# Contents

## Idea

The notion of weak bialgebra is a generalization of that of bialgebra in which the comultiplication $\Delta$ is weak in the sense that $\Delta(1)\neq 1\otimes 1$ in general. (Still a special case of sesquialgebra.)

Correspondingly weak Hopf algebras generalize Hopf algebras accordingly.

### Physical motivation

This kind of structures naturally comes in CFT models relation to quantum groups a root of unity: the full symmetry algebra is not quite a quantum group at root of unity, because if it were one would have to include the nonphysical quantum dimension zero finite-dimensional quantum group representations into the (pre)Hilbert space; those are the zero norm states which do not contribute to physics (like ghosts). If one quotients by these states then the true unit of a quantum group becomes an idempotent (projector), hence one deals with weak Hopf algebras instead as a price of dealing with true, physical, Hilbert space.

## Properties

### Relation to fusion categories

Under Tannaka duality (semisimple) weak Hopf algebras corespondo to (multi-)fusion categories (Ostrik).

## Literature

Weak comultiplications were introduced in

• G. Mack, Volker Schomerus, Quasi Hopf quantum symmetry in quantum theory, Nucl. Phys. B370(1992) 185.

where also weak quasi-bialgebras are considered and physical motivation is discussed in detail. Further work in this vain is in

• G. Böhm, K. Szlachányi, A coassociative $C^\ast$-quantum group with non-integral dimensions, Lett. Math. Phys. 35 (1996) 437–456, arXiv:q-alg/9509008g/abs/q-alg/9509008); Weak $C*$-Hopf algebras: the coassociative symmetry of non-integral dimensions, in: Quantum groups and quantum spaces (Warsaw, 1995), 9-19, Banach Center Publ. 40, Polish Acad. Sci., Warszawa 1997.
• Florian Nill, Axioms for weak bialgebras, math.QA/9805104
• G. Böhm, F. Nill, K. Szlachányi, Weak Hopf algebras. I. Integral theory and $C^\ast$-structure, J. Algebra 221 (1999), no. 2, 385-438, math.QA/9805116

Now these works are understood categorically from the point of view of weak monad theory: * Gabriella Böhm, Stefaan Caenepeel, Kris Janssen, Weak bialgebras and monoidal categories, Comm. Algebra 39 (2011), no. 12 (special volume dedicated to Mia Cohen), 4584-4607. arXiv:1103.226 * Gabriella Böhm, Stephen Lack, Ross Street, Weak bimonads and weak Hopf monads, J. Algebra 328 (2011), 1-30, arXiv:1002.4493
* Gabriella Böhm, José Gómez-Torrecillas, On the double crossed product of weak Hopf algebras, arXiv:1205.2163

The relation to fusion categories is discussed in

Revised on April 8, 2013 17:08:36 by Urs Schreiber (82.113.106.75)