Hopf algebras

A Hopf algebra is a weak Hopf algebra with $\cop(\unit) = \unit \otimes \unit$ ; equivalently the counit is multiplicative, $\At = K\unit$ , and the antipode axioms become

\ant(a_{(1)})\,a_{(2)} = \cou(a)\,\unit = a_{(1)}\,\ant(a_{(2)})

Its representations form a tensor category with a fiber functor, the forgetful functor to vector spaces. So a fusion category is $\operatorname{Rep}(H)$ of a Hopf algebra exactly when it has a fiber functor, which excludes Fibonacci and Ising; those need a weak Hopf algebra.

Facts used here

Group-likes $G(H) = \{g : \cop(g) = g \otimes g,\ \cou(g) = 1\}$ form a group, and $|G(H)|$ divides $\dim H$ (Nichols–Zoeller). The characters of $H$ are the group-likes of $H^{*}$ .
In characteristic 0, $H$ is semisimple exactly when it is cosemisimple, and then $\ant^{2} = \mathrm{id}$ (Larson–Radford).
$H$ is semisimple exactly when it has an integral $\Lambda$ with $\cou(\Lambda) = 1$ (Maschke). The traces $\operatorname{Tr}_{H}(\Lambda^{[n]})$ of its Sweedler powers sum the Frobenius–Schur indicators.
The commutative ones are $K^{G}$ and the cocommutative ones $K[G]$ over an algebraically closed field of characteristic 0 (Cartier–Kostant–Milnor–Moore, for finite dimension).

Examples in the database

Algebra	Dim	Semisimple	From
$K[G],\ K^{G}$	$\|G\|$	yes	$\|G\| \le 20$ (the commutative catalog)
Sweedler $H_{4}$ , Taft $T_{n}$	$n^{2}$	no	quantum linear spaces
Kac–Paljutkin $H_{8}$	8	yes	examples; abelian extensions of $D_{8}$
$u_{q}(\mathfrak{sl}_{2})$	27, 125	no	liftings
$D(H)$	$(\dim H)^{2}$	as H	Drinfeld doubles
$K^{F} \# K[H],\ K[G]^{J}$	$\|G\|$	yes	bismash products, extensions, twists

Open one below, or filter the full table under Algebras.