Weak Hopf algebra

Weak Hopf algebras are a generalization of Hopf algebras. They were introduced by Bohm, Nill and Szlachanyi in [1]. They are very often used in category theory since it was shown in [2] that any fusion category (which is a monoidal category with extra properties) is equivalent to a category of modules over a weak Hopf algebra.

Definition

A weak bialgebra $(H,\mu ,\eta ,\Delta ,\varepsilon )$ over a field $k$ is a vector space $H$ such that

• $(H,\mu ,\eta )$ forms an associative algebra with multiplication $\mu :H\otimes H\rightarrow H$ and unit $\eta :k\rightarrow H$,
• $(H,\Delta ,\varepsilon )$ forms a coassociative coalgebra with comultiplication $\Delta :H\rightarrow H\otimes H$ and counit $\varepsilon :H\rightarrow k$,

for which the following compatibility conditions hold :

1. Multiplicativity of the Coproduct :

   $\Delta \circ \mu =(\mu \otimes \mu )\circ (\mathrm {id} _{H}\otimes \sigma _{H,H}\otimes \mathrm {id} _{H})\circ (\Delta \otimes \Delta )$,

2. Weak Multiplicativity of the Counit :

   $\varepsilon \circ \mu \circ (\mu \otimes \mathrm {id} _{H})=(\varepsilon \otimes \varepsilon )\circ (\mu \otimes \mu )\circ (\mathrm {id} _{H}\otimes \Delta \otimes \mathrm {id} _{H})=(\varepsilon \otimes \varepsilon )\circ (\mu \otimes \mu )\circ (\mathrm {id} _{H}\otimes \Delta ^{op}\otimes \mathrm {id} _{H})$,

3. Weak Comultiplicativity of the Unit :

   $(\Delta \otimes \mathrm {id} _{H})\circ \Delta \circ \eta =(\mathrm {id} _{H}\otimes \mu \otimes \mathrm {id} _{H})\circ (\Delta \otimes \Delta )\circ (\eta \otimes \eta )=(\mathrm {id} _{H}\otimes \mu ^{op}\otimes \mathrm {id} _{H})\circ (\Delta \otimes \Delta )\circ (\eta \otimes \eta )$,

where $\sigma _{V,W}:V\otimes W\rightarrow W\otimes V:v\otimes w\mapsto w\otimes v$ flips the two tensor factors. Moreover $\mu ^{op}=\mu \circ \sigma _{H,H}$ is the opposite multiplication and $\Delta ^{op}=\sigma _{H,H}\circ \Delta$ is the opposite comultiplication. Note that we also implicitly use Mac Lane's coherence theorem for the monoidal category of vector spaces, identifying $(U\otimes V)\otimes W\cong U\otimes (V\otimes W)$ as well as $V\otimes k\cong V\cong k\otimes V$.

A weak Hopf algebra $(H,\mu ,\eta ,\Delta ,\varepsilon ,S)$ is a weak bialgebra $(H,\mu ,\eta ,\Delta ,\varepsilon )$ with a linear map $S:H\to H$, called the antipode, that satisfies:

• $\mu \circ (\mathrm {id} _{H}\otimes S)\circ \Delta =(\varepsilon \otimes \mathrm {id} _{H})\circ (\mu \otimes \mathrm {id} _{H})\circ (\mathrm {id} _{H}\otimes \sigma _{H,H})\circ (\Delta \otimes \mathrm {id} _{H})\circ (\eta \otimes \mathrm {id} _{H})$,
• $\mu \circ (S\otimes \mathrm {id} _{H})\circ \Delta =(\mathrm {id} _{H}\otimes \varepsilon )\circ (\mathrm {id} _{H}\otimes \mu )\circ (\sigma _{H,H}\otimes \mathrm {id} _{H})\circ (\mathrm {id} _{H}\otimes \Delta )\circ (\mathrm {id} _{H}\otimes \eta 0$,
• $S=\mu \circ (\mu \otimes \mathrm {id} _{H})\circ (S\otimes \mathrm {id} _{H}\otimes S)\circ (\Delta \otimes \mathrm {id} _{H})\circ \Delta$.

References

• Bohm, Gabriella; Nill, Florian; Szlachanyi, Kornel (1999). "Weak Hops algebras. I. Integral theory and $C^{*}$-stucture". Journal of Algebra. 221 (2): 385--438. doi:10.1006/jabr.1999.7984. ISSN 0021-8693.

• Etingof, Pavel; Nikshych, Dimitri; Ostrik, Viktor (2005). "On fusion categories". Annals of Mathematics. Second Series. 162 (2): 581--642. doi:10.4007/annals.2005.162.581. ISSN 0003-486X.