nLab Hopf C-star-algebra

under construction

Idea

A Hopf $C^\ast$-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 $C^\ast$-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 $C^\ast$-representation category of a (weak) $C^\ast$-Hopf algebra is a rigid monoidal C-star-category with fiber functor. (Böhm-Szlachanyi).

References

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

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

• J.-M. Vallin, $C^\ast$-algèbres de Hopf et $C^\ast$-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 $C^\ast$-context is discussed in

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

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

• Gabi Böhm, Kornel Szlachanyi, A Coassociative C-Quantum Group with Non-Integral Dimensions_ (arXiv:q-alg/9509008)

  Weak $C^\ast$-Hopf algebras (pdf)