Noncommutative Geometry of Finite Groups

Abstract

A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left-, right- and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and, moreover, for the formulation of field theories and dynamics on finite sets. Associated with each bicovariant first order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures. For a covariant calculus, there are notions of invariance for linear connections and tensors. All these concepts are explored for finite groups and illustrated with examples. Some results are formulated more generally for arbitrary associative (Hopf) algebras. In particular, the problem of extension of a connection on a bimodule (over an associative algebra) to tensor products is investigated, leading to the class of `extensible connections'. It is shown that invariance properties of an extensible connection on a bimodule over a Hopf algebra are carried over to the extension. Furthermore, an invariance property of a connection is also shared by a `dual connection' which exists on the dual bimodule (as defined in this work).

34 pages, Latex

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Cited by in corpus (27)

• On Curvature in Noncommutative Geometry
• Noncommutative geometry and physics: a review of selected recent results
• Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms
• Discrete Riemannian Geometry
• Electromagnetism and Gauge Theory on the Permutation Group $S_3$
• Noncommutative Riemannian Geometry of the Alternating Group A_4
• Nonassociative geometry in quasi-Hopf representation categories II: Connections and curvature
• Gravity on Finite Groups
• Differential Geometry of Group Lattices
• Advances in Quantum and Braided Geometry
• Cartan Pairs
• Riemannian Geometry of a Discretized Circle and Torus
• Discretized Yang-Mills and Born-Infeld actions on finite group geometries
• Twisting all the way: from algebras to morphisms and connections
• Quantum Double and Differential Calculi
• Finite group discretization of Yang-Mills and Einstein actions
• Pseudo-Riemannian Metrics in Models Based on Noncommutative Geometry
• Linear Connections on Graphs
• Dirac-Connes Operator on Discrete Abelian Groups and Lattices
• Riemannian Geometry of Bicovariant Group Lattices
• Moduli of quantum Riemannian geometries on <= 4 points
• Noncommutative de Rham cohomology of finite groups
• Classification of differentials and Cartan calculus on bicrossproducts
• A note on Connections and Bimodules
• Levi-Civita Connections on the Quantum Groups SL_{q}(N), O_{q}(N) and Sp_{q}(N)
• Quantum Calculi: differential forms and vector fields in noncommutative geometry
• The Exponential Map for Hopf Algebras