under construction
symmetric monoidal (∞,1)-category of spectra
algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
A Hopf -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 -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 -representation category of a (weak) -Hopf algebra is a rigid monoidal C-star-category with fiber functor. (Böhm-Szlachanyi).
algebras equipped with a suitable coproduct, but without an antipode – hence just -bialgebras – , are considered in
S. Baaj, Georges Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de -algèbres. Ann. scient. Ec. Norm. Sup., 4e série, 26 (1993), 425–488.
J.-M. Vallin, -algèbres de Hopf et -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 -context is discussed in
Weak -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 -Hopf algebras (pdf)
Last revised on April 8, 2013 at 16:21:43. See the history of this page for a list of all contributions to it.