nLab
Hopf C-star-algebra

Contents

under construction

Context

Algebra

Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

A Hopf C *C^\ast-algebra according to (Vaes-VanDale) is a C-star algebra equipped with structure and property analogous to that of a Hopf algebra structure on the underlying associative algebra.

A weak C *C^\ast-Hopf algebra according to (Böhm-Szlachanyi) is a star-weak Hopf algebra such that has a faithful star-representation on a Hilbert space.

With suitable definitions the central Tannaka duality-property of Hopf algebras (that their representation category is a rigid monoidal category with fiber functor) is lifted to the operator algebra context: the C *C^\ast-representation category of a (weak) C *C^\ast-Hopf algebra is a rigid monoidal C-star-category with fiber functor. (Böhm-Szlachanyi).

References

C *C^\ast algebras equipped with a suitable coproduct, but without an antipode – hence just C *C^\ast-bialgebras – , are considered in

  • S. Baaj, Georges Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de C *C^\ast-algèbres. Ann. scient. Ec. Norm. Sup., 4e série, 26 (1993), 425–488.

  • J.-M. Vallin, C *C^\ast-algèbres de Hopf et C *C^\ast-algèbres de Kac. Proc. London Math. Soc. (3)50 (1985), 131–174.

The issue of how to add the definition of the antipode in the C *C^\ast-context is discussed in

  • Stefaan Vaes, Alfons Van Daele, Hopf C *C^\ast-algebras, Proc. London Math. Soc. (3) 82, 2001, 337-384. (arXiv:math/9907030)

Weak C *C^\ast-Hopf algebras and their C-star categories of representations are discussed in

Last revised on April 8, 2013 at 16:21:43. See the history of this page for a list of all contributions to it.