Algebra / Topology Seminar

Speaker: Réamonn Ó Buachalla

Title: Noncommutative Kaehler Structures on Quantum Homogeneous Spaces

Abstract: Building on the definition of a noncommutative complex structure
for a general algebra A, I introduce the notion of a noncommutative
Kaehler structure for A. In the special case where A is a quantum
homogeneous space, I show that many of the fundamental results of
classical Kaehler geometry follow from the existence of such a structure:
Hodge decomposition, Serre duality, the Hard Lefschetz theorem, the
Kaehler identities, and collapse of the Froelicher spectral sequence at
the first page. We then apply these results to Heckenberger and Kolb's
differential calculus for quantum projective space, and show that they
have cohomology groups of at least classical dimension. Time permitting, I
will also discuss the relationship of this work to Connes proposal  to
study positive Hochschild cocycles as a starting point for noncommutative
complex geometry, and Froechlich, Grandjean, and Recknagel's definition
of a Kaehler spectral tuple