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VAKSMAN-SOIBELMAN ODD QUANTUM SPHERES AND COMPLEX QUANTUM PROJECTIVE SPACES REVISITED
- Speaker(s)
- FRANCESCO D'ANDREA
- Affiliation
- Università di Napoli Federico II, Italy
- Language of the talk
- English
- Date
- Oct. 16, 2024, 5:15 p.m.
- Link
- https://uw-edu-pl.zoom.us/j/95105055663?pwd=TTIvVkxmMndhaHpqMFUrdm8xbzlHdz09
- Information about the event
- ZOOM
- Seminar
- North Atlantic Noncommutative Geometry Seminar
Odd-dimensional quantum spheres were introduced in 1990 by Vaksman and Soibelman as quantum homogeneous spaces of Woronowicz's quantum unitary groups. Since then, they have served as major examples for testing ideas about non-commutative spaces. In 1997, Sheu proved that the C*-algebra of an odd quantum sphere is independent of the deformation parameter q (for non-negative q less than 1), and is isomorphic to a groupoid C*-algebra. In 2002, Hong and Szymański discovered a directed graph rendering the C*-algebra of an odd quantum sphere as a graph C*-algebra. In the first part of this talk, I will sketch a proof that the Vaksman-Soibelman polynomial algebra of a quantum sphere does depend on the deformation parameter. This is an expected result as a similar dependence was already proven by Krähmer for Podleś quantum spheres. Next, any graph C*-algebra is the convolution algebra of a groupoid called the "path groupoid" of a graph. I will show that the path groupoid of the directed graph of Hong and Szymański is isomorphic to the groupoid discovered by Sheu, which cannot be simply inferred from the fact that the two groupoids have isomorphic convolution C*- algebras. Finally, if time permits, I will discuss the AF core of the graph C*-algebra of an odd quantum sphere, which we interpret as the C*-algebra A of a complex quantum projective space. In particular, using a suitable morphism of graphs, I shall present some "geometric" ideas on how to construct a *-homomorphism from the tensor square of A to A inducing an ordered ring structure on the K-theory of A. (Based on arxiv:2406.17288 and arxiv:2407.02169.)
