Schedule

**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 *G* → *G/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/L*_{S}, 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 *R*_{q}(G/L_{S}) 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 *u*_{q}(sl(∞)). In particular the categorification of *u*_{q}(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 *S*_{3}. 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 *U*_{q}(k) of the quantized enveloping algebra *U*_{q}(g). More recently M. Balagović and S. Kolb showed that *U*_{q}(k) is quasitriangular (with respect to the category of finite-dimensional representations of *U*_{q}(g)). In particular, associated to *U*_{q}(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 *U*_{q}(k) of *U*_{q}(g) which is quasitriangular. Conjecturally, all quasitriangular coideal subalgebras of *U*_{q}(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 G_{2}-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.