Seminar za teoriju reprezentacija ZCI QuantiXLie

PMF Matematički odsjek
02.07.2019 - 16:00 - 19:00
Representation theory seminar and Center of Excellence QuantiXLie
Tuesday, July 2, room 109
16:00 Andrey Krutov, Independent University of Moscow: Dolbeault-Dirac Operators
on the Standard Podle's Sphere I
17:45 Réamonn Ó Buachalla, l’Université Libre de Bruxelles: Dolbeault-Dirac
Operators on the Standard Podle's Sphere II
We begin by recalling the definition and elementary structure of
the quantised enveloping algebra $U_q(\frak{sl}_2)$, along with its
dually paired quantised coordinate algebra $\mathcal{O}_q(SU_2)$. We
then introduce the (standard) Podle\'s sphere, a subalgebra of
$\mathcal{O}_q(SU_2)$ $q$-deforming the coordinate algebra of the
classical two sphere $S^2$. We show how the de Rham complex, and
Dolbeault double complex, of $S^2$ directly $q$-deform to this setting,
allowing us to construct a $q$-deformed Dolbeault—Dirac for the
Podle\'s sphere. Finally, we relate the square of this operator to the
quantum Casimir of $U_q(\frak{sl}_2)$, allowing us to calculate its
spectrum. Time permitting, we will present this operator as an example
of Alain Connes' notion of a spectral triple.
Everybody is invited.
Karmen Grizelj
Share this