Quantum cluster C*-algebras and positive representations

Lie algebras play a prominent role in physics, representing the infinitesimal symmetries of physical systems. Tools to unravel the symmetries of such models are found in the area of representation theory. This theory seeks symmetries, in order to shed light on the observed objects and collect them into abstract entities, which aid in distinguishing and comparing objects.
In our project, we will contribute to the representation theory of associated quantized real semisimple Lie groups. To do so, we will mainly study cluster algebras, i.e., algebras with a combinatorial flavor that encode information about a distinguished basis of the quantum group. The existence of a (quantum) cluster structure yields a strong tool to study the tensor product of representations.
One of the two main goals of this project is to show that cluster-type structures are not restricted to the finite-dimensional realm. More precisely, using the theory of positive representations, we will introduce and develop the theory of quantum cluster algebras in the analytic setting, i.e., for C*- algebras. As a byproduct, this will allow the application of powerful tools from the operator algebraic
framework to the understanding of positive representations. The second aim is to develop the structure theory of split real quantum groups and, by doing so, introduce new concepts for the local study of positive representations, and generalize results on tensor product representations to more general branching rules.