Noncommutative K\"ahler Structures on Quantum Homogeneous Spaces

By R. O. Buachalla. 2011

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Building on the theory of noncommutative complex structures, the notion of a noncommutative Kahler structure is introduced. In the quantum homogeneous space case many of the fundamental results of classical Kahler geometry are shown to follow from the existence of such a structure, allowing for the definition of noncommutative Lefschetz, Hodge, Kahler-Dirac, and Laplace operators. Quantum projective space, endowed with its Heckenberger-Kolb calculus, is taken as the motivating example. The general theory is then used to show that the calculus has cohomology groups of at least classical dimension.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/14068bb29efa36a2664e9acbc69ec02c125b7857