Universal K-matrix for quantum symmetric pairs
-
Martina Balagović
and
Stefan Kolb
Abstract
Let
In the late nineties T. tom Dieck and R. Häring-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
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
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
and applied it to establish Kazhdan–Lusztig theory for the category
appear via categorification using parabolic category
One of the fundamental properties of the quantized enveloping algebra
denote the coproduct of
In
The relations
hold. Here we use the usual leg notation for threefold tensor products.
The universal R-matrix gives rise to a family
In
By definition the family of commutativity isomorphisms
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
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
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
Recall that a subalgebra B of
In the present paper we introduce the notion of a universal K-matrix for a right coideal subalgebra B of
In
The relation
(1.1)holds in the completion
See Definition 4.12 for details. By the definition of the completion
in
The notion of a universal K-matrix can be defined for any coideal subalgebra of a braided bialgebra H with universal R-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
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
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
The construction in the present paper is significantly inspired by the example classes
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
We call the element
Recall from [18, Theorem 2.7] that the involutive automorphism
where
for every integrable
For
For the symmetric pairs
for all finite-dimensional
The construction of the bar involution for
In the final Section 9 we address the crucial problem to determine the coproduct
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
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
Section 5 is a brief summary of the construction and properties of the quantum symmetric pair coideal subalgebras
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
In Section 7 we consider the element
The map ξ involved in the definition of
In Section 9 we restrict to the finite case. We first perform some preliminary calculations with the quasi R-matrices of
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
Let
The Weyl group W is the subgroup of
of
Hence, under the resulting identification of
Remark 2.1.
The abelian groups
Define
and let
is the coweight lattice. Let
By construction, one has
Lemma 2.2.
The pairing
2.2 Quantized enveloping algebras
With the above notations we are ready to introduce the quantized enveloping algebra
Remark 2.3.
The choice of ground field is dictated by two reasons. Firstly, by Lemma 2.2 it makes sense to consider
Following [26, Section 3.1.1] the quantized enveloping algebra
the quantum Serre relations given in [26, Section 3.1.1 (e)].
We will use the notation
we will use the notation
We make the quantum-Serre relations (v) more explicit. Let
By [26, Section 33.1.5] the quantum Serre relations can be written in the form
The algebra
for all
As usual we write
denote the corresponding weight space. We can apply this notation in particular to
2.3 The bilinear pairing 〈 ⋅ , ⋅ 〉
Let k be any field, let A and B be k-algebras, and let
In the following we will use this convention for
There exists a unique
such that for all
Here we follow the conventions of [13, Section 6.12] in the finite case. In the Kac–Moody case the existence of the pairing
The pairing
Lemma 2.4.
Let
then
Proof.
We may assume that
Consider
By the nondegeneracy of the pairing on
2.4 Lusztig’s skew derivations r i r i
Let
As in (2.2) one obtains a
The natural projection
for any
For all
(2.10)see [26, Proposition 3.1.6].
For all
(2.11)where
For all
(2.12)see [26, Section 1.2.13]
Property (3) and the original definition of
Property (3) above and the nondegeneracy of the pairing
see also [26, Lemma 1.2.15]. Moreover, property (2) and the coassociativity of the coproduct imply that for any
see [13, Lemma 10.1]. Note that this includes the case
Similarly to the situation for the algebra
for all
As in [26, Section 3.1.3] let
The map σ intertwines the skew derivations
Recall that the bar involution on
defined by
for all
see [26, Lemma 1.2.14].
3 The completion 𝒰 U q ( 𝔤 )
It is natural to consider completions of the infinite-dimensional algebra
3.1 The algebra 𝒰
Let
Let
is a tensor functor. Let
such that the diagram
commutes for any
Example 3.1.
The action of
Example 3.2.
Let
is well defined. In this way the element
In view of (3.1) this is compatible with the inclusions
Example 3.3.
Let
Lusztig showed that
The family
By [26, Section 39.43] the elements
where
if
We also use the symbol
Hence
Furthermore, the bar involution for
see [26, Section 37.2.4]
3.2 The coproduct on 𝒰
To define a coproduct on
Let
is an injective algebra homomorphism. However, it is not surjective, as the following example shows.
Example 3.4.
For
The collection
The element κ is an important building block of the universal R-matrix for
Any natural transformation
We call
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
denote the flip of tensor factors. Then
is a
3.3 Quasi R-matrix and commutativity isomorphisms
Let
The element
gives a well-defined element
Remark 3.5.
The element
The quasi R-matrix has a second characterization in terms of the bar involution (2.18) of
By [26, Theorem 4.1.2] the quasi-R-matrix is the uniquely determined element
with
Moreover, R is invertible, with
Remark 3.6.
If
Then
and for
The quasi-R-matrix R and the transformation κ defined in Example 3.4 give rise to a family of commutativity isomorphisms. Define
in
are isomorphisms of
for all
Remark 3.7.
In the construction ([26, Chapter 32]) of the commutativity isomorphisms
It follows from the definitions of the completion
In the proof of the next lemma we will use this property for
where
Lemma 3.8.
Assume that
hold in
Proof.
First observe that equations (3.12) and (3.13) are equivalent. Indeed, equation (3.10) for
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
where
where
4 Braided tensor categories with a cylinder twist
As explained in Section 3.3 the commutativity isomorphisms (3.9) turn
In [34] T. tom Dieck outlined a program to extend the graphical calculus to braids or ribbons in the cylinder
4.1 Cylinder twists and the reflection equation
To define cylinder twists, let
on objects
which satisfy the pentagon and the triangle axiom given in [34, (2.1), (2.2)]. A category
Example 4.1.
As seen in Section 3.3, the category
Let
From now on, following [34], we will consider the following data:
We call
Definition 4.2.
Let
is called a
In other words, a
for all
Example 4.3.
The pair
The following definition provides the main structure investigated by tom Dieck and Häring-Oldenburg in [34, 35, 12].
Definition 4.4 ([34]).
Let
for all
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
holds for all
Proof.
As
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
4.2 Twisted cylinder twists
Let
Let
This means that
for all
Definition 4.6.
Let
for all
Let
As in the proof of Proposition 4.5 one now obtains
Example 4.7.
Consider the setting of Example 4.1. Let
given by
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
For all
(4.9)where
The element R satisfies the relations
(4.10)where we use the usual leg-notation.
In this case the element
Let H be a braided bialgebra with universal R-matrix
for all
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
Let B be a right coideal subalgebra of H. As in Example 4.3 define
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
In this case we call
The bialgebra automorphism φ defines a braided tensor equivalence
By construction the natural transformation
Remark 4.11.
Observe the parallel between Definition 4.8 and Definition 4.10 in the case
Definition 4.10 can be extended to include the quantized universal enveloping algebra
In the following definition we reformulate condition (4.13) in terms of
Definition 4.12.
Let
holds in
holds in
Similarly to the discussion for the bialgebra H above, a cylinder-braided coideal subalgebra of
Remark 4.13.
Let
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
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
To address the first difference, recall from [17, Definition XIV.6.1] that the braided Hopf algebra
If such a ribbon element
satisfies the relation
in
To eliminate the additional tensor factor in (4.17) and (4.18) let
be a character, that is a one-dimensional representation. Define
and observe that
commutes with all elements of
which coincides with relation (4.13) in Definition 4.10. We summarize the above discussion.
Proposition 4.14.
Let
By the above proposition the element
Remark 4.15.
Assume that
where
The constructions in [9] and in this subsection, however, do not answer the question how to find characters of
5 Quantum symmetric pairs
In the remainder of this paper we will show that quantum symmetric pair coideal subalgebras of
5.1 Involutive automorphisms of the second kind
Let
Involutive automorphisms of the second kind of
For any subset X of I let
Involutive automorphisms of
Definition 5.1.
A pair
The action of τ on X coincides with the action of
If
We briefly recall the construction of the involutive automorphisms
Such a function always exists. The map s gives rise to a group homomorphism
This in turn allows us to define a Lie algebra automorphism
Remark 5.2.
In [18, (2.7)] and in [2, (3.2)] we chose the values
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
gives a bijection between the set of
Let
where
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
Note that the derived Lie subalgebra
of θ, see [18, Definition 4.3] for details. The quantum involution
To shorten notation define
Quantum symmetric pair coideal subalgebras depend on a choice of parameters
Define
In [18, (5.9), (5.11)]) the following parameter sets appeared:
see also [2, Remark 3.3].
Let
Definition 5.4.
Let
for all
Remark 5.5.
The conditions
For
The algebra
see [18, Proposition 5.2]. One can calculate the coproduct of the generators
for some
In view of (5.9) it makes sense to define
The elements
5.3 The bar involution for quantum symmetric pairs
The bar involution for
In [2, Section 3.2] the algebras
If
If
The existence of the bar involution
Theorem 5.6 ([2, Theorem 3.11]).
Assume that conditions (i) and (ii) hold. The following statements are equivalent.
There exists a
(5.12)In particular,
The relation
holds for all
It is conjectured that Theorem 5.6 holds without assumptions (i) and (ii).
In [2, Proposition 3.5] it was proved that for all
for some
5.4 Assumptions
For later reference we summarize our setting. As before
The Cartan matrix
The parameters
(5.14)The parameters
(5.15)One has
If (4) holds, then using (5.13) and (5.3) one sees that equation (5.14) is equivalent to
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
Remark 5.8.
Observe that assumption (2) is a stronger statement then what is needed for the existence of the bar-involution
6 The quasi K-matrix 𝔛
The bar involution
In view of (3.3), the element
6.1 A recursive formula for 𝔛
As a first step towards the construction of
Proposition 6.1.
Let
The following are equivalent:
For all
For all
For all
(6.2)(6.3)
If these equivalent conditions hold then additionally
For all
Proof.
(1)
(2)
Now compare the
By (2.10), this is equivalent to
relations (6.2) and (6.3) for all
(3)
(3)
Let
and consequently
Finally, let
Thus
This proves that the relation
The proof of the implication (3)
6.2 Systems of equations given by skew derivations
By Proposition 6.1 the quasi K-matrix
Proposition 6.3.
Let
There exists an element
(6.5)The elements
For all
(6.6)For all
(6.7)
Moreover, if the system of equations (6.5) has a solution
Proof.
(1)
and hence (6.6) holds for all
Moreover, using the quantum Serre relation
which proves relation (6.7). Hence property (2) holds.
(2)
for all
As
We claim that relation (6.7) implies that
Similarly, if
Assume now that
Hence
Let
for any
To see uniqueness, assume that
6.3 Three technical lemmas
We will use Proposition 6.3 in Section 6.4 to inductively construct
Lemma 6.4.
Let
Proof.
Assume that
Hence
Assume that
Hence
But then
which contradicts condition (3) in Definition 5.1 of an admissible pair.
Hence
lies in
Lemma 6.5.
Let
Proof.
We prove this by induction on
Recall that σ denotes the involutive antiautomorphism of
Lemma 6.6.
Let
for all
Proof.
The space
for all
The assumption
By the induction hypothesis one has
As
equation (6.11) follows from the induction hypothesis. ∎
6.4 Constructing 𝔛 μ
We are now ready to construct
for all
Lemma 6.7.
The relation
Proof.
This is a direct calculation. Note that all computations include the case
and
We see that the first and fifth summands in the above expansions of
By (5.10) and (2.19) we may assume that
By equation (5.3) one has
If
this implies that
Using the definition (5.11) of
This proves that the elements
Lemma 6.8.
For all
Proof.
We may assume that
Case 1:
Therefore the left hand side of (6.17) is equal to
Using
Case 2:
By (6.12) and (6.13) one has
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
Set
Using the skew derivation property (2.9) and the assumption that
Using relations (2.15) and the property
Using the fact that
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. ∎
6.5 Constructing 𝔛
We are now ready to prove the main result of this section, namely the existence of the quasi K-matrix
Theorem 6.10.
There exists a uniquely determined element
holds in
Proof.
We construct
as
Set
By Proposition 6.1 the element
7 Construction of the universal K-matrix
Using the quasi K-matrix
7.1 A pseudo longest element of W
If
Moreover, in this case the Lusztig automorphism
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
for all
Lemma 7.1.
For all
Proof.
For
for all
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
Assumption (τ 0
We are given an additional involutive element
The parameters
(7.4)
Remark 7.2.
Assume that
If
The composition
defines an algebra automorphism. By (7.2) the automorphism
To obtain an analog of this Lusztig action on modules in
We will apply this notation in particular in the case where φ is one of
Remark 7.3.
If the algebra automorphism
7.2 The twisted universal K-matrix in the Kac–Moody case
We keep our assumptions from Section 5.4 and Assumption
and note that by (7.4) and (7.5) one has
Such a function exists. Indeed, we may take an arbitrary map on any set of representatives of
Lemma 7.4.
Let
Proof.
We prove this by induction on the height of λ. Assume that (7.9) holds for a given
which completes the induction step. ∎
As in Example 3.3 we may consider ξ as an element of
Theorem 7.5.
Let
for any
holds in
Proof.
It suffices to check that
where x is one of the elements
Case 1:
Case 2:
Using this and the recursion (7.8) one obtains
This confirms relation (7.10) for
Case 3:
To simplify the last term, recall from (5.7) that
To simplify the above expression observe that
for
Now set
In view of [26, Section 37.2.4] one has
see also the proof of [2, Lemma 2.9]. Hence (7.12) gives
In view of the relation
one now obtains
which completes the proof of the theorem. ∎
For later reference we note that relation (7.13) implies that the element
see also (5.1), (5.2) and property (3) in Definition 5.1 of an admissible pair.
Remark 7.6.
The function ξ is an important ingredient in the construction of the twisted K-matrix
7.3 The universal K-matrix in the finite case
We now assume that
If M is a finite-dimensional
Composing the inverse of this isomorphism with the isomorphism
Corollary 7.7.
Assume that
for any finite-dimensional
holds in
Remark 7.8.
As before let
8 A special choice of ξ
In the following we want to determine the coproduct of the element
8.1 Choosing ξ
Recall that ξ has to satisfy the recursion (7.8) which involves the function
For any
Observe that both
Remark 8.1.
A priori one only has
To this end it is useful to reformulate the above condition as
and to work with the weight lattice
We claim that ξ satisfies the recursion (7.8).
Lemma 8.2.
The function
for all
Proof.
For any
which proves (8.3). Choosing
As
8.2 The coproduct of ξ
Recall the invertible element
As in Example 3.4 the collection
Remark 8.3.
In the following we will apply this notion in the case
Recall from Example 3.3 that the function ξ defined by (8.1) may be considered as an element in
Proof.
Let
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 U q ( 𝔤 )
Conjugation by the invertible element
For any
Recall that we consider
Lemma 8.5.
The automorphism (8.6) restricts to an automorphism of
for all
Proof.
By definition the elements ξ and
which proves formula (8.7). Formula (8.8) is obtained analogously using the relation
which also follows from (8.3). ∎
For
Lemma 8.6.
Assume that
Proof.
Consider
Moreover, if
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
The group homomorphism
The weight lattice P is spanned by the fundamental weights
and we want to extend γ from Q to P.
Choice 1
Let
Choice 2
Let
and γ can be extended to P by
The advantage of Choice 1 is that the parameters
For any quantum symmetric pair of finite type it is possible to find analogs of Choice 1 and Choice 2 above. We can choose
9 The coproduct of the universal K-matrix 𝒦
For the remainder of this paper we assume that
given in Corollary 7.7. This will show that
9.1 Preliminary calculations with the quasi R-matrix
Let
In view of relation (3.4) one obtains that
Define
We will see in Theorem 9.4 that the element
Lemma 9.1.
The following relation holds:
Proof.
For any
Using (9.1) we get that
This implies that
Using (7.2) we get
As
which completes the proof of the lemma. ∎
Lemma 9.2.
For any
in
Proof.
It follows from the intertwining property (3.3) of the quasi R-matrix for
and that
Hence one gets
Applying Lemma 8.6 to the second factor in the above expression we get
As
Inserting this expression into (9.5) gives
Finally, inserting the above formula into (9.4) produces the desired result. ∎
9.2 The coproduct of the quasi K-matrix 𝔛
As in Section 6 write the quasi K-matrix as
This element will appear as a factor in the coproduct of
Lemma 9.3.
For any
in
Proof.
By equation (6.2) one has
By the explicit expression (7.14) for
Inserting (9.8) into (9.7) one obtains the desired formula. ∎
By Proposition 6.1 (4) the element
Now we are ready to compute the coproduct
Theorem 9.4.
The intertwiner
in
Proof.
By the definition (2.1) of the coproduct of
for all
For
where as before
On the other hand, using Lemma 9.2 and Lemma 9.3 one gets
Using the above formula, equation (9.9), and again Proposition 6.1 (4) one obtains
Now we want to compare equation (9.12) with equation (9.13). If
and hence
Inserting the above relation into (9.12) and comparing the outcome with (9.13) one obtains
also for
9.3 The coproduct of 𝒦
We apply construction (8.4) to the map
and
Recall from Corollary 7.7 that
Theorem 9.5.
The coproduct of
Proof.
Using Proposition 6.1 (4) one verifies that in
Moreover, the following relation holds:
Combining the above two formulas with all our previous preparations we calculate
which gives the desired formula. ∎
Recall Definition 4.12 of a φ-universal K-matrix. Combining Corollary 7.7 with Theorem 9.5 we obtain the first statement of the following corollary.
Corollary 9.6.
The element
in
Proof.
The second statement follows from Remark 4.13 and equation (4.8). Here one needs to observe that
Remark 9.7.
The approach to quantum symmetric pairs in the papers [27, 29, 28, 8] is based on explicit solutions of the reflection equation. In [27] Noumi first found a reflection equation for the symmetric pairs of type
For the symmetric pairs of type
a different reflection equation appeared in [28]. The differing reflection equations are unified by equation (9.16). Indeed, the diagram automorphism
Funding source: Engineering and Physical Sciences Research Council
Award Identifier / Grant number: EP/K025384/1
Funding statement: Research supported by Engineering and Physical Sciences Research Council grant EP/K025384/1.
Acknowledgements
The authors are grateful to Weiqiang Wang for comments and advice on referencing.
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© 2016 Martina Balagović and Stefan Kolb, published by Walter de Gruyter GmbH, Berlin/Boston
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Articles in the same Issue
- Frontmatter
- Limits in 𝒫ℳℱ of Teichmüller geodesics
- Rational connectivity and analytic contractibility
- Uniformization of p-adic curves via Higgs–de Rham flows
- Néron models of jacobians over base schemes of dimension greater than 1
- Cuspidal curves, minimal models and Zaidenberg’s finiteness conjecture
- Moduli of Bridgeland semistable objects on 3-folds and Donaldson–Thomas invariants
- ⌢𝒟-modules on rigid analytic spaces I
- Tilings of amenable groups
- Universal K-matrix for quantum symmetric pairs
Articles in the same Issue
- Frontmatter
- Limits in 𝒫ℳℱ of Teichmüller geodesics
- Rational connectivity and analytic contractibility
- Uniformization of p-adic curves via Higgs–de Rham flows
- Néron models of jacobians over base schemes of dimension greater than 1
- Cuspidal curves, minimal models and Zaidenberg’s finiteness conjecture
- Moduli of Bridgeland semistable objects on 3-folds and Donaldson–Thomas invariants
- ⌢𝒟-modules on rigid analytic spaces I
- Tilings of amenable groups
- Universal K-matrix for quantum symmetric pairs
