AG Mathematische Physik: Réamonn Ó Buachalla (Prag): Noncommutative Kahler Structures

Réamonn Ó Buachalla (Prag)

Noncommutative Kahler Structures

Abstract: We begin by reviewing the general framework of differential

calculi over *-algebras. Building on this, we introduce the notion of a

complex structure, understood as a noncommutative analogue of the

decomposition of complexified differential forms on a complex manifold.

We then proceed to the concept of a (positive definite) noncommutative

Kähler structure and outline several fundamental results that arise

from its existence, including the Lefschetz decomposition, the Kähler

identities, the Hodge decomposition, and the refinement of de Rham

cohomology into Dolbeault cohomology. If time permits, we will conclude

with a discussion of the motivating example of the standard quantum

sphere and its Dolbeault-Dirac spectral triple.