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## Project

# Quantum symmetric spaces, operator algebras and quantum cluster algebras (FWOAL900)

The mathematician's notion of space has continuously evolved throughout the history of the subject. For the ancient Greeks, space was seen as a background continuum in which certain structures such as lines, triangles, circles and planes can be housed and studied. Their synthetic approach was subsequently supplemented by the Cartesian method in which algebraic methods were used to study for example curves in a plane. Gradually, it was realized that there is such a thing as an abstract space, which can have many different forms and inherent structure. A highlight in this development is the work of B. Riemann, which proved to provide the appropriate mathematical

framework in which to develop the physical theory of general relativity. At the same time, the theory of quantum mechanics has shown the need for a notion of quantum space, to be understood by the methods of abstract non-commutative algebra.

The goal of this project is to study in depth a particular class of such quantum spaces. In the realm of Riemannian spaces, there is a class with a very high degree of symmetry, known as symmetric spaces. These were studied and classified by E. Cartan at the beginning of the 20th century. Only recently, particularly through the work of G. Letzter, has it become apparent that one has a corresponding notion of quantum symmetric space. We will investigate these quantum symmetric spaces both with analytic methods (operator algebras) and algebraic methods (quantum cluster algebras).

framework in which to develop the physical theory of general relativity. At the same time, the theory of quantum mechanics has shown the need for a notion of quantum space, to be understood by the methods of abstract non-commutative algebra.

The goal of this project is to study in depth a particular class of such quantum spaces. In the realm of Riemannian spaces, there is a class with a very high degree of symmetry, known as symmetric spaces. These were studied and classified by E. Cartan at the beginning of the 20th century. Only recently, particularly through the work of G. Letzter, has it become apparent that one has a corresponding notion of quantum symmetric space. We will investigate these quantum symmetric spaces both with analytic methods (operator algebras) and algebraic methods (quantum cluster algebras).

Date:1 Jan 2019
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Keywords:operator algebras, quantum symmetric spaces, cluster algebras

Disciplines:Abstract harmonic analysis, Associative rings and algebras , Special functions