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; 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.
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.
A weak bialgebra is a tuple $(A,\mu,\eta,\Delta,\epsilon)$ such that $(A,\mu,\eta)$ is an associative unital algebra, $(A,\Delta,\epsilon)$ is a coassociative counital coalgebra and the following compatibilities hold:
(i) the coproduct $\Delta$ is multiplicative $\Delta(x)\Delta(y)= \Delta(x y)$
(ii) the counit $\epsilon$ satisfies weak multiplicativity
(iii) Weak comultiplicativity of the unit:
As usually in the context of coassociative coalgebras, we denoted $\Delta^{(2)} := (id\otimes\Delta)\Delta = (\Delta\otimes id)\Delta$.
A weak $k$-bialgebra $A$ is a weak Hopf algebra if it has a $k$-linear map $S:A\to A$ (which is then called an antipode) such that for all $x\in A$
For every weak bialgebra there are $k$-linear maps $\Pi^L,\Pi^R:A\to A$ with properties $\Pi^R\Pi^R = \Pi^R$ and $\Pi^L\Pi^L = \Pi^L$ and defined by
These expressions are already met above in two of the axioms for the antipode. Then $\epsilon(x z) = \epsilon(x 1 z) = \epsilon(x 1_({2}))\epsilon(1_{(1)}z)) = \epsilon(x \epsilon(1_{(1)}z))1_{(2)} = \epsilon(x\Pi^L(z)) = \epsilon(\Pi^R(x)z)$. The images of the idempotents $A^R = \Pi^R(A)$ and $A^L = \Pi^L(R)$ are dual as $k$-linear spaces: there is a canonical nondegenerate pairing $A^L\otimes A^R\to k$ given by $(x,y) \mapsto \epsilon(y x)$.
Also $\Pi^L(x\Pi^L(y)) = \Pi^L(x y)$ and $\Pi^R(\Pi^R(x)y) = \Pi^R(x y)$, and dually $\Delta(A^L)\subset A\otimes A^L$ and $\Delta(A^R)\subset A^R\otimes A$, and $\Delta(1)\in A^R\otimes A^L$.
Under Tannaka duality (semisimple) weak Hopf algebras correspond 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:
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)