Program 2019

In this talk, I will discuss techniques to study irreducible
representations of various quantum algebras at roots of unity, focussing in
particular on the example of quantum determinantal varieties. This is based
on joint works with Jason Bell, Samuel Lopes and Alexandra Rogers.

I will present some reformulation of geometric-topological notions in terms of representation theory of compact topological groups, which may be used to extend such notions to the quantum setting. I will then explain how some natural issues in compact quantum group theory are related to difficult problems in the context of discrete groups.

The notion of a covariant Hermitian structure was recently introduced as an algebraic framework in which to perform noncommutative Hermitian geometry on quantum homogeneous spaces. Combining covariant Hermitian structures with Woronowicz's theory of compact quantum groups produces a canonical Hilbert completion carrying a beautiful interaction of analysis, geometry, and algebra. We highlight two aspects of this completion: Firstly, the associated $*$-algebra of smooth functions, and secondly the interaction of Dirac operator index theory with noncommutative Dolbeault cohomology and noncommutative Fano structures. Time permitting, we will discuss the relationship of these structures with Connes' notion of a spectral triple. Throughout, the irreducible quantum flag manifolds, endowed with their Heckenberger--Kolb differential calculus, are presented as motivating examples, focusing in particular on the quantum Grassmannians.

The theory of quantum symmetric pairs provides coideal subalgebras of quantum groups which give rise to braided module categories over braided monoidal categories. In this talk I will outline a program to extend the theory of quantum symmetric pairs to a setting of (pre-)Nichols algebras (of diagonal type). I will explain how the resulting coideal subalgebras are obtained via star products on partial bosonizations. This new perspective allows a conceptual, bar-involution free interpretation of the quasi K-matrix, which is the crucial ingredient in the construction of the braiding on the corresponding module category. The talk is based on joint work with Milen Yakimov.

In this talk I shall sketch a construction of generalised Kahler 4-manifold for toric fano manifolds. It has been shown that certain type of generalised Kahler structures can be encoded in terms of a Morita equivalence of two holomorphic Poisson structures. We combine this with a construction due to Hitchin of bi-hermitian 4 manifolds in the toric setting, where, thanks to the action angle coordinates, one can be very explicit. We show that one obtains a generalised Kahler potential expressed in terms of Weistrass elliptic functions.

Both the string topology bracket developed by Chas-Sullivan and Menichi on negative cyclic cohomology groups and the dual bracket found by de Thanhoffer de Voelcsey-Van den Bergh on negative cyclic homology groups are examples of the brackets arising from the general noncommutative differential (or Cartan) calculus up to homotopy on the (co)cyclic bicomplex in case a certain Poincare' duality is given. For negative cyclic cohomology, this in particular leads to a Batalin-Vilkovisky algebra structure on the underlying Hochschild cohomology. In the special case in which this BV bracket vanishes, one obtains an e_3-algebra structure on Hochschild cohomology. The results are given in the general and unifying setting of (opposite) cyclic modules over (cyclic) operads. Joint work with Niels Kowalzig (arXiv:1712.09717)

I will explain how the graph C*-algebras of a trimmable graph can be decomposed as U(1)-equivariant pullback of two simpler C*-algebras. A main example is given by the algebra of Vaksman-Soibelman quantum sphere, that can be realized as pushout of a lower dimensional quantum sphere and the product of a quantum ball with a circle (we understand the pullback of C*-algebras as pushout of the underlying "noncommutative spaces"). The U(1)-invariant part of this pullback diagram gives a "CW complex" realization of quantum projective spaces that allows to give an explicit description of the K-theory generators. Further examples include quantum lens spaces, one-loop extensions of Cuntz algebras of the Toeplitz algebra. This is a joint work with Francesca Arici, Piotr M. Hajac and Mariusz Tobolski.

Cyclic Gerstenhaber-Schack cohomology" "In this talk, we answer a long-standin question by explaining how the diagonal complex computing the Gerstenhaber-Schack cohomology of a bialgebra (that is, the cohomology theory governing bialgebra deformations) can be given the structure of an operad with multiplication if the bialgebra is a (not necessarily finite dimensional) Hopf algebra with invertible antipode; if the antipode is involutive, the operad is even cyclic. Therefore, the Gerstenhaber-Schack cohomology of any such Hopf algebra carries a Gerstenhaber resp. Batalin-Vilkovisky algebra structure; in particular, one obtains a cup product and a cyclic boundary B that generate the Gerstenhaber bracket, and that allows to define cyclic Gerstenhaber-Schack cohomology. In case the Hopf algebra in question is finite dimensional, the Gerstenhaber bracket turns out to be zero in cohomology and hence the interesting structure is not given by this e2-algebr