Let be a symmetrizable Kac–Moody algebra and an involutive Lie algebra automorphism. Let denote the fixed Lie subalgebra. We call the pair of Lie algebras a symmetric pair. Assume that θ is of the second kind, which means that the standard Borel subalgebra of satisfies . In this case the universal enveloping algebra has a quantum group analog which is a right coideal subalgebra of the Drinfeld–Jimbo quantized enveloping algebra , see [22, 23, 18]. We call 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 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 .
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 . They performed their program for the symmetric pairs
and applied it to establish Kazhdan–Lusztig theory for the category of the ortho-symplectic Lie superalgebra . Bao and Wang developed the theory for these two examples in astonishing similarity to Lusztig’s exposition of quantized enveloping algebras in . In a closely related program M. Ehrig and C. Stroppel showed that quantum symmetric pairs for
appear via categorification using parabolic category of type D (see ). 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 is the existence of a universal R-matrix which gives rise to solutions of the quantum Yang–Baxter equation for suitable representations of . 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 . Let
denote the coproduct of and let denote the opposite coproduct obtained by flipping tensor factors. The universal R-matrix of is an element in a completion of , see Section 3.2. It has the following two defining properties:
In the element satisfies the relation for all .
hold. Here we use the usual leg notation for threefold tensor products.
The universal R-matrix gives rise to a family of commutativity isomorphisms for all category representations of . In our conventions one has where denotes the flip of tensor factors. The family can be considered as an element in an extension of the completion of , see Section 3.3 for details. In property (1) of can be rewritten as follows:
In the element commutes with for all .
By definition the family of commutativity isomorphisms is natural in M and N. The above relations mean that turns category for 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  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 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. Häring-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 as in  to the setting of braids and ribbons in the cylinder , see . It hence corresponds to an extension of the theory from the classical braid group of type to the braid group of type . Tom Dieck and Häring-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 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 are coideal subalgebras of 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 is called a right coideal subalgebra if
In the present paper we introduce the notion of a universal K-matrix for a right coideal subalgebra B of . A universal K-matrix for B is an element in a suitable completion of with the following properties:
In the universal K-matrix commutes with all .
holds in the completion of .
See Definition 4.12 for details. By the definition of the completion , a universal K-matrix is a family of linear maps for all integrable -modules in category . Moreover, this family is natural in M. The defining properties (1) and (2) of are direct analogs of the defining properties (1’) and (2) of the universal R-matrix . The fact that commutes with immediately implies that satisfies the reflection equation
in . By (1.1) and the naturality of a universal K-matrix for B gives rise to the structure of a universal cylinder twist on the braided tensor category of integrable -modules in category . Universal K-matrices, if they can be found, hence provide examples for the theory proposed by tom Dieck and Häring-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 . This works in complete analogy to the above definition for B and , 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 . Let denote the element obtained from by flipping the tensor factors. Under some technical assumptions the universal K-matrix in  is just the element . 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 , 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 . In that paper universal K-matrices were found for quantum symmetric pairs corresponding to the symmetric pairs and . 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 of for 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 and considered by Bao and Wang in . The papers  and  both observed the existence of a bar involution for quantum symmetric pair coideal subalgebras in this special case. Bao and Wang then constructed an intertwiner between the new bar involution and Lusztig’s bar involution. The element Υ 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 -module homomorphism for any finite-dimensional representation M of . If M is the vector representation, they show that satisfies the reflection equation and they establish Schur–Jimbo duality between the coideal subalgebra and a Hecke algebra of type acting on .
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
for the quantum symmetric pair coideal subalgebra was already established in . 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 which satisfies the relation
We call the element the quasi K-matrix for . It corresponds to the intertwiner Υ in the setting of .
Recall from [18, Theorem 2.7] that the involutive automorphism is determined by a pair up to conjugation. Here X is a subset of the set of nodes of the Dynkin diagram of and τ is a diagram automorphism. The Lie subalgebra corresponding to X is required to be of finite type. Hence there exists a longest element in the parabolic subgroup of the Weyl group W. The Lusztig automorphism may be considered as an element in the completion of , see Section 3. We define
where denotes a suitably chosen element which acts on weight spaces by a scalar. The element defines a linear isomorphism
for every integrable -module M in category . In Theorem 7.5 we show that is a -module homomorphism if one twists the -module structure on both sides of (1.4) appropriately. The element exists in the general Kac–Moody case.
For of finite type there exists a longest element and a corresponding family of Lusztig automorphisms . In this case we define
For the symmetric pairs and the construction of coincides with the construction of the -module homomorphisms in  up to conventions. The longest element induces a diagram automorphism of and of . Any -module M can be twisted by an algebra automorphism if we define for all , . We denote the resulting twisted module by . We show in Corollary 7.7 that the element defines a -module isomorphism
for all finite-dimensional -modules M. Alternatively, this can be written as
The construction of the bar involution for , the intertwiner , and the -module homomorphism are three expected key steps in the wider program of canonical bases for quantum symmetric pairs proposed in . 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 . Weiqiang Wang has informed us that he and Huanchen Bao have constructed and independently in the case , see .
In the final Section 9 we address the crucial problem to determine the coproduct in . The main step to this end is to determine the coproduct of the quasi K-matrix in Theorem 9.4. Even for the symmetric pairs and , this calculation goes beyond what is contained in . It turns out that if , then the coproduct is given by formula (1.1). Hence, in this case is a universal K-matrix as defined above for the coideal subalgebra . If , 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 is a -universal K-matrix for . The fact that 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.
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 of and the completion of . In particular, we consider Lusztig’s braid group action and the commutativity isomorphisms 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 Häring-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 to take into account the need for completions. Finally, in Section 4.4 we recall the different definition of a universal K-matrix from  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 in the conventions of . In Section 5.3 we recall the existence of the bar involution for following . The quantum symmetric pair coideal subalgebra 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 .
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 . The defining condition (1.2) gives rise to an overdetermined recursive formula for the weight components of . 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 , and it is verified in Section 6.4. This allows us to prove the existence of in Section 6.5. A similar argument is contained in [3, Section 2.4] for the special examples and . However, the explicit formulation of the conditions in Proposition 6.3 seems to be new.
In Section 7 we consider the element defined by (1.3). In Section 7.1 we define a twist of which reduces to the Lusztig action if is of finite type. We also record an additional Assumption () on the parameters. In Section 7.2 this assumption is used in the proof that is a -module isomorphism of twisted -modules. In the finite case this immediately implies that the element defined by (1.5) gives rise to an -module isomorphism (1.6). Up to a twist this verifies the first condition in the definition of a universal K-matrix for .
The map ξ involved in the definition of is discussed in more detail in Section 8. So far, the element ξ was only required to satisfy a recursion which guarantees that is a -module homomorphism. In Section 8.1 we choose ξ explicitly and show that our choice satisfies the required recursion. In Section 8.2 we then determine the coproduct of this specific ξ considered as an element in the completion . Moreover, in Section 8.3 we discuss the action of ξ on 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 and . This allows us in Section 9.2 to determine the coproduct of the quasi K-matrix , see Theorem 9.4. Combining the results from Sections 8 and 9 we calculate the coproduct and prove a -twisted version of formula (1.1) in Section 9.3. This shows that is a -universal K-matrix in the sense of Definition 4.12.
2 Preliminaries on quantum groups
2.1 The root datum
Let I be a finite set and let be a symmetrizable generalized Cartan matrix. By definition there exists a diagonal matrix with coprime entries such that the matrix DA is symmetric. Let be a minimal realization of A as in [15, Section 1.1]. Here and denote the set of simple roots and the set of simple coroots, respectively. We write to denote the Kac–Moody Lie algebra corresponding to the realization of A as defined in [15, Section 1.3].
Let be the root lattice and define . For we write if . For let denote the height of μ. For any the simple reflection is defined by
The Weyl group W is the subgroup of generated by the simple reflections for all . For simplicity set . Extend to a basis
of and set . Assume additionally that for all , . By [15, Section 2.1] there exists a nondegenerate, symmetric, bilinear form on such that
Hence, under the resulting identification of and we have . The induced bilinear form on is also denoted by the bracket . It satisfies for all . Define the weight lattice by
The abelian groups and together with the embeddings , and , form an X-regular and Y-regular root datum in the sense of [26, Section 2.2].
Define by , set
and let . Then
is the coweight lattice. Let for denote the basis vector of dual to . Let B denote the -matrix with entries . Define an matrix by
By construction, one has . The pairing induces -valued pairings on and . The above conventions lead to the following result.
The pairing takes values in on and on .
2.2 Quantized enveloping algebras
With the above notations we are ready to introduce the quantized enveloping algebra . Let be the smallest positive integer such that . Let be an indeterminate and let a field of characteristic zero. We will work with the field of rational functions in with coefficients in .
The choice of ground field is dictated by two reasons. Firstly, by Lemma 2.2 it makes sense to consider as an element of for any weights . This will allow us to define the commutativity isomorphism , see Example 3.4 and formula (3.8). Secondly, in the construction of the function ξ in Section 8 we will require factors of the form for and , see formula (8.1). Again, Lemma 2.2 shows that such factors lie in .
Following [26, Section 3.1.1] the quantized enveloping algebra is the associative -algebra generated by elements , , for all and satisfying the following defining relations:
and for all .
for all , .
for all , .
for all where and .
the quantum Serre relations given in [26, Section 3.1.1 (e)].
We will use the notation and all through this text. Moreover, for
we will use the notation
By [26, Section 33.1.5] the quantum Serre relations can be written in the form
The algebra is a Hopf algebra with coproduct , counit ε, and antipode S given by
for all , . We denote by the Hopf subalgebra of generated by the elements , and for all . Moreover, for any let be the subalgebra of generated by and . The Hopf algebra is isomorphic to up to the choice of the ground field.
As usual we write , , and to denote the -subalgebras of generated by , , and , respectively. We also use the notation and for the positive and negative Borel part of . For any -module M and any let
denote the corresponding weight space. We can apply this notation in particular to , , and which are -modules with respect to the left adjoint action. We obtain algebra gradings
2.3 The bilinear pairing
Let k be any field, let A and B be k-algebras, and let be a bilinear pairing. Then can be extended to by setting
In the following we will use this convention for , , , and and 3 without further remark.
There exists a unique -bilinear pairing
such that for all , , , and the following relations hold
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 , and one has
The pairing respects weights in the following sense. For with the restriction of the pairing to vanishes identically. On the other hand, the restriction of the paring to is nondegenerate for all . The nondegeneracy of this restriction implies the following lemma, which we will need in the proof of Theorem 9.4.
Let . If
We may assume that . Write , with
Consider . For any , , and we then have
By the nondegeneracy of the pairing on it follows that . Consequently, for all , and hence as claimed. ∎
2.4 Lusztig’s skew derivations and
Let be the free associative -algebra generated by elements for all . The algebra is a -module algebra with
As in (2.2) one obtains a -grading
The natural projection , respects the -grading. There exist uniquely determined -linear maps such that
for any and . The above equations imply in particular that . By [26, Section 1.2.13] the maps and factor over , that is there exist linear maps , denoted by the same symbols, which satisfy relations (2.8) and (2.9) for all , and with replaced by . The maps and on satisfy the following three properties, each of which is equivalent to the definition given above.
For all and all one has
see [26, Proposition 3.1.6].
For all one has
where and , see [13, Section 6.14].
For all , , and one has
see [26, Section 1.2.13]
Property (3) and the original definition of and 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 , respectively.
Property (3) above and the nondegeneracy of the pairing imply that for any with one has
see also [26, Lemma 1.2.15]. Moreover, property (2) and the coassociativity of the coproduct imply that for any one has
see [13, Lemma 10.1]. Note that this includes the case .
Similarly to the situation for the algebra , the maps also factor over the canonical projection , which maps to for all . The maps satisfy (2.8) and (2.9) for all , with replaced by . Moreover, the maps can be equivalently described by analogs of properties (1)–(3) above. For example, in analogy to (3) one has
for all , , and .
As in [26, Section 3.1.3] let denote the -algebra antiautomorphism determined by
The map σ intertwines the skew derivations and as follows:
Recall that the bar involution on is the -algebra automorphism
for all , . The bar involution on also intertwines the skew derivations and in the sense that
see [26, Lemma 1.2.14].
3 The completion of
It is natural to consider completions of the infinite-dimensional algebra and related algebras. The quasi R-matrix for , for example, lies in a completion of , and the universal R-matrix lies in a completion of , see Section 3.3. Similarly, the universal K-matrix we construct in this paper lies in a completion of . 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 in quite some detail. This allows us to introduce further concepts, such as the Lusztig automorphisms, as elements of . It also provides a more conceptual view on the quasi R-matrix and the commutativity isomorphisms.
3.1 The algebra
Let denote the category of integrable -modules in category . Recall that category consists of -modules M which decompose into finite-dimensional weight spaces and on which the action of is locally finite. Moreover, the weights of M are contained in a finite union for some . Objects in are additionally locally finite with respect to the action of for all . Simple objects in are irreducible highest weight modules with dominant integral highest weight [26, Corollary 6.2.3]. If is of finite type, then is the category of finite-dimensional type 1 representations.
Let be the category of vector spaces over . Both and are tensor categories, and the forgetful functor
is a tensor functor. Let be the set of natural transformations from to itself. The category is equivalent to a small category and hence is indeed a set. More explicitly, elements of are families of vector space endomorphisms
such that the diagram
commutes for any -module homomorphism . Natural transformations of 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 which turns into a -algebra.
Let and let . Let and . As the action of on M is locally finite there exist only finitely many such that . Hence the expression
is well defined. In this way the element defines an endomorphism of , and we may thus consider as a subalgebra of . We sometimes write elements of additively as
In view of (3.1) this is compatible with the inclusions .
Let be any map. For define a linear map by for all . Then the family is an element in . By slight abuse of notation we denote this element by ξ as well.
Lusztig showed that contains a homomorphic image of the braid group corresponding to W. For any and any the Lusztig automorphism is defined on with by
The family defines an element in . By [26, Proposition 5.2.3] the elements of are invertible with inverse given by
By [26, Section 39.43] the elements for satisfy the braid relations
where denotes the order of . Hence, for any there is a well-defined element given by
if is a reduced expression.
We also use the symbol for to denote the corresponding algebra automorphism of denoted by in [26, Section 37.1]. This is consistent with the above notation, in the sense that for any , any , and any we have
Hence , as an automorphism of , is nothing but conjugation by the invertible element . In this way we obtain automorphisms of for all .
Furthermore, the bar involution for intertwines and . More explicitly, for one has
see [26, Section 37.2.4]
3.2 The coproduct on
To define a coproduct on consider the functor
Let denote the set of natural transformations from to itself. Again, is an algebra for which the multiplication is given by composition of natural transformations. The map
is an injective algebra homomorphism. However, it is not surjective, as the following example shows.
For define a linear map
The collection lies in . However, one can show that κ is not of the form for any and any collection . Hence κ does not lie in the image of the map described above.
The element κ is an important building block of the universal R-matrix for , see Section 3.3. For κ to be well defined the ground field needs to contain for all . This gives one of the reasons why we work over the field .
Any natural transformation can be restricted to all , . Moreover, restriction is compatible with composition and linear combinations of natural transformations. Hence we obtain an algebra homomorphism
We call the coproduct of . The restriction of to coincides with the coproduct of from Section 2.2. For this reason we will drop the subscript and just denote the coproduct on by .
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
Define . For let
denote the flip of tensor factors. Then is an element of . The direct sum
is a -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 and let be a basis of . Let be the dual basis of with respect to the pairing (2.3). Define
The element is independent of the chosen basis . The quasi R-matrix
gives a well-defined element . Indeed, for only finitely many summands act nontrivially on any element of .
The element 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 defined in Section 5.1.
The quasi R-matrix has a second characterization in terms of the bar involution (2.18) of . Define a bar involution on by
By [26, Theorem 4.1.2] the quasi-R-matrix is the uniquely determined element
with and for which
Moreover, R is invertible, with
If is of finite type, then the quasi-R-matrix R can be factorized into a product of R-matrices for . Choose a reduced expression for the longest element of W. For set and define
Then is the set of positive roots of , and (3.5) are the root vectors used in the construction of the PBW basis corresponding to the chosen reduced expression for . For define
and for set if . By [13, Remark 8.29] one has
The quasi-R-matrix R and the transformation κ defined in Example 3.4 give rise to a family of commutativity isomorphisms. Define
in . By [26, Theorem 32.1.5] the maps
are isomorphisms of -modules for all . Moreover, the isomorphisms satisfy the hexagon property
In the construction ([26, Chapter 32]) of the commutativity isomorphisms it is assumed that 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 . We retain the assumption of integrability so that the Lusztig automorphisms given by (3.2) are well defined. The restrictions imposed by and force us to work with the category .
It follows from the definitions of the completion and the coproduct that in one has
In the proof of the next lemma we will use this property for . Moreover, by [26, Proposition 5.3.4] the Lusztig automorphisms satisfy
where 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 of finite type denotes the longest element. Define, moreover, .
Assume that is of finite type. Then the relations
hold in .
where . Hence for one has
4 Braided tensor categories with a cylinder twist
As explained in Section 3.3 the commutativity isomorphisms (3.9) turn into a braided tensor category. For any there exists a graphical calculus for the action of on in terms of braids in , see [17, Corollary XIII.3.8]. If is finite dimensional, then has a duality in the sense of [17, Section XIV.2] and there exists a ribbon element which turns 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  T. tom Dieck outlined a program to extend the graphical calculus to braids or ribbons in the cylinder . The underlying braid group corresponds to a Coxeter group of type B. In the papers [34, 35, 12] tom Dieck and R. Häring-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 . 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 be a tensor category as defined in [17, Definition XI.2.1]. Let be another category and assume that there exists a functor which we write as
on objects and morphisms in and , respectively. The functor is called a right action of on if there exist natural isomorphisms α and ρ with
which satisfy the pentagon and the triangle axiom given in [34, (2.1), (2.2)]. A category together with a right action of on is called a right -module category.
As seen in Section 3.3, the category is a braided tensor category. Let be a right coideal subalgebra, that is a subalgebra satisfying
Let be the category with and for all . Then is a right -module category with given by .
From now on, following , we will consider the following data:
is a braided tensor category with braiding for all .
is a right -module category.
is a subcategory of with . In other words, is a subset of for all .
, α, ρ restrict to , a, r on .
We call a tensor pair if the above conditions (1)–(4) are satisfied. By condition (3) there exists a forgetful functor
Let be a tensor pair. A natural transformation
is called a -endomorphism of . If is an automorphism for all , then t is called a -automorphisms of .
In other words, a -endomorphism of is a family of morphisms such that
for all .
The pair from Example 4.1 is a tensor pair. In this setting a -endomorphism of is an element which commutes with all elements of the coideal subalgebra . In other words, the maps are B-module homomorphisms for all .
Definition 4.4 ().
Let be a tensor pair. A cylinder twist for consists of a -automorphism of such that
for all .
Let be a tensor pair with a cylinder twist . Then the relation
holds for all .
As is a morphism in , relation (4.1) implies that
In  equation (4.3) is called the four-braid relation. Here we follow the mathematical physics literature  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 and of the reflection equation for M and N, respectively, to a new solution for the tensor product .
4.2 Twisted cylinder twists
Let 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 . This generalization involves a second twist which suggests the slightly repetitive terminology.
Let be braided tensor equivalence given by
This means that is a braided tensor functor as defined in [17, Definition XIII.3.6] and an equivalence of categories. A family of morphisms is called a --endomorphism of if
for all . In other words, a --endomorphism of is a natural transformation .
Let be a tensor pair and a braided tensor equivalence. A -cylinder twist for consists of a --automorphism of such that
for all .
Let be a tensor pair with a -cylinder twist. The relation and (4.4) imply that
As in the proof of Proposition 4.5 one now obtains
Consider the setting of Example 4.1. Let be a Hopf algebra automorphism. For any let be the integrable representation with left action given by for all , . By [36, Theorem 2.1] one has and and hence . Moreover, as the map φ induces a group isomorphism . We assume additionally that is an isometry, that is for all . Then one obtains an auto-equivalence of braided tensor categories
respectively, for any .
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
is called braided (or quasitriangular) if there exists an invertible element such that the following two properties hold:
For all one has
where denotes the opposite coproduct.
The element R satisfies the relations
where we use the usual leg-notation.
In this case the element is called a universal R-matrix for H.
Let H be a braided bialgebra with universal R-matrix . In this situation the category - of H-modules is a braided tensor category with braiding
for all , see [17, Section VIII.3].
The conventions in Definition 4.8 slightly differ from the conventions in . The reason for this is that following  we use the braiding for and hence the braiding for H-mod. To match conventions observe that in Definition 4.8 coincides with in [17, Definition VIII.2.2].
Let B be a right coideal subalgebra of H. As in Example 4.3 define to be the category with and for all . Then is a tensor pair. For any bialgebra automorphism define
In analogy to the notion of a universal R-matrix the following definition is natural.
Let H be a braided bialgebra with universal R-matrix and let 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 such that
In this case we call a φ-universal K-matrix for the coideal subalgebra B. If , then we simply say that B is cylinder-braided and that is a universal K-matrix for B.
The bialgebra automorphism φ defines a braided tensor equivalence given by where as before denotes the H-module which coincides with M as a vector space and has the left action . In the above setting a φ-universal K-matrix for the coideal subalgebra B defines a family of maps
By construction the natural transformation is a tw-cylinder twist for the tensor pair .
Observe the parallel between Definition 4.8 and Definition 4.10 in the case . Indeed, condition (4.9) means that the maps defined by (4.11) are H-module homomorphisms while condition (4.12) means that the maps defined by (4.14) are B-module homomorphisms if . 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 which is braided only in the completion. In this case we also need to allow for to lie in the completion . 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 define an element by
In the following definition we reformulate condition (4.13) in terms of and .
Let be a Hopf algebra automorphism. A right coideal subalgebra is called φ-cylinder-braided if there exists an invertible element such that the relation
holds in and the relation
holds in . In this case we call a φ-universal K-matrix for the coideal subalgebra B. If , then we simply say that B is cylinder-braided and that is a universal K-matrix for B.
Similarly to the discussion for the bialgebra H above, a cylinder-braided coideal subalgebra of naturally gives rise to a cylinder twist. For later reference we summarize the situation in the following remark.
Let be a right coideal subalgebra and let be the tensor pair from Example 4.1. Moreover, let be a Hopf-algebra automorphism and let be the corresponding braided tensor equivalence as in Example 4.7. An element is a φ-universal K-matrix for B if and only if is a -cylinder twist of . In this case, in particular, the element satisfies the fusion procedure (4.7) and the reflection equation (4.8) for all .
4.4 Cylinder braided coideal subalgebras via characters
In  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 , 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 be a braided Hopf algebra over a field k. We retain the conventions from Definition 4.8 and hence the symbol in  corresponds to in our conventions. Let denote the linear dual space of H. Recall from  that the braided Hopf H algebra is called factorizable if the linear map
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
the universal K-matrix of H. It follows from (4.10) that
where we label the tensor legs of by 0, 1, 2. The above formula is closely related to formula (4.13) for . There are two differences, however, namely the occurrence of the additional factor and the fact that (4.17) holds in while formula (4.13) holds in . Moreover, the element makes no reference to a coideal subalgebra of .
To address the first difference, recall from [17, Definition XIV.6.1] that the braided Hopf algebra is called a ribbon algebra if there exists a central element such that
If such a ribbon element exists, then the element
satisfies the relation
be a character, that is a one-dimensional representation. Define
and observe that is a right coideal subalgebra of H. The element
Let be a factorizable ribbon Hopf algebra over a field k and let be a character. Then the right coideal subalgebra defined by (4.19) is cylinder braided with universal K-matrix
By the above proposition the element satisfies the reflection equation in every tensor product of representations of H. As the ribbon element is central, the element also satisfies the reflection equation.
Assume that is of finite type. If one naively translates the construction of Proposition 4.14 to the setting of , then the resulting universal K-matrix is the identity element because does not have any interesting characters. However, in  a universal K-matrix is also defined for non-factorizable H. In this case one chooses to be the canonical element in where denotes a twisted version of the dual Hopf algebra . One obtains a universal K-matrix by application of a character f of . This framework translates to the setting of if one replaces by the braided restricted dual of . The braided restricted dual of is isomorphic as an algebra to the (right) locally finite part
where for denotes the right adjoint action. The locally finite part has many nontrivial characters, and a cylinder braiding for can be associated to each of them, see [19, Propositions 2.8, 3.14].
The constructions in  and in this subsection, however, do not answer the question how to find characters of . For this amounts to finding numerical solutions of the reflection equation which satisfy additional compatibility conditions. For this is a manageable problem, see [19, Remark 5.11]. It would be interesting to find a conceptual classification of characters of for all of finite type.
5 Quantum symmetric pairs
In the remainder of this paper we will show that quantum symmetric pair coideal subalgebras of are φ-cylinder-braided as in Definition 4.12 for a suitable automorphism φ of . 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  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 denote the positive Borel subalgebra of . An automorphism is said to be of the second kind if
For any subset X of I let denote the corresponding Lie subalgebra of . The sublattice of Q generated by is the root lattice of . If is of finite type, then let and denote the half sum of positive roots and positive coroots of , respectively. The Weyl group of is the parabolic subgroup of W generated by all with . If is of finite type, then let denote the longest element. Let denote the group of permutations such that the entries of the Cartan matrix satisfy for all . Let denote the subgroup of all which additionally satisfy .
Involutive automorphisms of of the second kind are parametrized by combinatorial data attached to the Dynkin diagram of . This combinatorial data is a generalization of Satake diagrams from the finite-dimensional setting to the Kac–Moody case, see , [18, Definition 2.3].
A pair consisting of a subset of finite type and an element is called admissible if the following conditions are satisfied:
The action of τ on X coincides with the action of .
If and then .
We briefly recall the construction of the involutive automorphisms corresponding to the admissible pair , see [18, Section 2] for details. Let denote the Chevalley involution as in [15, (1.3.4)]. Any can be lifted to a Lie algebra automorphism . Moreover, for of finite type let denote the corresponding braid group action of the longest element in . Finally, let be a function such that
Such a function always exists. The map s gives rise to a group homomorphism such that
This in turn allows us to define a Lie algebra automorphism such that the restriction of to any root space is given by multiplication by .
In [18, (2.7)] and in [2, (3.2)] we chose the values for to be certain fourth roots of unity. This had the advantage that commutes with the involutive automorphism corresponding to the admissible pair . 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 . This is more suitable for the categorification program in  and for the program of canonical bases for coideal subalgebras in .
With the above notations at hand we can now recall the classification of involutive automorphisms of the second kind in terms of admissible pairs.
gives a bijection between the set of -orbits of admissible pairs for and the set of -conjugacy classes of involutive automorphisms of the second kind.
Let denote the fixed Lie subalgebra of . We refer to as a symmetric pair. The involution leaves invariant. The induced map is given by
where for all , see [18, Section 2.2, (2.10)]. Hence Θ restricts to an involution of the root lattice. Let be the sublattice of Q consisting of all elements fixed by Θ. For later use we note that
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 . For the remainder of this paper let be an admissible pair and a function satisfying (5.1) and (5.2). Let denote the subalgebra of generated by the elements , , for all . Correspondingly, let and denote the subalgebras of generated by the elements in the sets and , respectively.
Note that the derived Lie subalgebra is invariant under the involutive automorphism . One can define a quantum group analog
of θ, see [18, Definition 4.3] for details. The quantum involution is a -algebra automorphism but it is not a coalgebra automorphism and . However, the map has the following desirable properties:
To shorten notation define
Quantum symmetric pair coideal subalgebras depend on a choice of parameters
In [18, (5.9), (5.11)]) the following parameter sets appeared:
see also [2, Remark 3.3].
Let be the subalgebra of generated by all with .
Let be an admissible pair. Further, let and let . The quantum symmetric pair coideal subalgebra is the subalgebra of generated by , , and the elements
for all .
The conditions and can be found in [18, (5.9) and (5.11)]. They are necessary to ensure that the intersection of the coideal subalgebra with is precisely . This in turn implies that the coideal subalgebra specializes to at with , see [18, Remark 5.12, Theorem 10.8].
For we set and . This convention will occasionally allow us to treat the cases and simultaneously.
The algebra is a right coideal subalgebra of , that is
see [18, Proposition 5.2]. One can calculate the coproduct of the generators for more explicitly and obtains
In view of (5.9) it makes sense to define
5.3 The bar involution for quantum symmetric pairs
The bar involution for defined in (2.18) does not map to itself. Inspired by the papers [11, 3], it was shown in  under mild additional assumptions that allows an intrinsic bar involution . We now recall these assumptions and the construction of the intrinsic bar involution for .
In [2, Section 3.2] the algebras are given explicitly in terms of generators and relations for all Cartan matrices and admissible pairs which satisfy the following properties:
If with and , then .
If with and , then .
The existence of the bar involution on 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.
There exists a -algebra automorphism , such that
In particular, .
holds for all for which or for which there exists such that .
for some . For of finite type it was moreover proved that for all , and this was conjectured to hold also in the Kac–Moody case [2, Proposition 2.3, Conjecture 2.7].
For later reference we summarize our setting. As before denotes the Kac–Moody algebra corresponding to the symmetrizable Cartan matrix and is an admissible pair. We fix parameters and and let denote the corresponding quantum symmetric pair coideal subalgebra of as given in Definition 5.4. Additionally, the following assumptions are made for the remainder of this paper.
The Cartan matrix satisfies conditions (i) and (ii) in Section 5.3.
The parameters satisfy the condition
The parameters satisfy the condition
One has for all , that is [2, Conjecture 2.7] holds true.
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 for all , all results of this paper hold for with and satisfying relations (5.14) and (5.15).
Observe that assumption (2) is a stronger statement then what is needed for the existence of the bar-involution 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.
6 The quasi K-matrix
The bar involution on defined by (2.18) and the internal bar involution on defined by (5.12) satisfy if . Hence the two bar involutions do not coincide when restricted to . The aim of this section is to construct an element which intertwines between the two bar involutions. More precisely, we will find with and such that satisfies
In view of (3.3), the element is an analog of the quasi-R-matrix R for quantum symmetric pairs. For this reason we will call the quasi K-matrix for . Examples of quasi K-matrices were first constructed in [3, Theorems 2.10, 6.4] for the coideal subalgebras corresponding to the symmetric pairs and .
6.1 A recursive formula for
As a first step towards the construction of we translate relation (6.1) into a recursive formula for the components .
The following are equivalent:
For all one has .
For all one has .
For all and all one has
If these equivalent conditions hold then additionally
For all such that , one has .
(1) (2) Property (2) is the special case of property (1).
Now compare the -homogeneous components for all . One obtains that equation (6.4) holds if and only if for all one has
(3) (4) We prove this implication by induction on . For there is nothing to show. Assume that . If , then by (2.13), there exists such that . By (6.2) we have either or . In the case , by induction hypothesis , which implies . In the case , the condition implies that , while the induction hypothesis implies that . Together, this gives .
(3) (1) We have already seen that (3) (2) and hence for .
Let and assume that . The implication gives . On the other hand and therefore . This implies that
and consequently for all .
Finally, let and again assume that . As and , we already know that and . Hence is the lowest weight vector for the left adjoint action of on . As is locally finite for the left adjoint action of , we conclude that is also a highest weight vector, and hence
Thus and consequently for all .
This proves that the relation holds for the generators of the algebra and hence it holds for all . ∎
The proof of the implication (3) (4) only refers to with . Hence we get the following corollary.
6.2 Systems of equations given by skew derivations
By Proposition 6.1 the quasi K-matrix 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.
Let with and fix elements for all . The following are equivalent:
There exists an element such that
The elements have the following two properties:
For all one has
For all one has
Moreover, if the system of equations (6.5) has a solution , then this solution is uniquely determined.
(1) (2) Assume that there exists and element which satisfies the equations (6.5). Then
and hence (6.6) holds for all .
Moreover, using the quantum Serre relation and the properties (2.12) of the bilinear form , we get
which proves relation (6.7). Hence property (2) holds.
(2) (1) Assume that the elements 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 via the canonical projection on the first factor. Fix with . As , there exist uniquely determined linear functionals such that
for all . For any and any we have
As , any element in can be written as a linear combination of elements of the form with for . Consequently, the above relation implies that the functionals and coincide on . To simplify notation we write .
We claim that relation (6.7) implies that descends from to a linear functional on . Recall that the kernel of the projection is the ideal generated by the elements for all . Hence it is enough to show that all elements of the form lie in the kernel of the linear functional . If , then the fact that lies in the radical of the bilinear form implies that
Similarly, if , then we get
Assume now that . Then
Hence does indeed descend to a linear functional .
Let be the element dual to with respect to the nondegenerate pairing on . In other words, for all we have . Then
6.3 Three technical lemmas
We will use Proposition 6.3 in Section 6.4 to inductively construct by solving the system of equations given by (6.2), (6.3) for all . 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.
Let and . If , then and one of the following two cases holds:
Assume that . Then which together with implies that
Hence and and . This would mean that which is impossible.
Assume that . Then
Hence and and . This is only possible if
which contradicts condition (3) in Definition 5.1 of an admissible pair.
Hence . As for any , it follows that
lies in . Using , it follows that . So, there are two possibilities: either (1) and , or (2) and . ∎
Let and let with . Assume that a collection with satisfies condition (6.2) for all and for all . If , then .
We prove this by induction on . If and , then by (2.13) there exists some i such that . Relation (6.2) implies that or . If , then the induction hypothesis on and implies the claim. If , then the induction hypothesis on implies the claim. ∎
Recall that σ denotes the involutive antiautomorphism of defined by (2.16).
for all .
The space is spanned by elements of the form with . As the antiautomorphism σ is involutive it is enough to verify equation (6.10) for elements of the form . We will prove that
for all and by induction on . It holds for . Let , and assume that (6.11) holds for all with . Without loss of generality assume that . Using the assumption that satisfies (6.2) and (6.3), we get that
By the induction hypothesis one has . Hence,
equation (6.11) follows from the induction hypothesis. ∎
We are now ready to construct inductively. Fix and assume that a collection with and has already been constructed and that this collection satisfies conditions (6.2) and (6.3) for all and for all . Define
for all . We will keep the above assumptions and the definition of and all through this subsection. We will prove that the elements and , 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 with the desired properties.
The relation holds for all .
This is a direct calculation. Note that all computations include the case . We expand both sides of the desired equation, using (2.8) and (2.9) and the assumption that the elements satisfy (6.2) and (6.3) for . We obtain
We see that the first and fifth summands in the above expansions of and coincide, the second summand of is the same as the fourth summand of , and the fourth summand of coincides with the second summand of . Therefore, the claim of the lemma, , is equivalent to the third summands being equal,
If , then both sides of the above equation vanish. Hence we assume that is nonzero. Corollary 6.2 states that then . Along with
this implies that . Hence (6.16) is equivalent to the relation
This proves that the elements satisfy the first condition from Proposition 6.3 (2). Next we prove that they also satisfy the second condition.
For all the elements given by (6.12) satisfy the relation
We may assume that and that , as otherwise all terms in the above sum vanish. By Lemma 6.4 it suffices to consider the following two cases.
Case 1: and In this case and by definition (5.7) of the parameter set . Hence
Therefore the left hand side of (6.17) is equal to
Case 2: and In this case by (5.5) one has . Hence, by the definition (5.7) of the parameter set , one has either or . If , then Lemma 6.5 implies that , which is not the case. If is even, then the left hand side of (6.17) can be written as
Set and observe that . Inserting the definition of into (6.20) one obtains in view of the relation
We are now ready to prove the main result of this section, namely the existence of the quasi K-matrix . Recall the assumptions from Section 5.4.
There exists a uniquely determined element , with and , such that the equality
holds in for all .
as by (5.15). In this case satisfies (6.2) and (6.3). This defines in the case . Assume now that and that the elements have been defined for all with such that they satisfy (6.2) and (6.3) for all . The elements and 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 has a unique solution . By the definition of and the element satisfies equations (6.2) and (6.3).
By Proposition 6.1 the element satisfies the relation (6.22) for all . The uniqueness of 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 . ∎
7 Construction of the universal K-matrix
Using the quasi K-matrix from the previous section we now construct a candidate for a universal K-matrix as in Definition 4.12. Our approach is again inspired by the special case considered in . 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 -module homomorphism between twisted versions of modules in . 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 --automorphism for as in Section 4.2 with as in Example 4.1. The coproduct of will be determined in Section 9.
7.1 A pseudo longest element of W
If is of finite type, then there exists such that the longest element satisfies
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 denote the algebra automorphism defined by
for all , .
For all one has on .
For one has . It remains to check that
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 . 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 and .
We are given an additional involutive element with the following properties:
The parameters and satisfy the relations
Assume that is of finite type. In this case we always choose 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 . Moreover, by the definition of the parameter set one can have only if for all . Hence property (2) reduces to in the finite case. By (5.1) and (5.2) one can have only if . Hence property (3) is always satisfied in the finite case.
If , then property (2) is an empty statement. It is possible that and , see the list in . In this case condition (5.16) implies that equals up to multiplication by a bar invariant scalar. The new condition forces this scalar to be equal to 1. Finally, only in type is it possible that and . In this case, however, the condition implies that . These arguments show that the new condition is consistent with the conditions imposed in Section 5.4 and that it is always possible to choose parameters and which satisfy all of the assumptions.
defines an algebra automorphism. By (7.2) the automorphism is a Kac–Moody analog of the inverse of the Lusztig action on corresponding to the longest element in the Weyl group in the finite case. As , one has . By Lemma 7.1 this implies that
To obtain an analog of this Lusztig action on modules in we will twist the module structure. In the following subsection we construct a -module homomorphism between twisted versions of modules in . As is a subalgebra of , it suffices to consider objects in as -modules. With this convention, for any algebra automorphism and any let denote the vector space M with the -module structure given by
We will apply this notation in particular in the case where φ is one of and , see Theorem 7.5.
If the algebra automorphism extends to a Hopf algebra automorphism of , then the notation for 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 from the previous subsection. To construct the desired -module homomorphism we require one additional ingredient. Consider the function defined by
Such a function exists. Indeed, we may take an arbitrary map on any set of representatives of and uniquely extend it to P using (7.8).
Let be any function which satisfies the recursion (7.8). Then one has
We prove this by induction on the height of λ. Assume that (7.9) holds for a given . Then one obtains for any the relation
which completes the induction step. ∎
As in Example 3.3 we may consider ξ as an element of . The next theorem shows that the element defines a -module isomorphism between twisted modules in .
Let be a function satisfying the recursion (7.8). Then the element defines an isomorphism of -modules
for any . In other words, the relation
holds in for all .
It suffices to check that
where x is one of the elements , , , or for , , and . Moreover, it suffices to prove the above relation for a weight vector . In the following we will suppress the subscript M for elements in acting on M.
Case 1: for some In this case we have . Moreover, as , one has . Hence one obtains
Case 2: for some By relation (7.2) applied to we have
Using this and the recursion (7.8) one obtains
This confirms relation (7.10) for where . The case for is treated analogously.
Case 3: for some We calculate
To simplify the last term, recall from (5.7) that unless , in which case . Additionally moving ξ to the right one obtains
To simplify the above expression observe that
In view of [26, Section 37.2.4] one has
In view of the relation
one now obtains
which completes the proof of the theorem. ∎
The function ξ is an important ingredient in the construction of the twisted K-matrix 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 ξ for to be a -module homomorphism.
7.3 The universal K-matrix in the finite case
If M is a finite-dimensional -module, then the Lusztig action satisfies for all , . In other words, the Lusztig action on M defines an -module isomorphism
Composing the inverse of this isomorphism with the isomorphism from Theorem 7.5, we get the following corollary.
Assume that is of finite type and let be a function satisfying the recursion (7.8). Then the element defines an isomorphism of -modules
for any finite-dimensional -module M. In other words, the relation
holds in for all .
8 A special choice of ξ
In the following we want to determine the coproduct of the element from Corollary 7.7. We aim to show that is a -universal K-matrix for , that is that the coproduct is given by (4.16). This, however, will only hold true for a suitable choice of ξ.
8.1 Choosing ξ
Recall that ξ has to satisfy the recursion (7.8) which involves the function given by (7.7). Extend the function γ to a group homomorphism . Depending on the choice of coefficients , it may be necessary to replace by a finite extension to do this. We will illustrate the situation and comment on the field extension in Section 8.4.
For any write
Observe that both and are contained in for all . Recall from Section 2.1 that for denote the fundamental coweights. Now define a function by
A priori one only has . However, for all of finite type one can show by direct calculation that
To this end it is useful to reformulate the above condition as
and to work with the weight lattice of the restricted root system Σ of the symmetric pair . The relation between and P is discussed in detail in [25, Section 2]. We expect (8.2) also to hold for infinite-dimensional . If it does not hold, then the definition of ξ requires an extension of also for the q-power to lie in the field.
We claim that ξ satisfies the recursion (7.8).
The function defined by (8.1) satisfies the relation
for all . In particular, ξ satisfies the recursion (7.8).
8.2 The coproduct of ξ
Recall the invertible element defined in Example 3.4. Let be any map. For every define a linear map
As in Example 3.4 the collection defines an element in .
In the following we will apply this notion in the case , see Section 5.1. To this end we need to assume that the minimal realization is compatible with the involution as in [18, Section 2.6]. This means that the map extends to a permutation such that . In this case τ may be considered as a map . We will make this assumption without further comment. In the finite case, which is our only interest in Section 9, it is always satisfied.
The element defined by (8.1) satisfies the relation
Let and , for some . Then lies in the weight space . Hence one gets
which proves formula (8.5). ∎
For the rest of this paper the symbol ξ will always denote the function given by (8.1) and the corresponding element of .
8.3 The action of ξ on
Conjugation by the invertible element gives an automorphism
For any one has
Recall that we consider as a subalgebra of .
The automorphism (8.6) restricts to an automorphism of . More explicitly, one has
for all and and all .
By definition the elements ξ and commute in . This proves (8.9). To verify the remaining two formulas let and for some . Then one has
which also follows from (8.3). ∎
For of finite type the above lemma allows us to identify the restriction of to the subalgebra of . Recall the conventions for the diagram automorphisms in the finite case from Remark 7.2.
Assume that is of finite type. Then one has
8.4 Extending γ from Q to P
In this final subsection, we illustrate how different choices of and s influence the extension of the group homomorphism to the weight lattice P. As an example consider the root datum of type with , that is , and the admissible pair given by and . In this case the constraints (5.16) and (5.2) reduce to the relations
The group homomorphism is defined by
The weight lattice P is spanned by the fundamental weights
and we want to extend γ from Q to P.
Let , , . Then , , and γ can be extended to P by