Recursive methods in compact quantum group theory

• Applicant: Dr. Alexander Mang
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 549208639

The proposal aims to develop and apply recursive methods in the theory of compact quantum gruops. More precisely, there are objectives on two levels. On the object level, the goal is to answer certain questions about quantisations of the general and special unitary groups. In general, compact quantum groups result from compact groups in two main ways, by deformation, esp. Drinfeld and Jimbo's q-deformation, or by liberation, esp. Banica and Speicher's so-called easy liberation. It is with the q-deformations of the general and special unitary groups that the theory of compact quantum group started in the 80s and those are well-understood today. In contrast, about the three families of easy liberations of the unitary group, for the most part discovered in 2017 by Weber and myself, virtually nothing is known besides a presentation of their function algebras in terms of generators and relations. Not even a definition has been put forward for a liberation of the special unitary group. And combining deformation and liberation has so far succeeded only in a single case. The project intends to change all that. In order to achieve the goals on the object level, particular recursive methods would be employed, methods which are in principle applicable to any algebra, in particular Gröbner bases and Anick resolutions. However, the idea is for that to be both inspiration and testing ground for, on the meta level, developing new recursive methods specially optimized for the function algebras of compact quantum groups. The complementary aims on the object and meta levels can be summarized as follows: 1. Object level: Describing the representation theories of the compact quantum groups which arise by easy liberation of the unitary group, in particular, computing their irreducible representations, the matrix elements, multiplicities and maybe Clebsch-Gordan coefficients of those. Meta level: Developing a specialized theory of Gröbner bases for function algebras of compact matrix quantum groups, in particular, a diamond lemma with, ideally, a large number of necessary conditions for Gröbner bases and a low number of ambiguities left to resolve, as well as conclusions about the representation theories based on that. 2. Object level: Computing homological invariants of the easy liberations of the unitary groups, in particular, Hochschild, bialgebra and maybe L2-cohomology. Meta level: Developing a specialized theory of projective resolutions of the co-units of the function algebras of compact quantum groups, based on Anick resolutions, but closer to minimal and with better convergence properties. 3. Object level: Constructing simultaneously q-deformed and liberated quantisations of the general and special unitary groups. Meta level: Developing a specialized elimination of variables theory for subalgebras of coproducts of function algebras of compact matrix quantum groups.

• DFG Programme: Research Grants