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fundamental quantum groupoid (Definition)
Definition 0.1   A fundamental quantum groupoid $F_{\mathcal Q}$ is defined as a functor $F_{\mathcal Q}: \H _B \to {\mathcal Q}_G$, where ${\H }_B$ is the category of Hilbert space bundles, and ${\mathcal Q}_G$ is the category of quantum groupoids and their homomorphisms.

Fundamental Groupoid Functors and Functor Categories

The natural setting for the definition of a quantum fundamental groupoid $F_{\mathcal Q}$ is in one of the functor categories– that of fundamental groupoid functors, $F_{\mathcal G}$, and their natural transformations defined in the context of quantum categories of quantum spaces ${\mathcal Q}$ represented by Hilbert space bundles or `rigged' Hilbert (or Frechét) spaces ${\H }_B$.

Other related functor categories are those specified with the general definition of the fundamental groupoid functor, $F_{\mathcal G}: \textbf{Top} \to \mathcal G_2$, where Top is the category of topological spaces and $\mathcal G_2$ is the groupoid category.

Example 0.1  

A specific example of a quantum fundamental groupoid can be given for spin foams of spin networks, with a spin foam defined as a functor between spin network categories. Thus, because spin networks or graphs are specialized one-dimensional CW-complexes whose cells are linked quantum spin states, their quantum fundamental groupoid is defined as a functor representation of CW-complexes on `rigged' Hilbert spaces (also called Frechét nuclear spaces).



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See Also: quantum groupoid, fundamental groupoid functors in quantum theories, quantum groupoids and C*-algebras

Other names:  quantum fundamental groupoid
Also defines:  quantum groupoid homomorphism
Keywords:  fundamental groupoid, quantum fundamental groupoid, functor category, natural transformations, spin networks, spin foams

Cross-references: rigged Hilbert spaces, functor representation, spin, graphs, spin foam, spin networks, spin foams, topological, fundamental groupoid functor, general definition, functor categories, Hilbert space bundles, quantum categories, quantum fundamental groupoid, homomorphisms, quantum groupoids, category, category of Hilbert space, functor

This is version 3 of fundamental quantum groupoid, born on 2009-02-13, modified 2009-02-20.
Object id is 513, canonical name is FundamentalQuantumGroupoid.
Accessed 887 times total.

Classification:
Physics Classification: 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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