Contents
Contents
Idea
The notion of weak bialgebra is a generalization of that of bialgebra in which the comultiplication is weak in the sense that in general; similarly the compatibility of counit with the multiplication map is weakened (counit might fail to be a morphism of algebras). (Still a special case of sesquialgebra.)
Correspondingly weak Hopf algebras generalize Hopf algebras accordingly. Every weak Hopf algebra defines a Hopf algebroid.
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.
Definitions
A weak bialgebra is a tuple such that is an associative unital algebra, is a coassociative counital coalgebra and the following compatibilities, (i),(ii) and (iii), hold:
(i) the coproduct is multiplicative . If only (i) is satisfied, following Böhm, Caenapeel and Janssen 2011, we may speak of a prebialgebra.
(ii) the counit satisfies weak multiplicativity
A prebialgebra satisfying the first (the second) of the above properties is said to be left (right) monoidal.
(iii) Weak comultiplicativity of the unit:
A prebialgebra satisfying the first (the second) of the above properties is said to be left (right) comonoidal.
As usually in the context of coassociative coalgebras, we denoted .
A weak -bialgebra is a weak Hopf algebra if it has a -linear map (which is then called an antipode) such that for all
It follows that the antipode is antimultiplicative, , and anticomultiplicative, .
Properties
Idempotents (“projections”)
For every weak bialgebra there are -linear maps defined by
Expressions for are already met above as the right hand sides in two of the axioms for the antipode. Maps are idempotents, and :
Notice . The images of the idempotents and are dual as -linear spaces: there is a canonical nondegenerate pairing given by .
Also and , dually and , and in particular .
Sometimes it is also useful to consider the idempotents defined by
Relation to fusion categories
Under Tannaka duality (semisimple) weak Hopf algebras correspond 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 -quantum group with non-integral dimensions, Lett. Math. Phys. 35 (1996) 437–456, arXiv:q-alg/9509008g/abs/q-alg/9509008); Weak -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 -structure, J. Algebra 221 (1999), no. 2, 385-438, math.QA/9805116 #{BohmNillSzlachanyi}
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