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Quantum Flag Manifolds in Prague

16-20 September 2019

Focusing on the noncommutative geometry of quantum groups and its interactions with noncommutative algebra geometry, spectral triples, and parabolic geometry


Poster (A3)

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Réamonn Ó Buachalla, Université Libre de Bruxelles
Petr Somberg, Charles University in Prague
Jan Šťovíček, Charles University in Prague
Karen Strung, Radboud University Nijmegen

Short Courses

To maximize interactions between research areas covered by the conference, we will have three introductory short courses and an introductory lecture on the following topics.

Noncommutative Algebraic Geometry, Adam-Christaan van Roosmalen, Hasselt University 

In algebraic geometry, one uses algebraic techniques to study geometry, and one uses geometric ideas and intuition to describe algebra.  Even though the same interplay cannot be duplicated naïvely to the noncommutative world, many techniques and definitions can still be carried over to the setting of noncommutative rings.
In these lectures, we will concentrate on projective algebraic geometry as introduced by Artin–Zhang.  Here, one starts with a reasonably nice graded ring R and studies the category qgr(R) as the category of coherent sheaves on a “nonexistent” noncommutative projective space Proj(R).

Around 2002, Polishchuk and Schwarz constructed a category of coherent sheaves on a noncommutative 2-torus, starting from a given Dolbeault dg algebra.  They subsequently showed that the category of coherent sheaves is (derived equivalent to) the category of coherent sheaves on an elliptic curve.  This provides an example of a link between noncommutative differential geometry and noncommutative algebraic geometry.

Recent progress in understanding the vector bundles on quantum flag manifolds (specifically, the introduction of a noncommutative Kähler structure and a noncommutative version of the Kodaira vanishing theorem) seem to make a similar result tractable.

In this lecture series, we will study some techniques of noncommutative algebraic geometry, in particular those related to quantum flag manifolds (based on Heckenberger and Kolb’s construction).  We will then discuss the category of coherent sheaves on quantum flag manifolds (or specifically, quantum projective space) and show how one can obtain this category as qgr(R) where R is the q-polynomial ring.  This establishes another instance where one has a correspondence between noncommutative algebraic geometry and noncommutative differential geometry

Parabolic Geometry, Katharina Neusser, Masaryk University 

This mini-course will give an introduction to Cartan geometries, which provide a uniform approach to a large variety of differential geometric structures.It will focus on parabolic geometries which are Cartan geometries infinitesimally modelled on flag manifolds. The most prominent examples of geometric structures admitting descriptions as parabolic geometries are conformal manifolds (dim>2), projective structures, almost quaternionic manifolds, and certain types of CR-structures. After having introduced the basic concepts and discussed some examples, we will (as time permits) give some applications of Cartan connections to symmetries and/or to the construction of invariant differential operators for parabolic geometries.

Quantum semisimple groups and quantum flag varieties, Robert Yuncken, Université Clermont Auvergne

Compact and complex semisimple Lie groups admit quantizations which have been discovered twice—first by Drinfeld and Jimbo in the form of quantized enveloping algebras, and later by Woronowicz in the form of quantum matrix pseudogroups.  These quantum groups have a rich representation theory which echoes that of their classical cousins.  At least in the compact case, setting the classical group in its family of quantum groups has given us new tools in representation theory.

Meanwhile, for the noncommutative geometer, quantized semisimple groups and their homogeneous spaces give some of the most natural examples of noncommutative spaces, although they have been frustratingly difficult to incorporate precisely into Connes’ philosophy of noncommutative differential geometry.  In these lectures, we’ll give a rapid tour of compact and complex semisimple quantum groups and their flag varieties, from both a representation theoretic and a geometric point of view. 

The plan of the lectures will be as follows, time permitting:

Lecture 1: A survey of quantum groups; definitions of quantized enveloping algebras, compact semisimple quantum groups, and quantum flag varieties; finite dimensional representation theory and Verma modules.

Lecture 2: Complex semisimple quantum groups as quantum doubles in the framework of algebraic quantum groups; bundles over flag varieties as principal series representations; duality with Verma modules; intertwining operators.

Lecture 3: The Bernstein–Gelfand–Gelfand complex, algebraically and geometrically; application to noncommutative geometry (the fundamental class of the flag variety of SLq(3,C)) and representation theory (the Plancherel formula for a complex semisimple Lie group).


Klimyk, A & Schmüdgen, K, Quantum groups and their representations.
Texts and Monographs in Physics. Springer-Verlag, Berlin, 1997

Voigt, C & Yuncken R, Complex semisimple quantum groups

Quaternion-Kähler Manifolds and Where to Find Them, Henrik Winther, Masaryk University 

In this lecture we discuss the history, origins and theory of quaternion-Kähler manifolds. We introduce the two main classes of examples, the so-called Wolf spaces and Alekseevskian spaces.  The former class consists of the quaternion-Kähler symmetric spaces. Slides




  • Ryan Aziz, London slides
  • Martina Balagovic, Newcastle slides
  • Edwin Beggs, Swansea slides
  • Suvrajit Bhattacharjee, Kolkata slides
  • Branimir Ćaćić, Fredericton slides
  • Fredy Diaz, Morelia slides
  • Rita Fioresi, Bologna slides
  • Lisa Glaser, Vienna slides
  • Ken Goodearl, Santa Barbara slides
  • Jan Grabowski, Lancaster slides
  • Ruben Henrard, Hasselt slides
  • Andrey Krutov, Moscow/Zagreb slides
  • Emanuele Latini, Bologna slides
  • Rubén Martos, Copenhagen slides
  • Pavle  Pandžić, Zagreb slides
  • Wenqing Tao, Hasselt slides
  • Vít Tuček, Zagreb slides
  • Kent B. Vashaw, Baton Rouge slides
  • Bart Vlaar, Edinburgh (Heriot-Watt) slides
  • Elmar Wagner, Morelia
  • Petr Zima, Prague slides

Titles and Abstracts

Ryan Aziz: Quantum differentials on cross product Hopf algebras

This talk is based on joint work with Shahn Majid. We introduce general methods to construct well-behaved quantum differential calculi or DGAs on the main types of cross product Hopf algebras by constructing their super version. The construction applies for double cross product Hopf algebras, double cross coproduct Hopf algebras, bicrossproduct Hopf algebras, and biproducts or Radford–Majid bosonisation. The resulting exterior algebra is strongly bicovariant and for the latter three cases, coacts differentiably on the canonical comodule algebra associated to the inhomogeneous quantum group.

If time permits, we will discuss some examples. In the case of the Drinfeld double of dually paired Hopf algebras A and H, we show that its canonical actions on A and H are differentiable. Other applications are a canonical differential calculus for the quantum group of affine transformations of the quantum plane, and a differential calculus for the bicrossproduct Poincaré quantum group in 2 dimensions.

Martina Balagovic: Universal K-matrices for quantum symmetric pairs

I will review some recent progress on quantum symmetric pair coideal subalgebras of quantum groups, in particular the construction of the universal K-matrix, which bears significant resemblance to the construction of the universal R-matrix for the quantum group.

Edwin Beggs: Noncommutative complex structures and covariant derivatives

This talk will begin with noncommutative differential calculi, and then specialise to calculi with integrable complex structures. As examples we consider the standard Podleś sphere and noncommutative stereographic projection, and also the noncommutative open unit disk with its hyperbolic Riemannian structure. We consider Dolbeault cohomology, holomorphic modules and holomorphic vector fields.

Suvrajit Bhattacharjee: Generalized Symmetry In Noncommutative Complex Geometry

In this talk, we discuss a framework that incorporates Hopf algebroid covariance into noncommutative Kähler structures. We begin by recalling the definition of a Hopf algebroid and present the two main examples. Then we define Hopf algebroid covariant Kähler structures. Finally, we present a version of Hodge decomposition theorem. If time permits, we will discuss how transversely Kähler foliations fit into this framework.

Branimir Ćaćić: Principal bundles in noncommutative Riemannian geometry

The cubic Dirac operator of a quadratic Lie algebra 𝖌 decomposes with respect to any quadratic subalgebra 𝖐 as the sum of the cubic Dirac operator of 𝖐 and the relative Dirac operator of the pair 𝖌, 𝖐; when 𝖌 integrates to a connected Lie group G with bi-invariant Riemannian metric and 𝖐 integrates to a compact subgroup K, this yields an encoding of the vertical geometry, principal connection, and horizontal geometry of the Riemannian principal K-bundle GG/K in terms of index-theoretically significant canonical first-order differential operators. In this talk, I’ll show how these considerations generalise to a theory of noncommutative Riemannian principal bundles with compact connected Lie structure group in terms of spectral triples and unbounded KK-theory; if time permits, I’ll discuss the possibility of generalising further to the case of q-deformed structure groups. This is joint work with Bram Mesland.

Fredy Diaz: Borel–Weil Theorem for the Irreducible Quantum Flag Manifolds

The Borel–Weil theorem is an elegant geometric procedure for constructing all unitary irreducible representations of a compact Lie group. The construction realises each representation as the space of holomorphic sections of a line bundle over a flag manifold. In this talk we present a direct quantum group generalisation of this result for the special case of the Hermitian symmetric spaces G/LS, generalising earlier work of a number of authors on the quantum Grassmannians. As a result we give a noncommutative geometric realisation of the homogeneous coordinate ring Rq(G/LS) in terms of its tautological line bundle, again directly q-deforming the classical case.

Rita Fioresi: Quantum flags, quantum line bundles and the Quantum Duality Principle

The concept of quantum section, introduced by F. Gavarini and Ciccoli in 2009, is a key ingredient to produce projective embeddings of quantum homogeneous varieties G/P. In this talk we shall describe this construction also taking into account the case where P is a coisotropic subgroup of G. This will lead to a generalization of the quantum duality principle originally due to Drinfeld and later on developed by Gavarini. In the end we will discuss some recent generalizations of this theory to supergeometry and also examples of physical applications.

Lisa Glaser: Simulating spectral triples

A spectral triple consists of an algebra, a Hilbert space and a Dirac operator, and if these three fulfill certain relations to each other they contain the entire information of a compact Riemannian manifold. Using the language of spectral triples makes it possible to generalize the concept of manifold to include non-commutativity and genuine discreteness.

We have used computer simulations to examined how well truncated spectral triples, in the sense of containing only a finite part of the spectrum still fulfill the Heisenberg relation introduced by Chamseddine, Connes and Mukhanov. Doing so we found another type of solution to the equation and got motivated to delve into trying to visualize non-commutative geometries.

Ken Goodearl:  Quantum Cluster Algebras and Quantized Varieties

This talk will focus on the concept of quantum cluster algebras and the appearance of these structures in large families of quantized coordinate rings, such as quantum Schubert cells, quantum double Bruhat cells, and quantum flag varieties. Many quantized coordinate rings are tightly related to iterated skew polynomial algebras, through either direct isomorphisms or mechanisms such as localization or dehomogenization. We will describe how quantum cluster structures develop from skew polynomial structures.

Jan Grabowski: Leveraging cluster structures

We now know that many important families of algebras admit quantum cluster algebra structures, including many related to quantum flag manifolds.  So, what can we learn about these algebras from their cluster structures?

There are theorems telling us that certain behaviours in cluster structures imply ring-theoretic properties: perhaps the best known of these is that acyclic cluster algebras are Noetherian. What new information can we get from the cluster structure? 

If our cluster algebra is of finite type, we know essentially everything about the cluster structure. Infinite types are more challenging, and most geometrically-derived examples are infinite type. We need ways of looking at cluster algebras that can see global properties: two such ways are gradings and categorification.  We will discuss some results in these directions and pose some questions for further study.

Ruben Henrard: Hall algebras of directed categories

Hall algebras are decategorifications of quiver representations. For Dynkin quivers, one recovers the quantized universal enveloping algebra of the associated Kac–Moody Lie algebra via Hall algebras. We use this framework to obtain and compare multiple Hopf algebra structures on uq(sl(∞)). In particular the categorification of uq(sl(∞))  allows us to produce quantum symmetries. This is joint work with Guillaume Pouchin and Adam-Christiaan van Roosmalen.

Andrey Krutov: Nichols algebras from quantum principal bundles over quantum flag manifolds

We present sufficient conditions allowing us to associate to any quantum principal bundle over a quantum homogeneous space a Yetter–Drinfeld module structure on the cotangent space of the base calculus. This allows us to show that the holomorphic and anti-holomorphic Heckenberger–Kolb calculi of the quantum Grassmannians could be expressed as Nichols algebras. Joint work with  Réamonn Ó Buachalla (Bruxelles) and Karen Strung (Nijmegen).

Emanuele Latini: Quantum principal bundle on a projective base

A quantum principal bundle is usually described trough the notion of faithfully flat Hopf–Galois extension while local triviality is encompassed by the locally cleft property.

In the commutative setting, this picture proves to be extremely effective when the base space is affine, while for a projective base the ring of coinvariants consists of just the constants, so it is not the object of interest anymore.

In this talk we aim thus to describe and propose a sheaf theoretic approach to the theory of quantum principal bundles over non affine bases. We then describe in details relevant and enlightening examples.

Rubén Martos: An overview of the Baum–Connes conjecture for quantum groups

In this talk we will discuss the Baum–Connes conjecture for quantum groups. After recalling the notion of compact quantum group (in the sense of Woronowicz) and explaining the Meyer–Nest’s categorical reformulation of the Baum-Connes conjecture, we will see why this approach is appropriate for (at least, torsion-free) discrete quantum groups. Next, we will present the current status of the conjecture for concrete examples of discrete quantum groups. Finally, we will summarize some stability properties of BC for relevant constructions of quantum groups (subgroups, (semi-)direct products and wreath products) as generalizations of the corresponding stability properties of BC for classical discrete groups.

Pavle  Pandžić: Dirac cohomology for modules over quantum sl(n)

I will start by briefly explaining the notion and properties of Dirac cohomology for (g,K)-modules. Then I will describe the analogous notion and results for quantum sl(2) (joint work with P. Somberg). At the end I will discuss on going joint work with P. Somberg aiming at generalization to quantum sl(n).

Wenqing Tao: Noncommutative Riemannian geometry on finite groups and Hopf quivers

We explore the differential geometry of finite groups where the differential structure is given by a Hopf quiver (in the sense of C. Cibils and M. Rosso) rather than as more usual by a Cayley graph. In this setting, the duality between the geometries on the function algebra and group algebra emerges and is realised as an extension of duality between path algebra and path coalgebra on the same quiver. Interestingly, linear (left) connections are given by quiver representations. We show how quiver geometries arise naturally in the context of quantum principal bundles. If time permits, I will illustrate constructions on the symmetric group S3. The talk is based on the joint works with Shahn Majid.

Vít Tuček: Invariant differential operators and singular vectors

Invariant differential operators on homogeneous bundles over G/P are classified by homomorphisms of induced representations. If P is a parabolic subgroup of G then these homomorphisms are determined by so called singular vectors. The task of finding singular vectors can be translated to solving systems of PDEs on polynomials. After explaining these relationships I will illustrate them with the Grassmannian case.

Kent Vashaw: Prime spectra of abelian 2-categories and categorification of Richardson varieties 

We describe a general theory of the prime, semiprime, completely prime, and primitive spectra of an abelian 2-category, providing a noncommutative version of Balmers prime spectrum of a tensor triangulated category. The prime ideals of an abelian 2-category can be described in terms of containment conditions of either thick or Serre ideals of the 2-category. The Serre prime spectrum of a 2-category is linked to the set of Serre primes of its Grothendieck ring. We construct a categorification of the quantized coordinate rings of open Richardson varieties for symmetric Kac–Moody groups, by constructing Serre completely prime ideals of monoidal categories of modules of the KLR algebras, and by taking Serre quotients with respect to them. This is a joint work with Milen Yakimov.

Bart Vlaar: Quantized pseudo-fixed-point subalgebras—towards a classification of quasitriangular coideal subalgebras 

Let g be a finite-dimensional simple Lie algebra over the complex numbers and let k be the fixed-point subalgebra of an involutive automorphism of g. In the 1990s, M. Noumi and collaborators and independently G. Letzter studied a coideal subalgebra Uq(k) of the quantized enveloping algebra Uq(g). More recently M. Balagović and S. Kolb showed that Uq(k) is quasitriangular (with respect to the category of finite-dimensional representations of Uq(g)). In particular, associated to Uq(k) there is a solution of the reflection equation (the quartic or type B braid relation). 

This story can be told in a more general setting. Roughly speaking, a pseudo-fixed-point subalgebra of g is a Lie subalgebra k which intersects the root spaces of g in the same way as does the fixed-point subalgebra of an involution of g. However (as opposed to fixed-point subalgebras) the “new” Lie subalgebras are typically not reductive. For any pseudo-fixed-point subalgebra k there exists a coideal subalgebra Uq(k) of Uq(g) which is quasitriangular. Conjecturally, all quasitriangular coideal subalgebras of Uq(g) arise this way. Joint work with V. Regelskis (part in arXiv:1807.02388, part in progress).

Elmar Wagner: Dirac operator on quantum flag manifolds: the quantum tangent space point of view 

The aim of the talk is to present an explicit construction of Dirac operators on irreducible quantum flag manifolds using the quantum tangent space approach of I. Heckenberger and S. Kolb.

Petr Zima: Generalized Killing spinors on 3-Sasakian manifolds

Special Riemannian structures are characterized by a reduction of the structure group to a subgroup G together with suitable integrability properties. A convenient way to understand the reduction is describing G as the stabilizer of a suitable tensor or tensor-spinor. The tensor or tensor-spinor then gives rise to a distinguished section on the manifold which solves an invariant system of PDEs related to the required integrability properties.

A particular case of interest are the Killing spinors in dimension 7 which are related to G2-structures. There is also a notion of generalized Killing spinor and an example is the so called canonical spinor of a 3-Sasakian manifold in dimension 7. We introduce a different generalization called the 2nd order Killing spinor and show that the same canonical spinor is also a nontrivial solution of our system of PDEs. The advantage of our approach is that our system is invariant and makes sense even without prior assumption of the special Riemannian structure.