Titles and Abstracts
Francesca Arici Noncommutative sphere bundles: beyond Cuntz--Pimsner algebras
Cuntz-Pimsner algebras are universal C*-algebras associated to a C*- correspondence and they encode dynamical information. In the case of a self Morita equivalence bimodule they can be thought of as total spaces of a noncommutative circle bundle, and the associated six term exact sequence in K-theory can be interpreted as an operator algebraic version of the classical Gysin sequence for circle bundles. In this talk I will review known results in this direction and report on work in progress concerning the construction of higher dimensional noncommutative sphere bundles in terms of Cuntz-Pimsner algebras of sub-product systems. Based on (ongoing) joint work with G. Landi and J. Kaad.
Andrey Krutov Schubert Calculus for Quantum Grassmannians
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.)
Marco Matassa On the Parthasarathy formula for quantized irreducible flag manifolds
We consider Dolbeault-Dirac operators on quantized irreducible flag manifolds, as defined by Krähmer and Tucker-Simmons. We show that in general these operators do not satisfy a formula of Parthasarathy-type. This is a consequence of two results: 1) we always have quadratic commutation relations for the quantum root vectors, up to terms in the quantized Levi factor, 2) there are examples of quantum Clifford algebras where the relations are not of quadratic-constant type.
Réamonn Ó Buachalla Spectral Triples and Noncommutative Fano Structures
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.)
Adam-Christaan van Roosmalen Coherent and Quasi-coherent Sheaves for Quantum Projective Space
In 2006, Heckenberger and Kolb defined a de Rham complex and a Dolbeault complex for the quantized irreducible flag varieties. Based on the category of differential graded modules of the Dolbeault differential graded algebra, we construct a category of coherent sheaves on quantum projective space. We show that this category is equivalent to the proj-category of the q-commutative polynomial algebra, a category often studied in noncommutative algebraic geometry. (Joint work with R. Ó Buachalla and J. Šťovíček.)
TBA Quantum (Symmetric Pairs, Harish-Chandra Modules and GNS-representation)
In this talk I introduce the notion of Quantum symmetric pairs which enables us to develop the beginning of a theory of (Quantum) Harish-Chandra modules. Classically, this theory was one of the main ingredients in the decomposition of the left regular representation of a real reductive Lie group. We then describe briefly how a very similar decomposition holds for the GNS representation of the locally compact quantum group N(SU_q(1, 1)).