Weak Hopf algebras

A weak Hopf algebra over a field $K$ , on a basis $e_{1}, \dots, e_{D}$ , is given by five arrays, stored sparsely with exact entries in $\mathbb{Q}$ or a number field:

Map	On the basis	Size
$\mul$	$\cmul{e_{i} e_{j}} = \sum_{k} \nabla_{ij}^{k}\, e_{k}$	$D^{3}$
$\unit$	$\unit = \sum_{i} u_{i}\, e_{i}$	$D$
$\cop$	$\cop(e_{i}) = \sum_{j,k} \Delta_{i}^{jk}\, e_{j} \otimes e_{k}$	$D^{3}$
$\cou$	$\cou(e_{i})$	$D$
$\ant$	$\ant(e_{i}) = \sum_{j} S_{i}^{j}\, e_{j}$	$D^{2}$

In string diagrams, read from top to bottom, the five maps are:

\mul

\cop

\unit

\cou

\ant

product, coproduct, unit, counit, antipode

Axioms

Böhm, Nill and Szlachányi (1999). $(\mul, \unit)$ is an associative unital algebra and $(\cop, \cou)$ a coassociative counital coalgebra, and the coproduct is multiplicative:

\cop(ab) = \cop(a)\,\cop(b)

A bialgebra would also have $\cop(\unit) = \unit \otimes \unit$ and a multiplicative counit. A weak bialgebra asks only for the weak unit axiom and the weak counit axiom (each with its mirror image):

(\cop \otimes \mathrm{id})\,\cop(\unit) = (\cop(\unit) \otimes \unit)(\unit \otimes \cop(\unit))

\cou(abc) = \cou(a b_{(1)})\,\cou(b_{(2)} c)

The counital maps $\epst(a) = \cou(\unit_{(1)} a)\,\unit_{(2)}$ and $\epss(a) = \unit_{(1)}\,\cou(a\, \unit_{(2)})$ have images the counital subalgebras $\At$ and $\As$ . An antipode satisfies

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

together with $\ant(a_{(1)})\, a_{(2)} = \epss(a)$ and $\ant(a_{(1)})\, a_{(2)}\, \ant(a_{(3)}) = \ant(a)$ .

Weak or genuine

H \text{ is a Hopf algebra} \iff \cop(\unit) = \unit \otimes \unit \iff \dim \At = 1.

Then $\epst(a) = \cou(a)\unit$ and the axioms are those of a Hopf algebra; see Hopf algebras.

Checking

classify(H) returns the tags that hold, each decided by an exact zero test: algebra, coalgebra, weak_bialgebra, bialgebra, weak_hopf, hopf, and qt_weak_hopf when an $R$ -matrix is stored. A tag whose prerequisite fails is reported as not tested.