Loop groups and noncommutative geometry

Quantum Groups and Noncommutative Geometry

Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalisation of symmetry groups for certain integrable systems, and on the other as part of a generalisation of geometry itself powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century sees at its end the birt...

Noncommutative Geometry and Quantum Groups

Bounded noncommutative geometry and boundary actions of hyperbolic groups

We use the canonical Lipschitz geometry available on the boundary of a hyperbolic group to construct finitely summable Fredholm modules over the crossed product of the boundary action. The procedure yields a full set of representatives of the K-homology of the crossed product, and all of them are p-summable for p in an appropriate range related to the geometry of the boundary. By restricting th...

Noncommutative Differential Geometry and Twisting of Quantum Groups

We outline the recent classification of differential structures for all main classes of quantum groups. We also outline the algebraic notion of ‘quantum manifold’ and ‘quantum Riemannian manifold’ based on quantum group principal bundles, a formulation that works over general unital algebras.

Noncommutative Geometry and Gauge Theory on Discrete Groups

We build and investigate a pure gauge theory on arbitrary discrete groups. A systematic approach to the construction of the differential calculus is presented. We discuss the metric properties of the models and introduce the action functionals for unitary gauge theories. A detailed analysis of two simple models based on Z2 and Z3 follows. Finally we study the method of combining the discrete an...