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.
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.
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Under Tannaka duality (semisimple) weak Hopf algebras corespondo to (multi-)fusion categories (Ostrik).
Weak comultiplications were introduced in
where also weak quasi-bialgebras are considered and physical motivation is discussed in detail. Further work in this vain is in
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
Takahiro Hayashi, A canonical Tannaka duality for finite seimisimple tensor categories (arXiv:math/9904073)
Victor Ostrik, Module categories, weak Hopf algebras and modular invariants (arXiv:math/0111139)