Differential Calculus on Quantum Complex Grassmann Manifolds I: Construction

TL;DR

This paper constructs and analyzes covariant first order differential calculi on quantum complex Grassmann manifolds, showing their classical-like behavior and providing explicit examples like the q-deformed Chern character.

Contribution

It demonstrates the existence of exactly two covariant differential calculi near classical Kähler differentials on quantum Grassmann manifolds and constructs explicit examples such as the q-deformed Chern character.

Findings

Exactly two covariant calculi exist near classical differentials.

Differential calculi behave similarly to classical counterparts.

Constructed the q-deformed Chern character of the tautological bundle.

Abstract

Covariant first order differential calculus over quantum complex Grassmann manifolds is considered. It is shown by a Pusz-Woronowicz type argument that under restriction to calculi close to classical Kaehler differentials there exist exactly two such calculi for the homogeneous coordinate ring. Complexification and localization procedures are used to induce covariant first order differential calculi over quantum Grassmann manifolds. It is shown that these differential calculi behave in many respects as their classical counterparts. As an example the q-deformed Chern character of the tautological bundle is constructed. Keywords: Quantum groups, quantum spaces, quantum Grassmann manifolds, differential calculus