Summer School

The chiralization in the title denotes a certain procedure that turns cluster X-varieties into q-W algebras. Many important notions from cluster and q-W worlds, such as mutations, global functions, screening operators, R-matrices, etc emerge naturally in this context.

Based on a joint project with J.E. Bourgine, B. Feigin, G. Schrader, A. Shapiro, and J. Shiraishi.

Prerequisites: Some basic facts about cluster algebras, cluster varieties, quantum groups.

1. Quantum cluster algebras, quantum parallel transports, cluster realization of quantum groups (R-matrices, affine type A in evaluation etc)

2. Loop realization of quantum affine, quantum toroidal, something on the representations

3. Chiralization of cluster structures

Abstract: We give a brief introduction to the theory of operator algebras and locally compact quantum groups, focusing on their representation theory and analytic structure. At the end of this lecture series, we will then sketch the construction of some totally positive locally compact quantum groups following recent work with G. Schrader, A. Shapiro and C. Voigt.

Tentative schedule, still susceptible to deformation, degeneration and contraction:

First lecture (1.5hrs)

• C*-algebras: representations, multipliers, affiliated elements

• von Neumann algebras: weight theory, affiliated elements

• Example: Rieffel deformation

Second lecture (1.5hrs)

• Locally compact quantum groups: structure theory, representation theory

• Multiplicative unitaries: modularity, regularity

• Examples

Third lecture (1hr)

• Totally positive quantum groups

Steinberg symbol is a skew-symmetric form on a multiplicative group of a ring. We will consider examples of symbols and show that such symbol defines on structure on cluster A-manifolds a structure (the so called K-symplectic structure) analogue to the (pre-)symplectic form. We will discuss many examples, mainly related to cluster integrable systems. In particular we will discuss the relation of the cluster integrable systems with the classical ones (like KdV).

In this lecture series I'll explain a contemporary point of view on the quantum trace relating skein and cluster structures on (quantum) character varieties of surfaces, and then I'll explain how this can be extended to 3-manifolds, with relations to the so-called quantum A-polynomial invariant of knots.

While this can be phrased in higher categories language, I will endeavour in this lecture series to only make concrete definitions and statements at the level of linear algebra and linear 1-categories.

This will cover joint works with Le, Schrader, Shapiro and separately with Brown.

I'll give an introduction to the cluster quantization of moduli spaces of framed local systems on surfaces with boundary discovered by Fock and Goncharov. The goal of the lectures will be to describe the behavior of the resulting quantum cluster algebras under the operation of gluing of surfaces along a pair of boundary circles. This description is the content of joint work with Alexander Shapiro.

Here is a rough outline of the plan for the course:

First lecture (1hr)

• definition of the moduli spaces and their cluster coordinate systems

• computing the monodromy representation in cluster coordinates

Second lecture (1.5hrs)

• quantization of cluster structures via the quantum dilogarithm; the universal Laurent ring

• the conjecture of Fock and Goncharov on gluing universal Laurent rings

• gluing for classical moduli spaces of local systems

Third lecture (1.5hrs)

• local description of gluing for quantum universal Laurent rings; relation with q-Whittaker polynomials

• Baxter operators and consistency of the gluing map under changes of cluster coordinate system