Colloquium Mathematics, Réamonn Ó Buachalla (Radboud University), Andrey Krutov (Independent University of Moscow) & Karen Strung (Radboud University)

Abstract 1

"Spectral Triples and Noncommutative Fano Structures"
Speaker = Réamonn Ó Buachalla (Radboud University)

The notion of a noncommutative Kähler structure was recently
introduced as a framework in which to understand the metric aspects of Heckenberger and Kolb's remarkable covariant differential calculi over the cominiscule quantum flag manifolds. Many of the fundamental results of classical Kähler geometry are shown to follow from the existence of such a structure, allowing in particular for the definition of noncommutative Dolbeault-Dirac operators. In this talk we will discuss how a Kähler structure can be used to complete a calculus to a Hilbert space, and show that when the calculus is of so called Fano minimal type, the holomorphic and anti-holomorphic Dolbeault-Dirac operators give even spectral triples of index 1.
Quantum projective space will be presented as the motivating example, and time permitting the extension to odd and even quantum quadrics discussed. (Joint work with B. Das, P. Somberg, J. Šťovíček and A.C. van Roosmalen.)


Abstract 2

"Schubert Calculus for Quantum Grassmannians"
Speaker = Andrey Krutov (Independent University of Moscow)

We discuss Nichols-Woronowicz cacluli on the quantum Grassmannians. The direct computations shows that the equivariant cohomology of quantum Grassmannians is isomorphic to that of the classical Grassmannians in low-dimensional examples. We conjecture that this is true for all quantum Grassmannians. (Joint work with R. Ó Buachalla and K. Strung.)


Abstract 3

"C*-algebras associated to quantum Grassmannians and weighted quantum Grassmannians"
Speaker = Karen Strung (Radboud University)

In this talk I will discuss the structure of the C*-algebraic completions of torus bundles over the quantum flag manifolds and show that they are so-called “Cuntz—Pimsner algebras”. In addition, I will introduce a weighted version of the total space of a circle bundle over a quantum Grassmannian. This in turn leads to a weighted quantum Grassmannian. In the classical case, these weighted space are no longer principal. However, in the quantum case, their C*-algebras are again Cuntz—Pimsner algebras and share many properties with their unweighted counterparts.