⁢(Fτ⁢(i))∈UΘ⁢(αi)-for all i∈I∖X,Ξq⁢(Fi⁢Ki)=-s⁢(τ⁢(i))⁢TwX⁢(Eτ⁢(i))∈UΘ⁢(-αi)+for all i∈I∖X.

To shorten notation define

Xi=Ξq⁢(Fi⁢Ki)=-s⁢(τ⁢(i))⁢TwX⁢(Eτ⁢(i)) for all i∈I∖X.(5.4)

Quantum symmetric pair coideal subalgebras depend on a choice of parameters

𝐜=(ci)i∈I∖X∈(𝕂⁢(q1/d)×)I∖X and 𝐬=(si)i∈I∖X∈𝕂⁢(q1/d)I∖X.

Define

Ins={i∈I∖X|τ⁢(i)=i⁢ and ⁢ai⁢j=0⁢ for all ⁢j∈X}.(5.5)

In [18, (5.9), (5.11)]) the following parameter sets appeared:

𝒞={𝐜∈(𝕂⁢(q1/d)×)I∖X∣ci=cτ⁢(i)⁢ if ⁢τ⁢(i)≠i⁢ and ⁢(αi,Θ⁢(αi))=0},(5.6)𝒮={𝐬∈𝕂(q1/d)I∖X∣sj≠0⇒(j∈Ins and ai⁢j∈-2ℕ0∀i∈Ins∖{j})},(5.7)

see also [2, Remark 3.3].

Let UΘ0′ be the subalgebra of U0 generated by all KÎŒ with Ό∈QΘ.

Definition 5.4.

Let (X,τ) be an admissible pair. Further, let 𝐜=(ci)i∈I∖X∈𝒞 and let 𝐬=(si)i∈I∖X∈𝒮. The quantum symmetric pair coideal subalgebra B𝐜,𝐬=B𝐜,𝐬⁢(X,τ) is the subalgebra of Uq⁢(𝔀′) generated by ℳX, UΘ0′, and the elements

Bi=Fi+ci⁢Xi⁢Ki-1+si⁢Ki-1(5.8)

for all i∈I∖X.

Remark 5.5.

The conditions 𝐜∈𝒞 and 𝐬∈𝒮 can be found in [18, (5.9) and (5.11)]. They are necessary to ensure that the intersection of the coideal subalgebra with U0 is precisely UΘ0′. This in turn implies that the coideal subalgebra B𝐜,𝐬 specializes to U⁢(𝔚′) at q=1 with 𝔚′={x∈𝔀′∣Ξ⁢(x)=x}, see [18, Remark 5.12, Theorem 10.8].

For i∈X we set ci=si=0 and Bi=Fi. This convention will occasionally allow us to treat the cases i∈X and i∉X simultaneously.

The algebra B𝐜,𝐬 is a right coideal subalgebra of Uq⁢(𝔀′), that is

Δ⁢(B𝐜,𝐬)⊆B𝐜,𝐬⊗Uq⁢(𝔀′),

see [18, Proposition 5.2]. One can calculate the coproduct of the generators Bi for i∈I∖X more explicitly and obtains

Δ⁢(Bi)=Bi⊗Ki-1+1⊗Fi+ci⁢rτ⁢(i)⁢(Xi)⁢Ki-1⁢Kτ⁢(i)⊗Eτ⁢(i)⁢Ki-1+Υ(5.9)

for some Υ∈ℳXUΘ0⊗′∑γ>ατ⁢(i)Uγ+Ki-1, see [18, Lemma 7.2]. By (2.11) this implies that

rj⁢(Xi)=0 whenever j≠τ⁢(i).(5.10)

In view of (5.9) it makes sense to define

𝒵i=rτ⁢(i)⁢(Xi)⁢Ki-1⁢Kτ⁢(i).(5.11)

The elements 𝒵i play a crucial role in the description of B𝐜,𝐬 in terms of generators and relations, see [18, Section 7], [2, Section 3.2].

5.3 The bar involution for quantum symmetric pairs

The bar involution for Uq⁢(𝔀) defined in (2.18) does not map B𝐜,𝐬 to itself. Inspired by the papers [11, 3], it was shown in [2] under mild additional assumptions that B𝐜,𝐬 allows an intrinsic bar involution ¯B:B𝐜,𝐬→B𝐜,𝐬. We now recall these assumptions and the construction of the intrinsic bar involution for B𝐜,𝐬.

In [2, Section 3.2] the algebras B𝐜,𝐬 are given explicitly in terms of generators and relations for all Cartan matrices A=(ai⁢j) and admissible pairs (X,τ) which satisfy the following properties:

  • (i)

    If i∈I∖X with τ⁢(i)=i and j∈X, then ai⁢j∈{0,-1,-2}.

  • (ii)

    If i∈I∖X with τ⁢(i)=i and i≠j∈I∖X, then ai⁢j∈{0,-1,-2,-3}.

The existence of the bar involution ¯B on B𝐜,𝐬 was then proved by direct computation based on the defining relations.

Theorem 5.6 ([2, Theorem 3.11]).

Assume that conditions (i) and (ii) hold. The following statements are equivalent.

  • (1)

    There exists a 𝕂 -algebra automorphism ¯B:B𝐜,𝐬→B𝐜,𝐬, x↩x¯B such that

    x¯B=x¯ for all x∈ℳXUΘ0,′Bi¯B=Bi for all i∈I∖X.(5.12)

    In particular, q1/d¯B=q-1/d.

  • (2)

    The relation

    ci⁢𝒵i¯=q(αi,ατ⁢(i))⁢cτ⁢(i)⁢𝒵τ⁢(i)

    holds for all i∈I∖X for which τ⁢(i)≠i or for which there exists j∈I∖{i} such that ai⁢j≠0.

It is conjectured that Theorem 5.6 holds without assumptions (i) and (ii). In [2, Proposition 3.5] it was proved that for all i∈I∖X one has

𝒵i¯=Îœi⁢q(αi,αi-wX⁢(αi)-2⁢ρX)⁢𝒵τ⁢(i)(5.13)

for some Îœi∈{-1,1}. For 𝔀 of finite type it was moreover proved that Îœi=1 for all i∈I∖X, and this was conjectured to hold also in the Kac–Moody case [2, Proposition 2.3, Conjecture 2.7].

5.4 Assumptions

For later reference we summarize our setting. As before 𝔀 denotes the Kac–Moody algebra corresponding to the symmetrizable Cartan matrix A=(ai⁢j) and (X,τ) is an admissible pair. We fix parameters 𝐜∈𝒞 and 𝐬∈𝒮 and let B𝐜,𝐬 denote the corresponding quantum symmetric pair coideal subalgebra of Uq⁢(𝔀′) as given in Definition 5.4. Additionally, the following assumptions are made for the remainder of this paper.

  • (1)

    The Cartan matrix A=(ai⁢j) satisfies conditions (i) and (ii) in Section 5.3.

  • (2)

    The parameters 𝐜∈𝒞 satisfy the condition

    ci⁢𝒵i¯=q(αi,ατ⁢(i))⁢cτ⁢(i)⁢𝒵τ⁢(i)  for all i∈I∖X.(5.14)

  • (3)

    The parameters 𝐬∈𝒮 satisfy the condition

    s¯i=si for all i∈I∖X.(5.15)

  • (4)

    One has Îœi=1 for all i∈I∖X, that is [2, Conjecture 2.7] holds true.

If (4) holds, then using (5.13) and (5.3) one sees that equation (5.14) is equivalent to

cτ⁢(i)=q(αi,Θ⁢(αi)-2⁢ρX)⁢ci¯.(5.16)

Remark 5.7.

Assumption (1) is only used in the proof of Theorem 5.6. Assumption (4) is only used to obtain equation (5.16). Once Theorem 5.6 is established without assuming conditions (i) and (ii), and once it is proved that Îœi=1 for all i∈I∖X, all results of this paper hold for B𝐜,𝐬 with 𝐜∈𝒞 and 𝐬∈𝒮 satisfying relations (5.14) and (5.15).

Remark 5.8.

Observe that assumption (2) is a stronger statement then what is needed for the existence of the bar-involution ¯B in Theorem 5.6. This stronger statement will be used in the construction of the quasi-K-matrix in Section 6.4, see the end of the proof of Lemma 6.7. It is moreover used in the calculation of the coproduct of the universal K-matrix in Section 9, see proof of Lemma 9.3. Assumption (3) is new and will be used in the proofs of Lemma 6.8 and Theorem 6.10.

Remark 5.9.

For every admissible pair there exist parameters ci∈𝕂⁢(q) satisfying equation (5.16), see [2, Remark 3.14].

6 The quasi K-matrix 𝔛

The bar involution x↩x¯ on Uq⁢(𝔀) defined by (2.18) and the internal bar involution x↩x¯B on B𝐜,𝐬 defined by (5.12) satisfy Bi¯≠Bi¯B if i∈I∖X. Hence the two bar involutions do not coincide when restricted to B𝐜,𝐬. The aim of this section is to construct an element 𝔛∈U+^ which intertwines between the two bar involutions. More precisely, we will find (𝔛Ό)Ό∈Q+ with 𝔛Ό∈UÎŒ+ and 𝔛0=1 such that 𝔛=∑Ό𝔛Ό satisfies

x¯B⁢𝔛=𝔛⁢x¯ for all x∈B𝐜,𝐬.(6.1)

In view of (3.3), the element 𝔛∈U+^⊆𝒰 is an analog of the quasi-R-matrix R for quantum symmetric pairs. For this reason we will call 𝔛 the quasi K-matrix for B𝐜,𝐬. Examples of quasi K-matrices 𝔛 were first constructed in [3, Theorems 2.10, 6.4] for the coideal subalgebras corresponding to the symmetric pairs (𝔰⁢𝔩2⁢n,𝔰⁢(𝔀⁢𝔩n×𝔀⁢𝔩n)) and (𝔰⁢𝔩2⁢n+1,𝔰⁢(𝔀⁢𝔩n+1×𝔀⁢𝔩n)).

6.1 A recursive formula for 𝔛

As a first step towards the construction of 𝔛 we translate relation (6.1) into a recursive formula for the components 𝔛Ό.

Proposition 6.1.

Let

𝔛=∑Ό∈Q+𝔛Ό∈U+^,with ⁢𝔛Ό∈UÎŒ+.

The following are equivalent:

  • (1)

    For all x∈B𝐜,𝐬 one has x¯B⁢𝔛=𝔛⁢x¯.

  • (2)

    For all i∈I one has Bi¯B⁢𝔛=𝔛⁢Bi¯.

  • (3)

    For all Ό∈Q+ and all i∈I one has

    ri⁢(𝔛Ό)=-(qi-qi-1)⁢(𝔛Ό+Θ⁢(αi)-αi⁢ci⁢Xi¯+si¯⁢𝔛Ό-αi),(6.2)ri⁢(𝔛Ό)=-(qi-qi-1)⁢(q-(Θ⁢(αi),αi)⁢ci⁢Xi⁢𝔛Ό+Θ⁢(αi)-αi+si⁢𝔛Ό-αi).(6.3)

If these equivalent conditions hold then additionally

  • (4)

    For all Ό∈Q+ such that 𝔛Ό≠0 , one has Θ⁢(ÎŒ)=-ÎŒ.

Proof.

(1) ⇒ (2) Property (2) is the special case x=Bi of property (1).

(2) ⇔ (3) Fix i∈I. Using the definition (5.8) of Bi, the definition (2.18) of Bi¯ and the definition (5.12) of Bi¯B, we see that (2) is equivalent to

(Fi+ci⁢Xi⁢Ki-1+si⁢Ki-1)⁢𝔛=𝔛⁢(Fi+ci⁢Xi¯⁢Ki+si¯⁢Ki).(6.4)

Now compare the (ÎŒ-αi)-homogeneous components for all Ό∈Q+. One obtains that equation (6.4) holds if and only if for all Ό∈Q+ one has

[𝔛Ό,Fi]=-(𝔛Ό-αi+Θ⁢(αi)⁢ci⁢Xi¯+si¯⁢𝔛Ό-αi)⁢Ki,+Ki-1⁢(q-(αi,Θ⁢(αi))⁢ci⁢Xi⁢𝔛Ό-αi+Θ⁢(αi)+si⁢𝔛Ό-αi).

By (2.10), this is equivalent to relations (6.2) and (6.3) for all Ό∈Q+.

(3) ⇒ (4) We prove this implication by induction on ht⁢(ÎŒ). For ÎŒ=0 there is nothing to show. Assume that ÎŒ>0. If 𝔛Ό≠0, then by (2.13), there exists i∈I such that ri⁢(𝔛Ό)≠0. By (6.2) we have either 𝔛Ό+Θ⁢(αi)-αi≠0 or si⁢𝔛Ό-αi≠0. In the case 𝔛Ό+Θ⁢(αi)-αi≠0, by induction hypothesis Θ⁢(ÎŒ+Θ⁢(αi)-αi)=-(ÎŒ+Θ⁢(αi)-αi), which implies Θ⁢(ÎŒ)=-ÎŒ. In the case si⁢𝔛Ό-αi≠0, the condition 𝐬∈𝒮 implies that Θ⁢(αi)=-αi, while the induction hypothesis implies that Θ⁢(ÎŒ-αi)=-(ÎŒ-αi). Together, this gives Θ⁢(ÎŒ)=-ÎŒ.

(3) ⇒ (1) We have already seen that (3) ⇒ (2) and hence x¯B⁢𝔛=𝔛⁢x¯ for x=Bi.

Let β∈QΘ and assume that 𝔛Ό≠0. The implication (3)⇒(4) gives Θ⁢(ÎŒ)=-ÎŒ. On the other hand Θ⁢(β)=β and therefore (β,ÎŒ)=0. This implies that

Kβ⁢𝔛Ό⁢Kβ-1=q(β,ÎŒ)⁢𝔛Ό=𝔛Ό

and consequently x¯B⁢𝔛=𝔛⁢x¯ for all x∈UΘ0.

Finally, let i∈X and again assume that 𝔛Ό≠0. As Ki∈UΘ0 and Fi=Bi, we already know that ad⁢(Ki)⁢(𝔛Ό)=𝔛Ό and ad⁢(Fi)⁢(𝔛Ό)=0. Hence 𝔛Ό is the lowest weight vector for the left adjoint action of Uqi⁢(𝔰⁢𝔩2)i on Uq⁢(𝔀). As U+ is locally finite for the left adjoint action of U+, we conclude that 𝔛Ό is also a highest weight vector, and hence

0=ad⁢(Ei)⁢(𝔛Ό)=Ei⁢𝔛Ό-Ki⁢𝔛Ό⁢Ki-1⁢Ei=Ei⁢𝔛Ό-𝔛Ό⁢Ei.

Thus Ei⁢𝔛Ό=𝔛Ό⁢Ei and consequently x¯B⁢𝔛=𝔛⁢x¯ for all x∈ℳX.

This proves that the relation x¯B⁢𝔛=𝔛⁢x¯ holds for the generators of the algebra B𝐜,𝐬 and hence it holds for all x∈B𝐜,𝐬. ∎

The proof of the implication (3) ⇒ (4) only refers to 𝔛Ό′ with Ό′≀Ό. Hence we get the following corollary.

Corollary 6.2.

Let Ό∈Q+ and let (XΌ′)Ό′≀Ό∈Q+, with XΌ′∈UΌ′+, be a collection of elements satisfying (6.2) and (6.3) for all Ό′≀Ό and all i∈I. If XΌ≠0, then Θ⁢(ÎŒ)=-ÎŒ.

6.2 Systems of equations given by skew derivations

By Proposition 6.1 the quasi K-matrix 𝔛 can be constructed inductively if in each step it is possible to solve the system of equations given by (6.2) and (6.3) for all i. In this subsection we derive necessary and sufficient conditions for such a system to have a solution.

Proposition 6.3.

Let Ό∈Q+ with ht⁢(ÎŒ)≥2 and fix elements Ai,Ai∈UÎŒ-αi+ for all i∈I. The following are equivalent:

  • (1)

    There exists an element X¯∈UÎŒ+ such that

    ri⁢(X¯)=Ai 𝑎𝑛𝑑 ri⁢(X¯)=Ai for all i∈I.(6.5)

  • (2)

    The elements Ai,Ai have the following two properties:

    • (a)

      For all i,j∈I one has

      ri(jA)=rj(Ai).(6.6)

    • (b)

      For all i≠j∈I one has

      -1qi-qi-1⁢∑s=11-ai⁢j[1-ai⁢js]qi⁢(-1)s⁢〈Fi1-ai⁢j-s⁢Fj⁢Fis-1,Ai〉(6.7)-1qj-qj-1⁢〈Fi1-ai⁢j,Aj〉=0.

Moreover, if the system of equations (6.5) has a solution X¯, then this solution is uniquely determined.

Proof.

(1) ⇒ (2) Assume that there exists and element X¯∈UÎŒ+ which satisfies the equations (6.5). Then

ri⁢(Aj)=ri⁢(rj⁢(X¯))⁢=(2.14)⁢rj⁢(ri⁢(X¯))=rj⁢(Ai)

and hence (6.6) holds for all i,j∈I.

Moreover, using the quantum Serre relation Si⁢j⁢(Fi,Fj)=0 and the properties (2.12) of the bilinear form 〈⋅,⋅〉, we get

0=〈Si⁢j⁢(Fi,Fj),X¯〉=∑s=01-ai⁢j[1-ai⁢js]qi⁢(-1)s⁢〈Fi1-ai⁢j-s⁢Fj⁢Fis,X¯〉=-1qi-qi-1⁢∑s=11-ai⁢j[1-ai⁢js]qi⁢(-1)s⁢〈Fi1-ai⁢j-s⁢Fj⁢Fis-1,Ai〉-1qj-qj-1⁢〈Fi1-ai⁢j,Aj〉,

which proves relation (6.7). Hence property (2) holds.

(2) ⇒ (1) Assume that the elements Ai,Ai satisfy relations (6.6) and (6.7). We first solve the system dual to (6.5) with respect to the bilinear form 〈⋅,⋅〉. By slight abuse of notation we consider 〈⋅,⋅〉 as a pairing on 𝐟′×U+ via the canonical projection 𝐟′→U- on the first factor. Fix Ό∈Q+ with ht⁢(ÎŒ)≥2. As ÎŒ>0, there exist uniquely determined linear functionals X¯L*,X¯R*:𝐟Ό′→𝕂⁢(q1/d) such that

X¯L*⁢(𝖿i⁢z)=-1qi-qi-1⋅〈z,Ai〉,(6.8)X¯R*⁢(z⁢𝖿i)=-1qi-qi-1⋅〈z,Ai〉(6.9)

for all z∈𝐟Ό-αi′. For any i,j∈I and any x∈𝐟Ό-αi-αj′ we have

X¯L*⁢(𝖿j⁢x⁢𝖿i)⁢=(6.6)⁢-1qj-qj-1⋅〈x⁢𝖿i,Aj〉⁢=(2.12)⁢-1qi-qi-1⋅-1qj-qj-1⁢〈x,ri⁢(Aj)〉=(6.6)⁢-1qi-qi-1⋅-1qj-qj-1⁢〈x,rj⁢(Ai)〉⁢=(2.12)⁢-1qi-qi-1⋅〈𝖿j⁢x,Ai〉=(6.9)⁢X¯R*⁢(𝖿j⁢x⁢𝖿i).

As ht⁢(ÎŒ)≥2, any element in 𝐟Ό′ can be written as a linear combination of elements of the form 𝖿j⁢x⁢𝖿i with x∈𝐟Ό-αi-αj′ for i,j∈I. Consequently, the above relation implies that the functionals X¯L* and X¯R* coincide on 𝐟Ό′. To simplify notation we write X¯*=X¯L*=X¯R*.

We claim that relation (6.7) implies that X¯* descends from 𝐟Ό′ to a linear functional on U-ÎŒ-. Recall that the kernel of the projection 𝐟′→U- is the ideal generated by the elements Si⁢j⁢(𝖿i,𝖿j) for all i,j∈I. Hence it is enough to show that all elements of the form x=𝖿a1⁢ ⁢𝖿al⋅Si⁢j⁢(𝖿i,𝖿j)⋅𝖿b1⁢ ⁢𝖿bk lie in the kernel of the linear functional X¯*. If l>0, then the fact that Si⁢j⁢(𝖿i,𝖿j) lies in the radical of the bilinear form 〈⋅,⋅〉 implies that

X¯*⁢(x)=X¯L*⁢(𝖿a1⁢ ⁢𝖿al⋅Si⁢j⁢(𝖿i,𝖿j)⋅𝖿b1⁢ ⁢𝖿bk)=-1qa1-qa1-1⋅〈𝖿a2⁢ ⁢𝖿al⋅Si⁢j⁢(𝖿i,𝖿j)⋅𝖿b1⁢ ⁢𝖿bk,Aa1〉=0.

Similarly, if k>1, then we get

X¯*⁢(x)=X¯R*⁢(x)=0.

Assume now that l=k=0. Then

X¯*⁢(Si⁢j⁢(𝖿i,𝖿j))⁢=⁢∑s=01-ai⁢j[1-ai⁢js]qi⁢(-1)s⋅X¯R*⁢(𝖿i1-ai⁢j-s⁢𝖿j⁢𝖿is)=⁢-1qi-qi-1⁢∑s=11-ai⁢j[1-ai⁢js]qi⁢(-1)s⁢〈𝖿i1-ai⁢j-s⁢𝖿j⁢𝖿is-1,Ai〉-1qj-qj-1⁢〈𝖿i1-ai⁢j,Aj〉=(6.7)⁢0.

Hence X¯* does indeed descend to a linear functional X¯*:U-ÎŒ-→𝕂⁢(q1/d).

Let X¯∈UÎŒ+ be the element dual to X¯* with respect to the nondegenerate pairing 〈⋅,⋅〉 on UÎŒ-×UÎŒ+. In other words, for all z∈U- we have X¯*⁢(z)=〈z,X¯〉. Then

〈z,ri⁢(X¯)〉=-(qi-qi-1)⁢〈z⁢Fi,X¯〉=-(qi-qi-1)⁢X¯*⁢(z⁢Fi)⁢=(6.9)⁢〈z,Ai〉

for any z∈UÎŒ-αi- and hence ri⁢(X¯)=Ai for all i∈I. Similarly, (6.8) implies that ri⁢(X¯)=Ai for all i∈I. This completes the proof of relation (6.5) and hence (1) holds.

To see uniqueness, assume that X¯ and X¯′ both satisfy the system of equations (6.5). Then ri⁢(X¯-X¯′)=0 for all i∈I, so by (2.13), we have that X¯=X¯′. ∎

6.3 Three technical lemmas

We will use Proposition 6.3 in Section 6.4 to inductively construct 𝔛Ό by solving the system of equations given by (6.2), (6.3) for all i∈I. To simplify the proof that the right hand sides of equations (6.2), (6.3) satisfy the conditions from Proposition 6.3 (2), we provide several technical lemmas. These results are auxiliary and will only be used in the proof of Lemma 6.8.

Lemma 6.4.

Let i≠j∈I and ÎŒ=(1-ai⁢j)⁢αi+αj. If Θ⁢(ÎŒ)=-ÎŒ, then i,j∈I∖X and one of the following two cases holds:

  • (1)

    Θ⁢(αi)=-αj and ai⁢j=0.

  • (2)

    Θ⁢(αi)=-αi and Θ⁢(αj)=-αj.

Proof.

Assume that i∈X. Then Θ⁢(αi)=αi which together with Θ⁢(ÎŒ)=-ÎŒ implies that

wX⁢(ατ⁢(j))=-Θ⁢(αj)=-Θ⁢(ÎŒ-(1-ai⁢j)⁢αi)=αj+2⁢(1-ai⁢j)⁢αi.

Hence τ⁢(j)=j and σi⁢(αj)=wX⁢(αj) and -ai⁢j=2⁢(1-ai⁢j). This would mean that ai⁢j=2 which is impossible.

Assume that j∈X. Then

wX⁢(ατ⁢(i))=-Θ⁢(αi)=-1(1-ai⁢j)⁢Θ⁢(ÎŒ-αj)=αi+2(1-ai⁢j)⁢αj.

Hence τ⁢(i)=i and σj⁢(αi)=wX⁢(αi) and aj⁢i=-2(1-ai⁢j). This is only possible if

aj⁢i=ai⁢j=-1.

But then

αi⁢(ρX√)=12⁢αi⁢(hj)=-12∉℀

which contradicts condition (3) in Definition 5.1 of an admissible pair.

Hence i,j∈I∖X. As (wX-id)⁢(αk)∈QX for any k∈I, it follows that

(1-ai⁢j)⁢(αi-ατ⁢(i))+(αj-ατ⁢(j))=-Θ⁢(ÎŒ)-τ⁢(ÎŒ)=(wX-id)⁢(τ⁢(ÎŒ))

lies in QX. Using i,j∈I∖X, it follows that (1-ai⁢j)⁢(αi-ατ⁢(i))+(αj-ατ⁢(j))=0. So, there are two possibilities: either (1) τ⁢(i)=j and ai⁢j=0, or (2) τ⁢(i)=i and τ⁢(j)=j. ∎

Lemma 6.5.

Let Ό∈Q+ and let j∈I∖X with sj=0. Assume that a collection (XΌ′)Ό′≀Ό with XΌ′∈UΌ′+ satisfies condition (6.2) for all Ό′≀Ό and for all i∈I. If XΌ≠0, then Ό∈spanN0⁢{αj-Θ⁢(αj)}⊕spanN0⁢{αk∣k≠j}.

Proof.

We prove this by induction on ht⁢(ÎŒ). If ÎŒ>0 and 𝔛Ό≠0, then by (2.13) there exists some i such that ri⁢(𝔛Ό)≠0. Relation (6.2) implies that 𝔛Ό+Θ⁢(αi)-αi≠0 or si⁢𝔛Ό-αi≠0. If i≠j, then the induction hypothesis on ÎŒ+Θ⁢(αi)-αi and ÎŒ-αi implies the claim. If i=j, then the induction hypothesis on ÎŒ+Θ⁢(αi)-αi implies the claim. ∎

Recall that σ denotes the involutive antiautomorphism of Uq⁢(𝔀) defined by (2.16).

Lemma 6.6.

Let Μ∈Q+, and let (XÎŒ)ÎŒ<Μ∈Q+ be a collection with XΌ∈UÎŒ+ and X0=1. For all ÎŒ<Îœ assume that XÎŒ satisfies (6.2) and (6.3) for all i∈I. Let j,k∈I∖X be such that Θ⁢(αj)=-αj and Θ⁢(αk)=-αk. Assume that n≥0, and that x∈U-n⁢αk-αj- satisfies σ⁢(x)=-x. Then

〈x,𝔛Ό〉=0(6.10)

for all Ό<Μ.

Proof.

The space U-n⁢αk-αj- is spanned by elements of the form Fka⁢Fj⁢Fkb with a+b=n. As the antiautomorphism σ is involutive it is enough to verify equation (6.10) for elements of the form x=Fka⁢Fj⁢Fkb-σ⁢(Fka⁢Fj⁢Fkb)=Fka⁢Fj⁢Fkb-Fkb⁢Fj⁢Fka. We will prove that

〈Fka⁢Fj⁢Fkb-Fkb⁢Fj⁢Fka,𝔛Ό〉=0(6.11)

for all ÎŒ<Îœ and a,b≥0 by induction on n=a+b. It holds for n=0. Let a+b=n>0, and assume that (6.11) holds for all a′,b′ with a′+b′<n. Without loss of generality assume that b>0. Using the assumption that 𝔛Ό satisfies (6.2) and (6.3), we get that

〈FkaFjFkb-FkbFjFka,𝔛Ό〉=-1qk-qk-1(〈FkaFjFkb-1,rk(𝔛Ό)〉-〈Fkb-1FjFka,rk(𝔛Ό)〉)=〈Fka⁢Fj⁢Fkb-1,𝔛Ό-2⁢αk⁢ck⁢Xk¯+sk¯⁢𝔛Ό-αk〉-〈Fkb-1⁢Fj⁢Fka,q(αk,αk)⁢ck⁢Xk⁢𝔛Ό-2⁢αk+sk⁢𝔛Ό-αk〉.

The assumption Θ⁢(αk)=-αk implies that Xk=-Ek and by (5.16) and (5.15) one has ck¯=q(αk,αk)⁢ck and sk¯=sk. Hence the above equation turns into

〈Fka⁢Fj⁢Fkb-Fkb⁢Fj⁢Fka,𝔛Ό〉=sk⁢〈Fka⁢Fj⁢Fkb-1-Fkb-1⁢Fj⁢Fka,𝔛Ό-αk〉-ck¯(〈FkaFjFkb-1,𝔛Ό-2⁢αkEk〉-〈Fkb-1FjFka,Ek𝔛Ό-2⁢αk〉).

By the induction hypothesis one has 〈Fka⁢Fj⁢Fkb-1-Fkb-1⁢Fj⁢Fka,𝔛Ό-αk〉=0. Hence,

〈Fka⁢Fj⁢Fkb-Fkb⁢Fj⁢Fka,𝔛Ό〉=ck¯qk-qk-1⁢〈rk⁢(Fka⁢Fj⁢Fkb-1)-rk⁢(Fkb-1⁢Fj⁢Fka),𝔛Ό-2⁢αk〉.

As

σ⁢(rk⁢(Fka⁢Fj⁢Fkb-1)-rk⁢(Fkb-1⁢Fj⁢Fka))⁢=(2.17)⁢rk⁢(σ⁢(Fka⁢Fj⁢Fkb-1))-rk⁢(σ⁢(Fkb-1⁢Fj⁢Fka))=-(rk⁢(Fka⁢Fj⁢Fkb-1)-rk⁢(Fkb-1⁢Fj⁢Fka)),

equation (6.11) follows from the induction hypothesis. ∎

6.4 Constructing 𝔛Ό

We are now ready to construct 𝔛Ό inductively. Fix Ό∈Q+ and assume that a collection (𝔛Ό′)Ό′<Ό∈Q+ with 𝔛Ό′∈UΌ′+ and 𝔛0=1 has already been constructed and that this collection satisfies conditions (6.2) and (6.3) for all Ό′<ÎŒ and for all i∈I. Define

Ai=-(qi-qi-1)⁢(𝔛Ό+Θ⁢(αi)-αi⁢ci⁢Xi¯+si¯⁢𝔛Ό-αi),(6.12)Ai=-(qi-qi-1)⁢(q-(Θ⁢(αi),αi)⁢ci⁢Xi⁢𝔛Ό+Θ⁢(αi)-αi+si⁢𝔛Ό-αi)(6.13)

for all i∈I. We will keep the above assumptions and the definition of Ai and Ai all through this subsection. We will prove that the elements Ai and Ai, which are the right hand sides of equations (6.2) and (6.3), satisfy conditions (6.6) and (6.7). By Proposition 6.3 this will prove the existence of an element 𝔛Ό with the desired properties.

Lemma 6.7.

The relation ri⁢(Aj)=rj⁢(Ai) holds for all i,j∈I.

Proof.

This is a direct calculation. Note that all computations include the case i=j. We expand both sides of the desired equation, using (2.8) and (2.9) and the assumption that the elements 𝔛Ό′ satisfy (6.2) and (6.3) for Ό′<ÎŒ. We obtain

ri(jA)=-(qj-qj-1)q-(Θ⁢(αj),αj)cjXjri(𝔛Ό+Θ⁢(αj)-αj)-(qj-qj-1)⁢q-(Θ⁢(αj),αj)⁢q(αi,ÎŒ+Θ⁢(αj)-αj)⁢ri⁢(cj⁢Xj)⁢𝔛Ό+Θ⁢(αj)-αj-(qj-qj-1)⁢sj⁢ri⁢(𝔛Ό-αj)=(qj-qj-1)⁢q-(Θ⁢(αj),αj)⁢cj⁢Xj⁢(qi-qi-1)⁢𝔛Ό+Θ⁢(αj)-αj+Θ⁢(αi)-αi⁢ci⁢Xi¯+(qj-qj-1)⁢q-(Θ⁢(αj),αj)⁢cj⁢Xj⁢(qi-qi-1)⁢si¯⁢𝔛Ό+Θ⁢(αj)-αj-αi-(qj-qj-1)⁢q-(Θ⁢(αj),αj)⁢q(αi,ÎŒ+Θ⁢(αj)-αj)⁢ri⁢(cj⁢Xj)⁢𝔛Ό+Θ⁢(αj)-αj+(qj-qj-1)⁢sj⁢(qi-qi-1)⁢𝔛Ό-αj+Θ⁢(αi)-αi⁢ci⁢Xi¯+(qj-qj-1)⁢sj⁢(qi-qi-1)⁢si¯⁢𝔛Ό-αi-αj,

and

rj⁢(Ai)=-(qi-qi-1)⁢rj⁢(𝔛Ό+Θ⁢(αi)-αi)⁢ci⁢Xi¯-(qi-qi-1)⁢q(αj,ÎŒ+Θ⁢(αi)-αi)⁢𝔛Ό+Θ⁢(αi)-αi⁢rj⁢(ci⁢Xi¯)-(qi-qi-1)⁢si¯⁢rj⁢(𝔛Ό-αi)=(qi-qi-1)⁢(qj-qj-1)⁢q-(Θ⁢(αj),αj)⁢cj⁢Xj⁢𝔛Ό+Θ⁢(αi)-αi+Θ⁢(αj)-αj⁢ci⁢Xi¯+(qi-qi-1)⁢(qj-qj-1)⁢sj⁢𝔛Ό+Θ⁢(αi)-αi-αj⁢ci⁢Xi¯-(qi-qi-1)⁢q(αj,ÎŒ+Θ⁢(αi)-αi)⁢𝔛Ό+Θ⁢(αi)-αi⁢rj⁢(ci⁢Xi¯)+(qi-qi-1)⁢si¯⁢(qj-qj-1)⁢q-(Θ⁢(αj),αj)⁢cj⁢Xj⁢𝔛Ό-αi+Θ⁢(αj)-αj+(qi-qi-1)⁢si¯⁢(qj-qj-1)⁢sj⁢𝔛Ό-αi-αj.

We see that the first and fifth summands in the above expansions of ri(jA) and rj⁢(Ai) coincide, the second summand of ri(jA) is the same as the fourth summand of rj⁢(Ai), and the fourth summand of ri(jA) coincides with the second summand of rj⁢(Ai). Therefore, the claim of the lemma, ri(jA)=rj(Ai), is equivalent to the third summands being equal,

-(qj-qj-1)⁢q-(Θ⁢(αj),αj)⁢q(αi,ÎŒ+Θ⁢(αj)-αj)⁢ri⁢(cj⁢Xj)⁢𝔛Ό+Θ⁢(αj)-αj(6.14)=-(qi-qi-1)⁢q(αj,ÎŒ+Θ⁢(αi)-αi)⁢𝔛Ό+Θ⁢(αi)-αi⁢rj⁢(ci⁢Xi¯).

By (5.10) and (2.19) we may assume that i=τ⁢(j)∈I∖X because otherwise both sides of the above equation vanish. By (2.19) we have rj⁢(ci⁢Xi¯)=q(αj,-Θ⁢(αi)-αj)⁢rj⁢(ci⁢Xi)¯. Substituting this and using qi=qj, we see that (6.14) is equivalent to

q(αj,Θ⁢(αi)-Θ⁢(αj))+(αi,ÎŒ-αj)⁢ri⁢(cj⁢Xj)⁢𝔛Ό+Θ⁢(αj)-αj(6.15)=q(αj,ÎŒ-αi-αj)⁢𝔛Ό+Θ⁢(αi)-αi⁢rj⁢(ci⁢Xi)¯.

By equation (5.3) one has Θ⁢(αi)-αi=Θ⁢(αj)-αj and hence 𝔛Ό+Θ⁢(αj)-αj=𝔛Ό+Θ⁢(αi)-αi. Moreover, ri⁢(TwX⁢(Ei)) lies in ℳX and hence it commutes with 𝔛Ό+Θ⁢(αi)-αi. Using this, we can rewrite (6.15) as

q(αj,Θ⁢(αi-αj))+(αi,ÎŒ)⁢𝔛Ό+Θ⁢(αi)-αi⁢ri⁢(cj⁢Xj)(6.16)=q(αj,ÎŒ-αj)⁢𝔛Ό+Θ⁢(αi)-αi⁢rj⁢(ci⁢Xi)¯.

If 𝔛Ό+Θ⁢(αi)-αi=0, then both sides of the above equation vanish. Hence we assume that 𝔛Ό+Θ⁢(αi)-αi is nonzero. Corollary 6.2 states that then Θ⁢(ÎŒ)=-ÎŒ. Along with

Θ⁢(αi-αj)=αi-αj,

this implies that (αi-αj,Ό)=0. Hence (6.16) is equivalent to the relation

q(αj,αi)⁢ri⁢(cj⁢Xj)=rj⁢(ci⁢Xi)¯.

Using the definition (5.11) of 𝒵i the above formula follows from assumption (5.14) about the parameters 𝐜. ∎

This proves that the elements Ai,Ai satisfy the first condition from Proposition 6.3 (2). Next we prove that they also satisfy the second condition.

Lemma 6.8.

For all i≠j∈I the elements Ai,Aj given by (6.12) satisfy the relation

-1qi-qi-1⁢∑s=11-ai⁢j[1-ai⁢js]qi⁢(-1)s⁢〈Fi1-ai⁢j-s⁢Fj⁢Fis-1,Ai〉(6.17)-1qj-qj-1⁢〈Fi1-ai⁢j,Aj〉=0.

Proof.

We may assume that ÎŒ=(1-ai⁢j)⁢αi+αj and that Θ⁢(ÎŒ)=-ÎŒ, as otherwise all terms in the above sum vanish. By Lemma 6.4 it suffices to consider the following two cases.

Case 1: Θ⁢(𝛂𝐢)=-𝛂𝐣 and 𝐚𝐢⁢𝐣=0 In this case ÎŒ=αi+αj and si=sj=0 by definition (5.7) of the parameter set 𝒮. Hence

Ai=-s⁢(j)⁢(qi-qi-1)⁢ci¯⁢Ej and Aj=-s⁢(i)⁢(qj-qj-1)⁢cj¯⁢Ei.

Therefore the left hand side of (6.17) is equal to

1qi-qi-1⁢〈Fj,Ai〉-1qj-qj-1⁢〈Fi,Aj〉=-s⁢(j)⁢ci¯⁢〈Fj,Ej〉+s⁢(i)⁢cj¯⁢〈Fi,Ei〉.(6.18)

Using qi=qj, the fact that s⁢(i)=s⁢(j) by (5.2), and the relation ci=cj which holds by definition of the parameter set 𝒞, one sees that the right hand side of (6.18) vanishes.

Case 2: Θ⁢(𝛂𝐢)=-𝛂𝐢 and Θ⁢(𝛂𝐣)=-𝛂𝐣 In this case by (5.5) one has i,j∈Ins. Hence, by the definition (5.7) of the parameter set 𝒮, one has either sj=0 or ai⁢j∈-2⁢ℕ0. If sj=0, then Lemma 6.5 implies that (1-ai⁢j)⁢αi+αj=Ό∈ℕ0⁢αi⊕2⁢ℕ0⁢αj, which is not the case. If -ai⁢j is even, then the left hand side of (6.17) can be written as

1qi-qi-1⁢〈Fj⁢Fi-ai⁢j,Ai〉-1qj-qj-1⁢〈Fi1-ai⁢j,Aj〉(6.19)+-1qi-qi-1⁢∑s=1-ai⁢j/2[1-ai⁢js]qi⁢(-1)s⋅〈Fi1-ai⁢j-s⁢Fj⁢Fis-1-Fis⁢Fj⁢Fi-ai⁢j-s,Ai〉.

By (6.12) and (6.13) one has Aj=-(qj-qj-1)-1⁢sj⁢𝔛Ό-αj=Aj and hence

1qi-qi-1⁢〈Fj⁢Fi-ai⁢j,Ai〉-1qj-qj-1⁢〈Fi1-ai⁢j,Aj〉=-1qi-qi-1⁢1qj-qj-1⁢〈Fi-ai⁢j,rj⁢(Ai)-ri⁢(Aj)〉.

In view of Lemma 6.7 the above relation shows that the sum of the first two terms of (6.19) vanishes. Each of the remaining summands in (6.19) contains a factor of the form

〈Fi1-ai⁢j-s⁢Fj⁢Fis-1-Fis⁢Fj⁢Fi-ai⁢j-s,Ai〉(6.20)=-1qi-qi-1⁢〈Fi-ai⁢j-s⁢Fj⁢Fis-1-Fis-1⁢Fj⁢Fi-ai⁢j-s,ri⁢(Ai)〉.

Set x=Fi-ai⁢j-s⁢Fj⁢Fis-1-Fis-1⁢Fj⁢Fi-ai⁢j-s and observe that σ⁢(x)=-x. Inserting the definition of Ai into (6.20) one obtains in view of Xi=-Ei the relation

〈Fi1-ai⁢j-s⁢Fj⁢Fis-1-Fis⁢Fj⁢Fi-ai⁢j-s,Ai〉=〈x,ri⁢(-𝔛Ό-2⁢αi⁢ci¯⁢Ei+si¯⁢𝔛Ό-αi)〉.

Using the skew derivation property (2.9) and the assumption that 𝔛Ό′ satisfies (6.3) for all Ό′<ÎŒ one obtains

〈Fi1-ai⁢j-s⁢Fj⁢Fis-1-Fis⁢Fj⁢Fi-ai⁢j-s,Ai〉=-(qi-qi-1)⁢〈x,ci¯⁢q(αi,αi)⁢ci⁢Ei⁢𝔛Ό-4⁢αi⁢Ei〉+(qi-qi-1)⁢〈x,si⁢ci¯⁢𝔛Ό-3⁢αi⁢Ei〉-〈x,ci¯⁢q(αi,ÎŒ-2⁢αi)⁢𝔛Ό-2⁢αi〉+(qi-qi-1)⁢〈x,si⁢q(αi,αi)⁢ci⁢Ei⁢𝔛Ό-3⁢αi〉-(qi-qi-1)⁢〈x,si2⁢𝔛Ό-2⁢αi〉.

Using relations (2.15) and the property ci¯=q(αi,αi)⁢ci which holds by (5.16), the above equation becomes

〈Fi1-ai⁢j-s⁢Fj⁢Fis-1-Fis⁢Fj⁢Fi-ai⁢j-s,Ai〉(6.21)=-ci¯⁢q(αi,αi)⁢ci⁢1qi-qi-1⁢〈ri⁢(ri⁢(x)),𝔛Ό-4⁢αi〉-ci¯⁢q(αi,ÎŒ-2⁢αi)⁢〈x,𝔛Ό-2⁢αi〉-si⁢ci¯⁢〈ri⁢(x)+ri⁢(x),𝔛Ό-3⁢αi〉-(qi-qi-1)⁢si2⁢〈x,𝔛Ό-2⁢αi〉.

Using the fact that σ⁢(x)=-x we obtain from (2.17) and (2.14) that

σ⁢(ri⁢(ri⁢(x)))=-ri⁢(ri⁢(x)),σ⁢(ri⁢(x)+ri⁢(x))=-(ri⁢(x)+ri⁢(x)).

By Lemma 6.6 the above relations imply that all terms in (6.21) vanish. Therefore all summands in (6.19) vanish, which completes the proof of the Lemma in the second case. ∎

Remark 6.9.

If one restricts to quantum symmetric pair coideal subalgebras B𝐜,𝐬 with 𝐬=(0,0,
,0), then Case 2 in the proof of Lemma 6.8 simplifies significantly and Lemma 6.6 is not needed.

6.5 Constructing 𝔛

We are now ready to prove the main result of this section, namely the existence of the quasi K-matrix 𝔛. Recall the assumptions from Section 5.4.

Theorem 6.10.

There exists a uniquely determined element X=∑Ό∈Q+XΌ∈U+^, with X0=1 and XΌ∈UÎŒ+, such that the equality

x¯B⁢𝔛=𝔛⁢x¯(6.22)

holds in U for all x∈Bc,s.

Proof.

We construct 𝔛Ό by induction on the height of ÎŒ, starting from 𝔛0=1. If ÎŒ=αj, then equations (6.2) and (6.3) are equivalent to

ri⁢(𝔛Ό)=ri⁢(𝔛Ό)={0if i≠j,-(qi-qi-1)⁢siif i=j,

as sj=s¯j by (5.15). In this case 𝔛αj=-(qj-qj-1)⁢sj⁢Ej satisfies (6.2) and (6.3). This defines 𝔛Ό in the case ht⁢(ÎŒ)=1. Assume now that ht⁢(ÎŒ)≥2 and that the elements 𝔛Ό′ have been defined for all Ό′ with ht⁢(Ό′)<ht⁢(ÎŒ) such that they satisfy (6.2) and (6.3) for all i∈I. The elements Ai and Ai given by (6.12) and (6.13), respectively, are then well defined, and by Lemmas 6.7 and 6.8 they satisfy the conditions of Proposition 6.3 (2). By Proposition 6.3 the system of equations given by (6.5) for all i∈I has a unique solution X¯=𝔛Ό∈UÎŒ+. By the definition of Ai and Ai the element 𝔛Ό satisfies equations (6.2) and (6.3).

Set

𝔛=∑Ό∈Q+𝔛Ό∈U+^.

By Proposition 6.1 the element 𝔛 satisfies the relation (6.22) for all x∈B𝐜,𝐬. The uniqueness of 𝔛 follows by Propositions 6.1 and 6.3 from the uniqueness of the solution of the system of equations given by (6.5) for all i∈I. ∎

7 Construction of the universal K-matrix

Using the quasi K-matrix 𝔛 from the previous section we now construct a candidate 𝒊∈𝒰 for a universal K-matrix as in Definition 4.12. Our approach is again inspired by the special case considered in [3]. However, we are aiming for a comprehensive construction for all quantum symmetric Kac–Moody pairs. In this setting the Weyl group does not contain a longest element. We hence replace the Lusztig action in [3, Theorem 2.18] by a twist of the underlying module, see Section 7.1. In Section 7.2 we construct a B𝐜,𝐬-module homomorphism between twisted versions of modules in 𝒪int. This provides the main step of the construction in the general Kac–Moody case. In Section 7.3 we restrict to the finite case and obtain a ℬ-tw-automorphism 𝒊 for 𝒪int as in Section 4.2 with ℬ as in Example 4.1. The coproduct of 𝒊 will be determined in Section 9.

7.1 A pseudo longest element of W

If 𝔀 is of finite type, then there exists τ0∈Aut⁢(A) such that the longest element w0∈W satisfies

w0⁢(αi)=-ατ0⁢(i)  for all i∈I.(7.1)

Moreover, in this case the Lusztig automorphism Tw0 of Uq⁢(𝔀) corresponding to w0 can be explicitly calculated. Indeed, by [13, Proposition 8.20] or [18, Lemma 3.4] one has

Tw0⁢(Ei)=-Fτ0⁢(i)⁢Kτ0⁢(i),Tw0⁢(Fi)=-Kτ0⁢(i)-1⁢Eτ0⁢(i),Tw0⁢(Ki)=Kτ0⁢(i)-1,(7.2)Tw0-1⁢(Ei)=-Kτ0⁢(i)-1⁢Fτ0⁢(i),Tw0-1⁢(Fi)=-Eτ0⁢(i)⁢Kτ0⁢(i),Tw0-1⁢(Ki)=Kτ0⁢(i)-1.

In the Kac–Moody case we mimic the inverse of the Lusztig automorphism corresponding to the longest element of the Weyl group as follows. Let tw:Uq⁢(𝔀)→Uq⁢(𝔀) denote the algebra automorphism defined by

tw⁢(Ei)=-Ki-1⁢Fi,tw⁢(Fi)=-Ei⁢Ki,tw⁢(Kh)=K-h

for all i∈I, h∈Qext√.

Lemma 7.1.

For all i∈I one has tw∘Ti=Ti∘tw on Uq⁢(g).

Proof.

For h∈Qext√ one has Ti∘tw⁢(Kh)=K-si⁢(h)=tw∘Ti⁢(Kh). It remains to check that

Ti∘tw⁢(Ej)=tw∘Ti⁢(Ej) and Ti∘tw⁢(Fj)=tw∘Ti⁢(Fj)(7.3)

for all j∈I. For j=i relation (7.3) holds because Ti-1|Uqi⁢(𝔰⁢𝔩2)i=tw|Uqi⁢(𝔰⁢𝔩2)i. For j≠i relation (7.3) is verified by a direct calculation using the formulas

Ti⁢(Ej)=∑k=0-ai⁢j(-1)k⁢qi-k⁢Ei(-ai⁢j-k)⁢Ej⁢Ei(k),Ti⁢(Fj)=∑k=0-ai⁢j(-1)k⁢qik⁢Fi(k)⁢Fj⁢Fi(-ai⁢j-k)

which hold by [26, Section 37.1.3]. ∎

To mimic the Lusztig action of the longest element in the Kac–Moody case we additionally need an automorphism τ0∈Aut⁢(A,X). Recall our setting and assumptions from Section 5.4. For the construction of the universal K-matrix we need to make minor additional assumptions on the parameters 𝐜∈𝒞 and 𝐬∈𝒮.

Assumption (τ0)

We are given an additional involutive element τ0∈Aut⁢(A,X) with the following properties:

  • (1)

    τ∘τ0=τ0∘τ.

  • (2)

    The parameters 𝐜∈𝒞 and 𝐬∈𝒮 satisfy the relations

    cτ0⁢τ⁢(i)=ci,sτ0⁢(i)=si for all i∈I∖X.(7.4)

  • (3)

    The function s:I→𝕂 described by (5.1) and (5.2) satisfies the relation

    s⁢(τ⁢(i))=s⁢(τ0⁢(i)) for all i∈I.(7.5)

Remark 7.2.

Assume that 𝔀 is of finite type. In this case we always choose τ0 to be the diagram automorphism determined by equation (7.1). Then property (1) is automatically satisfied as follows by inspection from the list of Satake diagrams in [1]. Moreover, by the definition of the parameter set 𝒮 one can have si≠0 only if τ′⁢(i)=i for all τ′∈Aut⁢(A). Hence property (2) reduces to cτ0⁢τ⁢(i)=ci in the finite case. By (5.1) and (5.2) one can have s⁢(i)≠1 only if τ⁢(i)=τ0⁢(i). Hence property (3) is always satisfied in the finite case.

If τ0=τ, then property (2) is an empty statement. It is possible that τ=id and τ0≠id, see the list in [1]. In this case condition (5.16) implies that ci equals cτ0⁢(i) up to multiplication by a bar invariant scalar. The new condition cτ0⁢τ⁢(i)=ci forces this scalar to be equal to 1. Finally, only in type D2⁢n is it possible that τ0=id and τ≠id. In this case, however, the condition 𝐜∈𝒞 implies that cτ0⁢τ⁢(i)=ci. These arguments show that the new condition cτ0⁢τ⁢(i)=ci is consistent with the conditions imposed in Section 5.4 and that it is always possible to choose parameters 𝐜 and 𝐬 which satisfy all of the assumptions.

The composition

tw∘τ0:Uq⁢(𝔀′)→Uq⁢(𝔀′)

defines an algebra automorphism. By (7.2) the automorphism tw∘τ0 is a Kac–Moody analog of the inverse of the Lusztig action on Uq⁢(𝔀) corresponding to the longest element in the Weyl group in the finite case. As τ0∈Aut⁢(A,X), one has τ0∘TwX=TwX∘τ0. By Lemma 7.1 this implies that

τ0∘tw∘TwX=TwX∘τ0∘tw.(7.6)

To obtain an analog of this Lusztig action on modules in 𝒪int we will twist the module structure. In the following subsection we construct a B𝐜,𝐬-module homomorphism between twisted versions of modules in 𝒪int. As B𝐜,𝐬 is a subalgebra of Uq⁢(𝔀′), it suffices to consider objects in 𝒪int as Uq⁢(𝔀′)-modules. With this convention, for any algebra automorphism φ:Uq⁢(𝔀′)→Uq⁢(𝔀′) and any M∈Ob⁢(𝒪int) let Mφ denote the vector space M with the Uq⁢(𝔀′)-module structure u⊗m↩u∙φm given by

u∙φm=φ⁢(u)⁢m for all u∈Uq⁢(𝔀′), m∈M.

We will apply this notation in particular in the case where φ is one of τ0∘τ and tw∘τ0, see Theorem 7.5.

Remark 7.3.

If the algebra automorphism φ:Uq⁢(𝔀′)→Uq⁢(𝔀′) extends to a Hopf algebra automorphism of Uq⁢(𝔀), then the notation Mφ for M∈Ob⁢(𝒪int) coincides with the notation in Example 4.7.

7.2 The twisted universal K-matrix in the Kac–Moody case

We keep our assumptions from Section 5.4 and Assumption (τ0) from the previous subsection. To construct the desired B𝐜,𝐬-module homomorphism we require one additional ingredient. Consider the function γ:I→𝕂⁢(q1/d) defined by

γ⁢(i)={1if i∈X,ci⁢s⁢(τ⁢(i))if i∈I∖X,(7.7)

and note that by (7.4) and (7.5) one has γ⁢(τ⁢τ0⁢(i))=γ⁢(i) for all i∈I. Now assume that Ο:P→𝕂⁢(q1/d)× is a function satisfying the following recursion:

Ο⁢(ÎŒ+αi)=γ⁢(i)⁢q-(αi,Θ⁢(αi))-(ÎŒ,αi+Θ⁢(αi))⁢Ο⁢(ÎŒ) for all Ό∈P, i∈I.(7.8)

Such a function exists. Indeed, we may take an arbitrary map on any set of representatives of P/Q and uniquely extend it to P using (7.8).

Lemma 7.4.

Let Ο:P→K⁢(q1/d)× be any function which satisfies the recursion (7.8). Then one has

Ο⁢(ÎŒ+λ)=q-(λ,λ)-2⁢(ÎŒ,λ)⁢Ο⁢(ÎŒ) for all Ό∈P, λ∈QX.(7.9)

Proof.

We prove this by induction on the height of λ. Assume that (7.9) holds for a given λ∈QX. Then one obtains for any i∈X the relation

Ο⁢(Ό+λ+αi)=q-(αi,αi)-2⁢(Ό+λ,αi)⁢Ο⁢(Ό+λ)=q-(αi,αi)-2⁢(Ό+λ,αi)-(λ,λ)-2⁢(Ό,λ)⁢Ο⁢(Ό)=q-(λ+αi,λ+αi)-2⁢(Ό,λ+αi)⁢Ο⁢(Ό)

which completes the induction step. ∎

As in Example 3.3 we may consider Ο as an element of 𝒰. The next theorem shows that the element 𝔛⁢Ο⁢TwX-1∈𝒰 defines a B𝐜,𝐬-module isomorphism between twisted modules in 𝒪int.

Theorem 7.5.

Let Ο:P→K⁢(q1/d)× be a function satisfying the recursion (7.8). Then the element K′=X⁢Ο⁢TwX-1∈U defines an isomorphism of Bc,s-modules

𝒊M′:Mtw∘τ0→Mτ0⁢τ,m↊𝔛M∘ΟM∘(TwX-1)M⁢(m)

for any M∈Ob⁢(Oint). In other words, the relation

𝒊′⁢tw⁢(τ0⁢(x))=τ0⁢(τ⁢(x))⁢𝒊′

holds in U for all x∈Bc,s.

Proof.

It suffices to check that

𝒊M′⁢(x∙tw∘τ0m)=x∙τ⁢τ0𝒊M′⁢(m) for all m∈M(7.10)

where x is one of the elements Kλ, Ei, Fi, or Bj for λ∈QΘ, i∈X, and j∈I∖X. Moreover, it suffices to prove the above relation for a weight vector m∈MÎŒ. In the following we will suppress the subscript M for elements in 𝒰 acting on M.

Case 1: 𝐱=𝐊𝛌 for some 𝛌∈𝐐Θ In this case we have wX⁢(λ)=-τ⁢(λ). Moreover, as τ0∈Aut⁢(A,X), one has τ0⁢(wX⁢(λ))=wX⁢(τ0⁢(λ)). Hence one obtains

𝒊′⁢(Kλ∙tw∘τ0m)=𝔛∘Ο∘TwX-1⁢(tw⁢(τ0⁢(Kλ))⁢m)=𝔛∘Ο∘TwX-1⁢(K-τ0⁢(λ)⁢m)=𝔛⁢(K-wX⁢τ0⁢(λ)⁢(Ο∘TwX-1⁢(m)))=K-wX⁢τ0⁢(λ)⁢𝔛∘Ο∘TwX-1⁢(m)=Kτ0⁢τ⁢(λ)⁢𝒊′⁢(m)=Kλ∙τ0⁢τ𝒊′⁢(m).

Case 2: 𝐱=𝐄𝐢 for some 𝐢∈𝐗 By relation (7.2) applied to ℳX we have

TwX-1⁢(Fi)=-Eτ⁢(i)⁢Kτ⁢(i).

Using this and the recursion (7.8) one obtains

𝒊′⁢(Ei∙tw∘τ0m)⁢=⁢𝔛∘Ο∘TwX-1⁢(-Kτ0⁢(i)-1⁢Fτ0⁢(i)⁢m)=⁢𝔛∘Ο⁢(q(αi,αi)⁢Eτ⁢τ0⁢(i)⁢Kτ⁢τ0⁢(i)2⁢TwX-1⁢(m))=𝔛(Ο(wX(ÎŒ)+ατ⁢τ0⁢(i))Ο(wX(ÎŒ))-1⋅q(αi,αi)+2⁢(wX⁢(ÎŒ),ατ0⁢τ⁢(i))Eτ⁢τ0⁢(i)Ο∘TwX-1(m))=(7.8)⁢Eτ⁢τ0⁢(i)⁢𝔛∘Ο∘TwX-1⁢(m)=⁢Ei∙τ0⁢τ𝒊′⁢(m).

This confirms relation (7.10) for x=Ei where i∈X. The case x=Fi for i∈X is treated analogously.

Case 3: 𝐱=𝐁𝐣=𝐅𝐣-𝛄⁢(𝐣)⁢𝐓𝐰𝐗⁢(𝐄𝛕⁢(𝐣))⁢𝐊𝐣-1+𝐬𝐣⁢𝐊𝐣-1 for some 𝐣∈𝐈∖𝐗 We calculate

𝒊′⁢(Bj∙tw∘τ0m)⁢=⁢𝔛∘Ο∘TwX-1⁢(tw∘τ0⁢(Bj)⁢m)=(7.6)⁢𝔛∘Ο⁢(tw∘τ0⁢(TwX-1⁢(Bj))⁢TwX-1⁢(m))=(7.6)𝔛∘Ο((TwX-1(-Eτ0⁢(j)Kτ0⁢(j))+γ(j)Kτ⁢τ0⁢(j)-1Fτ⁢τ0⁢(j)KwX⁢(ατ0⁢(j))+sjKwX⁢(τ0⁢(j)))TwX-1(m))=𝔛∘Ο((Fτ⁢τ0⁢(j)γ(j)q(αj,αj)-(wX⁢(ÎŒ),ατ⁢τ0⁢(j)+Θ⁢(ατ⁢τ0⁢(j)))-TwX-1⁢(Eτ0⁢(j))⁢Kτ0⁢τ⁢(j)⁢q-(wX⁢(ÎŒ),ατ0⁢τ⁢(j)+Θ⁢(ατ0⁢τ⁢(j)))+sjKτ0⁢τ⁢(j)q-(wX⁢(ÎŒ),ατ0⁢τ⁢(j)+Θ⁢(ατ0⁢τ⁢(j))))TwX-1(m)).

To simplify the last term, recall from (5.7) that sτ0⁢τ⁢(j)=sj=0 unless Θ⁢(αj)=-αj, in which case ατ0⁢τ⁢(j)+Θ⁢(ατ0⁢τ⁢(j))=0. Additionally moving Ο to the right one obtains

𝒊′⁢(Bj∙tw∘τ0m)(7.11)=𝔛((Fτ⁢τ0⁢(j)γ(j)q(αj,αj)-(wX⁢(ÎŒ),ατ⁢τ0⁢(j)+Θ⁢(ατ⁢τ0⁢(j)))Ο⁢(wX⁢(ÎŒ)-ατ0⁢τ⁢(j))Ο⁢(wX⁢(ÎŒ))-TwX-1⁢(Eτ0⁢(j))⁢Kτ0⁢τ⁢(j)⁢q-(wX⁢(ÎŒ),ατ0⁢τ⁢(j)+Θ⁢(ατ0⁢τ⁢(j)))⁢Ο⁢(wX⁢(ÎŒ)+wX⁢ατ0⁢(j))Ο⁢(wX⁢(ÎŒ))+sτ0⁢τ⁢(j)Kτ0⁢τ⁢(j))Ο∘TwX-1(m)).

To simplify the above expression observe that

Ο⁢(wX⁢(ÎŒ)+wX⁢αi)⁢=(7.9)⁢q-(wX⁢αi-αi,wX⁢αi-αi)-2⁢(wX⁢(ÎŒ)+αi,wX⁢αi-αi)⁢Ο⁢(wX⁢(ÎŒ)+αi)=⁢q-2⁢(wX⁢(ÎŒ),wX⁢αi-αi)⁢Ο⁢(wX⁢(ÎŒ)+αi)=(7.8)⁢γ⁢(τ0⁢τ⁢(i))⁢q-(αi,Θ(αi))+(wX(ÎŒ),ατ⁢(i)+Θ(ατ⁢(i))⁢Ο⁢(wX⁢(ÎŒ))

for i∈I∖X. Inserting this formula for i=τ0⁢(j) into equation (7.11) and applying the recursion (7.8) also to the first summand one obtains

𝒊′(Bj∙tw∘τ0m)=𝔛((Fτ⁢τ0⁢(j)-γ(τ(j))q-(αj,Θ⁢(αj))TwX-1(Eτ0⁢(j))Kτ0⁢τ⁢(j)(7.12)+sτ0⁢τ⁢(j)Kτ0⁢τ⁢(j))Ο∘TwX-1(m)).

Now set

βi=(-1)2⁢αi⁢(ρX√)⁢q(2⁢ρX,αi) for i∈I∖X.

In view of [26, Section 37.2.4] one has

TwX⁢(Ei)¯=βi-1⁢TwX-1⁢(Ei) for all i∈I∖X,(7.13)

see also the proof of [2, Lemma 2.9]. Hence (7.12) gives

𝒊′(Bj∙tw∘τ0m)=𝔛((Fτ⁢τ0⁢(j)-γ(τ(j))q-(αj,Θ⁢(αj))βτ0⁢(j)TwX⁢(Eτ0⁢(j))⁢Kτ0⁢τ⁢(j)-1¯+sjKτ0τ(j)))Ο∘TwX-1(m)).

In view of the relation

γ⁢(τ⁢(j))⁢q-(αj,Θ⁢(αj))⁢βτ0⁢(j)=s⁢(τ⁢(j))⁢cj¯

one now obtains

𝒊′⁢(Bj∙tw∘τ0m)=𝔛⁢(Bτ0⁢τ⁢(j)¯⁢Ο∘TwX-1⁢(m))=Bτ0∘τ⁢(j)⁢𝒊′⁢(m)=Bj∙τ⁢τ0𝒊′⁢(m)

which completes the proof of the theorem. ∎

For later reference we note that relation (7.13) implies that the element Xi defined by (5.4) satisfies the relation

Xi¯=-s⁢(i)⁢q-(2⁢ρX,αi)⁢TwX-1⁢(Eτ⁢(i)) for all i∈I∖X,(7.14)

see also (5.1), (5.2) and property (3) in Definition 5.1 of an admissible pair.

Remark 7.6.

The function Ο is an important ingredient in the construction of the twisted K-matrix 𝒊′ and should be compared to the recursively defined function f involved in the construction of the commutativity isomorphisms [26, Section 32.1.3]. The recursion (7.8) is a necessary and sufficient condition on Ο for 𝔛∘Ο∘TwX-1:Mtw∘τ0→Mτ0⁢τ to be a B𝐜,𝐬-module homomorphism.

7.3 The universal K-matrix in the finite case

We now assume that 𝔀 is of finite type. In this case, following Remark 7.2, we always choose τ0∈Aut⁢(I,X) such that the longest element w0∈W satisfies w0⁢(αi)=-ατ0⁢(i) for all i∈I. By equation (7.2) this gives Tw0-1=τ0∘tw on Uq⁢(𝔀).

If M is a finite-dimensional Uq⁢(𝔀)-module, then the Lusztig action Tw0:M→M satisfies Tw0⁢(u⁢m)=Tw0⁢(u)⁢Tw0⁢(m) for all m∈M, u∈Uq⁢(𝔀). In other words, the Lusztig action on M defines an Uq⁢(𝔀)-module isomorphism

Tw0:Mtw∘τ0→M.

Composing the inverse of this isomorphism with the isomorphism 𝒊′ from Theorem 7.5, we get the following corollary.

Corollary 7.7.

Assume that g is of finite type and let Ο:P→K⁢(q1/d)× be a function satisfying the recursion (7.8). Then the element K=X⁢Ο⁢TwX-1⁢Tw0-1∈U defines an isomorphism of Bc,s-modules

𝒊M:M→Mτ⁢τ0,m↊𝔛M∘ΟM∘(TwX-1)M∘(Tw0-1)M⁢(m)

for any finite-dimensional Uq⁢(g)-module M. In other words, the relation

𝒊⁢b=τ0⁢(τ⁢(b))⁢𝒊

holds in U for all b∈Bc,s.

Remark 7.8.

As before let ℬ denote the category with objects in 𝒪int and morphisms Homℬ⁢(V,W)=HomB𝐜,𝐬⁢(V,W). In the terminology of Section 4.2 the above corollary states that 𝒊=𝔛⁢Ο⁢TwX-1⁢Tw0-1 is a ℬ-(τ∘τ0)-automorphism of 𝒪int. Equivalently, the element 𝒊 satisfies relation (4.15) in Definition 4.12 of a τ⁢τ0-universal K-matrix.

8 A special choice of Ο

In the following we want to determine the coproduct of the element 𝒊∈𝒰 from Corollary 7.7. We aim to show that 𝒊 is a τ⁢τ0-universal K-matrix for B𝐜,𝐬, that is that the coproduct Δ⁢(𝒊) is given by (4.16). This, however, will only hold true for a suitable choice of Ο.

8.1 Choosing Ο

Recall that Ο has to satisfy the recursion (7.8) which involves the function γ:I→𝕂⁢(q1/d) given by (7.7). Extend the function γ to a group homomorphism γ:P→𝕂⁢(q1/d)×. Depending on the choice of coefficients 𝐜∈𝒞, it may be necessary to replace 𝕂⁢(q1/d) by a finite extension to do this. We will illustrate the situation and comment on the field extension in Section 8.4.

For any λ∈P write

λ+=λ+Θ⁢(λ)2,λ~=λ-Θ⁢(λ)2.

Observe that both (λ+,λ+) and (λ~,λ~) are contained in 12⁢d⁢℀ for all λ∈P. Recall from Section 2.1 that ϖi√ for i∈I denote the fundamental coweights. Now define a function Ο:P→𝕂⁢(q1/d)× by

Ο⁢(λ)=γ⁢(λ)⁢q-(λ+,λ+)+∑k∈I(α~k,α~k)⁢λ⁢(ϖk√).(8.1)

Remark 8.1.

A priori one only has -(λ+,λ+)+∑k∈I(α~k,α~k)⁢λ⁢(ϖk√)∈12⁢d⁢℀. However, for all 𝔀 of finite type one can show by direct calculation that

-(λ+,λ+)+∑k∈I(α~k,α~k)⁢λ⁢(ϖk√)∈1d⁢℀ for all λ∈P.(8.2)

To this end it is useful to reformulate the above condition as

-(λ~,λ~)+∑k∈I(α~k,α~k)⁢λ⁢(ϖk√)∈1d⁢℀ for all λ∈P

and to work with the weight lattice P⁢(Σ) of the restricted root system Σ of the symmetric pair (𝔀,𝔚). The relation between P⁢(Σ) and P is discussed in detail in [25, Section 2]. We expect (8.2) also to hold for infinite-dimensional 𝔀. If it does not hold, then the definition of Ο requires an extension of 𝕂⁢(q1/d) also for the q-power to lie in the field.

We claim that Ο satisfies the recursion (7.8).

Lemma 8.2.

The function Ο:P→K⁢(q1/d)× defined by (8.1) satisfies the relation

Ο⁢(ÎŒ+Îœ)=Ο⁢(ÎŒ)⁢Ο⁢(Îœ)⁢q-(ÎŒ+Θ⁢(ÎŒ),Îœ)(8.3)

for all ÎŒ,Μ∈P. In particular, Ο satisfies the recursion (7.8).

Proof.

For any ÎŒ,Μ∈P one calculates

Ο⁢(ÎŒ+Îœ)=γ⁢(ÎŒ+Îœ)⁢q-((ÎŒ+Îœ)+,(ÎŒ+Îœ)+)+∑k∈I(α~k,α~k)⁢(ÎŒ+Îœ)⁢(ϖk√)=γ⁢(ÎŒ)⁢γ⁢(Îœ)⁢q-(ÎŒ+,ÎŒ+)-(Îœ+,Îœ+)-(ÎŒ,Îœ)-(Θ⁢(ÎŒ),Îœ)+∑k∈I(α~k,α~k)⁢(ÎŒ+Îœ)⁢(ϖk√)=Ο⁢(ÎŒ)⁢Ο⁢(Îœ)⁢q-(ÎŒ+Θ⁢(ÎŒ),Îœ)

which proves (8.3). Choosing Μ=αi one now obtains

Ο⁢(ÎŒ+αi)⁢=(8.3)⁢Ο⁢(ÎŒ)⁢Ο⁢(αi)⁢q-(ÎŒ+Θ⁢(ÎŒ),αi)=(8.1)⁢Ο⁢(ÎŒ)⁢γ⁢(i)⁢q-(αi+,αi+)+(α~i,α~i)-(ÎŒ,αi+Θ⁢(αi)).

As (αi+,αi+)-(α~i,α~i)=(αi,Θ⁢(αi)), the above formula implies that Ο satisfies the recursion (7.8). ∎

8.2 The coproduct of Ο

Recall the invertible element κ∈𝒰(2) defined in Example 3.4. Let f:P→P be any map. For every M,N∈Ob⁢(𝒪int) define a linear map

κM,Nf:M⊗N→M⊗N,(m⊗n)↩q(f⁢(ÎŒ),Îœ)⁢m⊗n if m∈MÎŒ, n∈NÎœ.(8.4)

As in Example 3.4 the collection κf=(κM,Nf)M,N∈Ob⁢(𝒪int) defines an element in 𝒰(2).

Remark 8.3.

In the following we will apply this notion in the case f=-Θ=wX∘τ, see Section 5.1. To this end we need to assume that the minimal realization (𝔥,Π,Π√) is compatible with the involution τ∈Aut⁢(I,X) as in [18, Section 2.6]. This means that the map τ:Π∹→Π∹ extends to a permutation τ:Πext∹→Πext√ such that ατ⁢(i)⁢(dτ⁢(s))=αi⁢(ds). In this case τ may be considered as a map τ:P→P. We will make this assumption without further comment. In the finite case, which is our only interest in Section 9, it is always satisfied.

Recall from Example 3.3 that the function Ο defined by (8.1) may be considered as an element in 𝒰 and hence we can take its coproduct, see Section 3.2. The coproduct Δ⁢(Ο)∈𝒰(2) can be explicitly determined.

Lemma 8.4.

The element Ο∈U defined by (8.1) satisfies the relation

Δ⁢(Ο)=(Ο⊗Ο)⋅κ-1⋅κ-Θ.(8.5)

Proof.

Let M,N∈Ob⁢(𝒪int) and m∈MÎŒ, n∈NÎœ for some ÎŒ,Μ∈P. Then m⊗n lies in the weight space (M⊗N)ÎŒ+Îœ. Hence one gets

Δ⁢(Ο)⁢(m⊗n)⁢=⁢Ο⁢(ÎŒ+Îœ)⁢m⊗n=(8.3)⁢Ο⁢(ÎŒ)⁢Ο⁢(Îœ)⁢q-(ÎŒ,Îœ)⁢q-(Θ⁢(ÎŒ),Îœ)⁢m⊗n=⁢(Ο⊗Ο)⋅κ-1⋅κ-Θ⁢(m⊗n)

which proves formula (8.5). ∎

For the rest of this paper the symbol Ο will always denote the function given by (8.1) and the corresponding element of 𝒰.

8.3 The action of Ο on Uq⁢(𝔀)

Conjugation by the invertible element Ο∈𝒰 gives an automorphism

Ad⁢(Ο):𝒰→𝒰,u↩Ad⁢(Ο)⁢(u)=Ο⁢u⁢Ο-1.(8.6)

For any M∈Ob⁢(𝒪int) one has

Ο⁢(u⁢m)=Ad⁢(Ο)⁢(u)⁢Ο⁢(m) for all u∈𝒰 and m∈M.

Recall that we consider Uq⁢(𝔀) as a subalgebra of 𝒰.

Lemma 8.5.

The automorphism (8.6) restricts to an automorphism of Uq⁢(g). More explicitly, one has

Ad⁢(Ο)⁢(EÎœ)=Ο⁢(Îœ)⁢EΜ⁢KÎœ+Θ⁢(Îœ)-1,(8.7)Ad⁢(Ο)⁢(FÎœ)=Ο⁢(Îœ)-1⁢KÎœ+Θ⁢(Îœ)⁢FÎœ,(8.8)Ad⁢(Ο)⁢(Ki)=Ki(8.9)

for all EΜ∈UÎœ+ and FΜ∈U-Îœ- and all i∈I.

Proof.

By definition the elements Ο and Ki commute in 𝒰. This proves (8.9). To verify the remaining two formulas let M∈Ob⁢(𝒪int) and m∈MÎŒ for some Ό∈P. Then one has

Ο⁢EΜ⁢Ο-1⁢m=Ο⁢(ÎŒ+Îœ)Ο⁢(ÎŒ)⁢EΜ⁢m⁢=(8.3)⁢Ο⁢(Îœ)⁢q-(ÎŒ+Θ⁢(ÎŒ),Îœ)⁢EΜ⁢m=Ο⁢(Îœ)⁢EΜ⁢KÎœ+Θ⁢(Îœ)-1⁢m

which proves formula (8.7). Formula (8.8) is obtained analogously using the relation

Ο⁢(Îœ)-1=Ο⁢(-Îœ)⁢q(Îœ+Θ⁢(Îœ),Îœ)

which also follows from (8.3). ∎

For 𝔀 of finite type the above lemma allows us to identify the restriction of Ad⁢(Ο) to the subalgebra ℳXUΘ0′ of Uq⁢(𝔀). Recall the conventions for the diagram automorphisms τ0 in the finite case from Remark 7.2.

Lemma 8.6.

Assume that g is of finite type. Then one has

Ad⁢(Ο)|ℳX⁢UΘ0=(Tw0⁢TwX⁢τ⁢τ0)|ℳX⁢UΘ0.

Proof.

Consider Ό∈QX+ and elements EΌ∈UÎŒ+ and FΌ∈U-ÎŒ-. By Lemma 8.5 and relations (7.2) one has

Ad⁢(Ο)⁢(EÎŒ)=q-(ÎŒ,ÎŒ)⁢EΌ⁢KÎŒ-2=Tw0⁢TwX⁢τ⁢τ0⁢(EÎŒ),Ad⁢(Ο)⁢(FÎŒ)=q(ÎŒ,ÎŒ)⁢KÎŒ2⁢FÎŒ=Tw0⁢TwX⁢τ⁢τ0⁢(FÎŒ).

Moreover, if Μ∈QΘ, then Lemma 8.5 implies that

Ad⁢(Ο)⁢(KÎœ)=KÎœ=Tw0⁢τ0⁢TwX⁢τ⁢(KÎœ)

which completes the proof of the lemma. ∎

8.4 Extending γ from Q to P

In this final subsection, we illustrate how different choices of 𝐜 and s influence the extension of the group homomorphism γ:Q→𝕂⁢(q1/d)× to the weight lattice P. As an example consider the root datum of type A3 with I={1,2,3}, that is 𝔀=𝔰⁢𝔩4⁢(ℂ), and the admissible pair (X,τ) given by X={2} and τ⁢(i)=4-i. In this case the constraints (5.16) and (5.2) reduce to the relations

c3=q2⁢c1¯,s⁢(3)=-s⁢(1).

The group homomorphism γ:Q→𝕂⁢(q1/d)× is defined by

γ⁢(α1)=s⁢(3)⁢c1,γ⁢(α2)=1,γ⁢(α3)=s⁢(1)⁢c3.

The weight lattice P is spanned by the fundamental weights

ϖ1=3⁢α1+2⁢α2+α34,ϖ2=α1+2⁢α2+α32,ϖ3=α1+2⁢α2+3⁢α34

and we want to extend γ from Q to P.

Choice 1

Let c1=c3=q, s⁢(1)=1, s⁢(3)=-1. Then γ⁢(α1)=-q, γ⁢(α3)=q, and γ can be extended to P by

γ⁢(ϖ1)=e3⁢π⁢i/4⁢q,γ⁢(ϖ2)=eπ⁢i/2⁢q,γ⁢(ϖ3)=eπ⁢i/4⁢q.

Choice 2

Let c1=1-q2, c3=q2-1, s⁢(1)=1, s⁢(3)=-1. Then

γ⁢(α1)=γ⁢(α3)=q2-1,

and γ can be extended to P by

γ⁢(ϖ1)=γ⁢(ϖ2)=γ⁢(ϖ3)=q2-1.

The advantage of Choice 1 is that the parameters ci specialize to 1 as q→1. This property is necessary to show that B𝐜,𝐬 specializes to U⁢(𝔚) for q→1, see [18, Section 10]. The drawback of Choice 1 is that, for γ to extend to P, the field 𝕂 must contain some 4-th root of -1. Choice 2, one the other hand, has the advantage that γ can be defined on P with values in ℚ⁢(q)×. The drawback of Choice 2 is that ci→0 as q→1 and hence B𝐜,𝐬 does not specialize to U⁢(𝔚).

For any quantum symmetric pair of finite type it is possible to find analogs of Choice 1 and Choice 2 above. We can choose ci=qai for some ai∈℀, see [2, Remark 3.14]. If X=∅ or τ=id, then γ extends to a group homomorphism P→𝕂⁢(q1/d) and no field extension is necessary. Now assume that X≠∅ and τ≠id. If we keep the choice ci=qai, then the extension of γ to P requires the field to contain certain roots of unity. Alternatively, as in Choice 2, one can choose ci∈{qai,(1-qbi)⁢qai} for some ai∈℀, bi∈℀ and s⁢(i)=±1 in such a way that γ can be extended from Q to P with values in ℚ⁢(q1/d)×.

9 The coproduct of the universal K-matrix 𝒊

For the remainder of this paper we assume that 𝔀 is of finite type. We keep the setting from Section 5.4 and Assumption (τ0) from Section 7.1. Recall that in the finite case