WG3: Noncommutative Geometry and Quantum Homogeneous Spaces

Quantum Groups

The main aim is to make significant advances in the understanding of the noncommutative geometry of quantum groups homogeneous spaces through the application of ideas from parabolic geometry.

Group Leader

Reamonn O Buachalla

Charles University in Prague, Czechia

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Goals and Tasks:

G3.1: Prove the quantum Baum-Connes conjecture for the discrete duals of all quantum groups.

G3.2: Produce a quantum realization of Schubert calculus in terms of the quantum Grassmannians.

G3.3: Define q-deformed version of noncommutative Weil algebra and the corresponding cubic Dirac operators for all Drinfeld-Jimbo quantum groups.

T3.1: Produce a unified construction of spectral triples for quantum flags and symmetric spaces.

T3.2: Produce a noncommutative differential geometric construction of the category of noncommutative coherent sheaves for the quantum Hermitian symmetric spaces.

T3.3: Via spectral triple type operators for flag manifolds and differential graded algebras techniques find a theoretical approach to the Baum-Connes conjecture.

T3.4: Develop a full understanding of the cohomology rings of the quantum Grassmannians in connection with the noncommutative Hodge theory of its Kähler structure.

T3.5: Through the classical techniques developed in WG1 and WG2 develop a theory of quantum connections and quantum principal bundles for quantum symmetric spaces. Explore the SUSY
generalizations and the applications to Gauge field theory.

T3.6: Produce techniques to study equivariant cohomology of quantum homogeneous spaces via the q-deformed version of noncommutative Weil algebras. Explore connections with activities of WG1 via applications of the corresponding cubic Dirac operator in representation theory.