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\begin{document}

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\section*{weak bialgebra}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{physical_motivation}{Physical motivation}\dotfill \pageref*{physical_motivation} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{idempotents_projections}{Idempotents (``projections'')}\dotfill \pageref*{idempotents_projections} \linebreak
\noindent\hyperlink{relation_to_fusion_categories}{Relation to fusion categories}\dotfill \pageref*{relation_to_fusion_categories} \linebreak
\noindent\hyperlink{literature}{Literature}\dotfill \pageref*{literature} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of \emph{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]].

\hypertarget{physical_motivation}{}\subsubsection*{{Physical motivation}}\label{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.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

A \textbf{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, (i),(ii) and (iii), hold:

(i) the coproduct $\Delta$ is multiplicative $\Delta(x)\Delta(y)= \Delta(x y)$. If only (i) is satisfied, following Böhm, Caenapeel and Janssen 2011, we may speak of a \textbf{prebialgebra}.

(ii) the counit $\epsilon$ satisfies weak multiplicativity

\begin{displaymath}
\epsilon(x y z) = \epsilon(x y_{(1)})\epsilon(y_{(2)} z),
\end{displaymath}
\begin{displaymath}
\epsilon(x y z) = \epsilon(x y_{(2)})\epsilon(y_{(1)} z).
\end{displaymath}
A prebialgebra satisfying the first (the second) of the above properties is said to be left (right) monoidal.

(iii) Weak comultiplicativity of the unit:

\begin{displaymath}
\Delta^{(2)} (1) = (\Delta(1) \otimes 1)(1\otimes \Delta(1))
\end{displaymath}
\begin{displaymath}
\Delta^{(2)} (1) = (1 \otimes\Delta(1))(\Delta(1) \otimes 1)
\end{displaymath}
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 $\Delta^{(2)} := (id\otimes\Delta)\Delta = 
(\Delta\otimes id)\Delta$.

A weak $k$-bialgebra $A$ is a \textbf{weak Hopf algebra} if it has a $k$-linear map $S:A\to A$ (which is then called an antipode) such that for all $x\in A$

\begin{displaymath}
x_{(1)} S(x_{(2)}) = \epsilon(1_{(1)} x)1_{(2)},
\end{displaymath}
\begin{displaymath}
S(x_{(1)})x_{(2)} = 1_{(1)} \epsilon(x 1_{(2)}),
\end{displaymath}
\begin{displaymath}
S(x_{(1)})x_{(2)} S(x_{(3)}) = S(x)
\end{displaymath}
It follows that the antipode is antimultiplicative, $S(x y)=S(y)S(x)$, and anticomultiplicative, $\Delta(S(x)) = S(x)_{(1)}\otimes S(x)_{(2)} = 
S(x_{(2)})\otimes S(x_{(1)})$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{idempotents_projections}{}\subsubsection*{{Idempotents (``projections'')}}\label{idempotents_projections}

For every weak bialgebra there are $k$-linear maps $\Pi^L,\Pi^R:A\to A$ defined by

\begin{displaymath}
\Pi^L(x) := \epsilon(1_{(1)} x) 1_{(2)},\,\,\,\,
\Pi^R(x) := 1_{(1)}\epsilon(x 1_{(2)}).
\end{displaymath}
Expressions for $\Pi^L(x),\Pi^R(x)$ are already met above as the right hand sides in two of the axioms for the antipode. Maps $\Pi^L,\Pi^R$ are idempotents, $\Pi^R\Pi^R = \Pi^R$ and $\Pi^L\Pi^L = \Pi^L$:

\begin{displaymath}
\itexarray{
\Pi^L(\Pi^L(x)) &=& \epsilon\left(1_{(1')}\epsilon(1_{(1)}x) 1_{(2)}\right)1_{(2')}
= \epsilon(1_{(1)}x)\epsilon(1_{(1')}1_{(2)}) 1_{(2')} 
\\ &=&\epsilon(1_{(1)}x)\epsilon(1_{(2)}) 1_{(3)} 
= \epsilon(1_{(1)}x)1_{(2)} = \Pi^L(x).
}
\end{displaymath}
Notice $\epsilon(x z) = \epsilon(x 1 z)  = \epsilon(x 1_{(2)})\epsilon(1_{(1)}z)) = \epsilon(x \epsilon(1_{(1)}z))1_{(2)} = \epsilon(x\Pi^L(z)) = \epsilon(\Pi^R(x)z)$. 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 $\Pi^R(\Pi^R(x)y) = \Pi^R(x y)$, dually $\Delta(A^L)\subset A\otimes A^L$ and $\Delta(A^R)\subset A^R\otimes A$, and in particular $\Delta(1)\in A^R\otimes A^L$.

Sometimes it is also useful to consider the idempotents $\bar\Pi^L,\bar\Pi^R:A\to A$ defined by

\begin{displaymath}
\bar\Pi^L(x) := \epsilon(1_{(2)} x) 1_{(1)},\,\,\,\,
\bar\Pi^R(x) := 1_{(2)}\epsilon(x 1_{(1)}).
\end{displaymath}
\begin{displaymath}
\itexarray{
\bar\Pi^L(\bar\Pi^L(x))&=&\epsilon(1_{(2')}\epsilon(1_{(2)}x)1_{(1)})1_{(1')} = \epsilon(1_{(2)}x)\epsilon(1_{(2')}1_{(1)})1_{(1')} \\
&=& \epsilon(1_{(3)}x)\epsilon(1_{(2)})1_{(1)}=
\epsilon(1_{(2)}x)1_{(1)} = \bar\Pi^L(x).
}
\end{displaymath}
\hypertarget{relation_to_fusion_categories}{}\subsubsection*{{Relation to fusion categories}}\label{relation_to_fusion_categories}

Under [[Tannaka duality]] (semisimple) weak Hopf algebras correspond to (multi-)[[fusion categories]] (\hyperlink{Ostrik}{Ostrik}).

\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

Weak comultiplications were introduced in

\begin{itemize}%
\item G. Mack, [[Volker Schomerus]], \emph{Quasi Hopf quantum symmetry in quantum theory}, Nucl. Phys. B370(1992) 185.

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

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

\end{itemize}
Now these works are understood categorically from the point of view of weak monad theory:

\begin{itemize}%
\item [[Gabriella Böhm]], Stefaan Caenepeel, Kris Janssen, \emph{Weak bialgebras and monoidal categories}, Comm. Algebra \textbf{39} (2011), no. 12 (special volume dedicated to Mia Cohen), 4584-4607. \href{http://arxiv.org/abs/1103.2261}{arXiv:1103.226}
\item [[Gabriella Böhm]], [[Stephen Lack]], [[Ross Street]], \emph{Weak bimonads and weak Hopf monads}, J. Algebra 328 (2011), 1-30, \href{http://arxiv.org/abs/1002.4493}{arXiv:1002.4493}
\item [[Gabriella Böhm]], Jos\'e{} G\'o{}mez-Torrecillas, \emph{On the double crossed product of weak Hopf algebras}, \href{arXiv/1205.2163}{arXiv:1205.2163}

\end{itemize}
The relation to [[fusion categories]] is discussed in

\begin{itemize}%
\item Takahiro Hayashi, \emph{A canonical Tannaka duality for finite semisimple tensor categories} (\href{http://arxiv.org/abs/math/9904073}{arXiv:math/9904073})


\item [[Victor Ostrik]], \emph{Module categories, weak Hopf algebras and modular invariants} (\href{http://arxiv.org/abs/math/0111139}{arXiv:math/0111139})



\end{itemize}
category: algebra

[[!redirects weak bialgebras]]

[[!redirects weak Hopf algebra]] [[!redirects weak Hopf algebras]]



\end{document}