Twisted Hochschild Homology of Quantum Flag Manifolds and Kähler Forms

Abstract

We study the twisted Hochschild homology of quantum flag manifolds, the twist being the modular automorphism of the Haar state. We prove that every quantum flag manifold admits a non-trivial class in degree two, with an explicit representative defined in terms of a certain projection. The corresponding classical two-form, via the Hochschild-Kostant-Rosenberg theorem, is identified with a Kähler form on the flag manifold.

Fubini-Study metrics and Levi-Civita connections on quantum projective spaces

“…This is completely analogous to the definition of the Kähler forms discussed in [Mat19]. See also [Mat20] for a discussion of general quantum flag manifolds, not necessarily irreducibles, from the point of view of (twisted) Hochschild homology. 6.2.…”