Weak Hopf algebra

Weak Hopf algebras are a generalization of Hopf algebras. They were introduced by Bohm, Nill and Szlachányi. The first motivations for studying weak Hopf algebras came from quantum field theory and operator algebras. It was later proved by Etingof, Nikshych and Ostrik 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 $V\otimes k\cong V\cong k\otimes V$.

The definition is fairly self-explanatory, one sees that it is the compatibility between the algebra and coalgebra structures that is weaken.

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:

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

Examples

1. Hopf algebra. Of course any Hopf algebra is a weak Hopf algebra.
2. Groupoid algebra. Suppose $G=(G_{0},G_{1})$ is a groupoid and let $K[G]$ be the groupoid algebra, in other words, the algebra generated by the morphisms $g\in G_{1}$. This becomes a weak Hopf algebra if we define
   • $\mu :K[G]\otimes K[G]\to K[G]~{\text{by}}~$ ...
   • $\eta :k\to K[G]~{\text{by}}~\eta (1)=\sum _{X\in G_{0}}\mathrm {id} _{X}$
   • $\Delta :K[G]\to K[G]\otimes K[G]~{\text{by}}~\Delta (g)=g\otimes g~{\text{for all}}~g\in G_{1}$
   • $\varepsilon :K[G]\to k~{\text{by}}~\varepsilon (g)=1~{\text{for all}}~g\in G_{1}$
   • $S:K[G]\to KG~{\text{by}}~S(g)=g^{-1}~{\text{for all}}~g\in G_{1}$.

Note that this second example is not a Hopf algebra.

References

• Bohm, Gabriella; Nill, Florian; Szlachányi, 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.