Summer school lectures & short talks

Schedule

Lectures

Combinatorial Hopf algebras and operads

Loïc Foissy (University of the Littoral Opal Coast)

In this lecture, we shall explore links between combinatorial Hopf algebras and operads: how to obtain a bialgebra or a Hopf algebra from an operad, how to make an algebra over an operad a Hopf algebra, and what the relations between the objects obtained in this way are.

The first talk will be about combinatorial Hopf algebras, with examples coming from quantum field theory, computer sciences, graph theory, combinatorics, etc. Operads will be considered in the second talk, where functors from operads to bialgebras will be described, with examples on rooted trees or posets. In the last talk, the notion of cointeraction of bialgebras will be discussed. We shall see how operads can give pairs of cointeracting bialgebras and applications of these results, involving Ehrhart polynomials and chromatic polynomials.

Morita theory for tensor categories

David Jordan (The University of Edinburgh)

The theory of tensor and braided tensor categories borrows examples from representation theory of finite groups, algebraic groups, and quantum groups, and it blends these with concepts from higher categories and categorification.  The emerging "higher Morita theory" — the study of tensor and braided categories via their module categories — forms the contemporary axiomatic cornerstone to many celebrated topological field theories: the Turaev–Viro, Witten–Reshetikhin–Turaev, and Crane–Yetter theories, as well as the more recently introduced character and quantum character field theories of Ben-Zvi–Gunningham–Nadler and Ben-Zvi–Brochier–Jordan–Snyder, respectively.

These latter theories require us to leave the world of finite tensor categories behind for the world of locally presentable categories, and to contend with issues of compactness, convergence, and continuity.  New inspiration comes now from functional analysis and from the theory of invariant distributions — hence the area is sometimes called "categorical harmonic analysis".  Remarkably, despite the extra categorical machinery required, the new theories lend themselves to considerable explicit computation approachable via linear algebra and many concrete applications and conjectures.

In these lectures, I'll give an introduction to tensor and braided tensor categories with an emphasis on their Morita theory, I'll discuss the necessary technical notions from presentable enriched category theory, and finally I'll explain the relationship to topological field theories in dimensions 2, 3, and 4.

Lecture note

Hopf algebras

Ulrich Krähmer (TU Dresden)

These lectures will form an elementary and general introduction to Hopf algebras (aka quantum groups) and provide algebraic prerequisites for the other lecture series. I will discuss basic definitions and examples of Hopf algebras not covered in by my colecturers, and the connection to monoidal categories. Some slightly more advanced topics will be Hopf modules, Yetter–Drinfeld modules and generalisations of Hopf algebras such as Hopf algebroids, Hopf monads or coideal subalgebras (aka quantum homogeneous spaces). 

Lecture note

Locally compact quantum groups

Christian Voigt (The University of Glasgow)

The theory of locally compact quantum groups has its origins in attempts to extend Pontrjagin duality to arbitrary locally compact groups. Besides groups and their duals, various prominent examples of deformations fit into the theory as well, all in an operator algebraic framework using C*-algebras and von Neumann algebras.

In these talks I will explain the definition and basic properties of locally compact quantum groups and illustrate this with some motivating examples. I will also discuss natural links to Hopf algebras and tensor categories, and indicate some open problems.

Reference materials

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor categories, Mathematical Surveys and Monographs, vol. 205, American Mathematical Society, Providence, RI, 2015.

• Martijn Caspers, Locally compact quantum groups, Topological quantum groups, Banach Center Publ., vol. 111, Polish Acad. Sci. Inst. Math., Warsaw, 2017, pp. 153–184.

• Michael Müger, Tensor categories: a selective guided tour, Rev. Un. Mat. Argentina 51 (2010), no. 1, 95–163.

• Vladimir Turaev, Homotopy quantum field theory, EMS Tracts in Mathematics, vol. 10, European Mathematical Society (EMS), Zürich, 2010, Appendix 5 by Michael Müger and Appendices 6 and 7 by Alexis Virelizier.

Short talks

Ryan Aziz (Queen Mary University of London)

Quantum differentials on cross product Hopf algebras

This talk is based on a joint work with Shahn Majid. We introduce general methods to construct well-behaved quantum differential calculi  or DGAs on all biproduct Hopf algebras \(A{\buildrel\hookrightarrow\over \twoheadleftarrow}A\cdot\kern-.6em\triangleright\!\!\!<B\) (where B is a braided Hopf algebra in the category of A-crossed modules). As examples, we construct \(\Omega(\mathbb{C}_q[GL_2\ltimes \mathbb{C}^2])\) for the quantum group of affine transformations of the plane. We use super versions of the biproduct. such that the resulting exterior algebra is strongly bicovariant and coacts differentiably on the canonical comodule algebra associated to the inhomogeneous quantum group. Our method works on all other flavours of cross (co)product Hopf algebras, namely double cross products  \(A\hookrightarrow A\bowtie H\hookleftarrow H\), double cross coproducts \(A\twoheadleftarrow A {\blacktriangleright\!\!\blacktriangleleft} H\twoheadrightarrow H\), and bicrossproducts \(A\hookrightarrow A {\blacktriangleright\!\!\triangleleft} H\twoheadrightarrow H\) where we use super versions of each of the constructions.

slides

Hadewijch De Clercq (Ghent University)

A quantum symmetric pair approach to generalized q-Onsager algebras

The q-Onsager algebra is a quantum algebra originating from statistical mechanics, which has found many applications in the theory of quantum integrable systems. It can be embedded inside \(U_q(\widehat{\mathfrak{sl}}_2)\) as a coideal subalgebra. In this talk, I will explain how to define generalizations of the q-Onsager algebra inside U_q(𝔤) for arbitrary Kac–Moody algebras 𝔤. Our constructions are rooted in the theory of quantum symmetric pairs. I will present two methods to derive the defining relations of such generalized q-Onsager algebras.

Daniel Gromada (Saarland University)

Interpolating products of quantum groups

There are two main generalizations of the group direct product in the theory of quantum groups – the tensor product and the free product.  We define a number of further products interpolating these two. Then we show how partition calculus can be conveniently used to describe more general extensions of quantum groups by Z₂. This is a report on a recent preprint arXiv:1907.08462.

slides

Lucia Rotheray (TU Dresden)

Incidence bialgebras of monoidal categories

We will look at bialgebras whose elements are (linear combinations of) morphisms of a monoidal category, with the product and coproduct defined using the monoidal product and composition. We will discuss some combinatorial bi-/Hopf algebras which provide examples and non-examples of this construction.  

slides