Advances in Quantum and Braided Geometry

Abstract

We demonstrate our recent general results on the Casimir construction and moduli space of all bicovariant calculi by means of some detailed examples, including finite-difference and 2-jet cacluli on ${\mathbb{R}}^n$ and full details of the Casimir construction of the 4D calculus of $S{U}_q(2)$. We likewise demonstrate our previous general constructions with T. Brzezinski of quantum group gauge theory with examples of such nonuniversal differential calculi on spacetime. We outline a notion of quantum homotopy of a quantum space. We indicate a possible application to classical integrable systems.

18 pages LATEX, no figures. Final version (cut the introductory material)

References in corpus (4)

• Coalgebra Gauge Theory
• Noncommutative Geometry of Finite Groups
• Differential Calculus in Braided Abelian Categories
• Some Remarks on Quantum and Braided Group Gauge Theory

Cited by in corpus (5)

• Quantum differentials and the q-monopole revisited
• Meaning of Noncommutative Geometry and the Planck-Scale Quantum Group
• Coalgebra Extensions and Algebra Coextensions of Galois Type
• Geometry of Quantum Homogeneous Vector Bundles and Representation Theory of Quantum Groups I
• Quantum geometry of field extensions