Editing Hopf algebra (section)

===Weak Hopf algebras===
[[Weak Hopf algebra]]s, or quantum groupoids, are  generalizations of Hopf algebras. Like Hopf algebras, weak Hopf algebras form a self-dual class of algebras; i.e., if ''H'' is a (weak) Hopf algebra, so is ''H''*, the dual space of linear forms on ''H'' (with respect to the algebra-coalgebra structure obtained from  the natural pairing with ''H'' and its coalgebra-algebra structure). A weak Hopf algebra ''H'' is usually taken to be a
*finite-dimensional algebra and coalgebra with coproduct Δ: ''H'' → ''H'' ⊗ ''H''  and counit ε: ''H'' → ''k'' satisfying all the axioms of Hopf algebra except possibly Δ(1) ≠ 1 ⊗ 1 or ε(''ab'') ≠ ε(''a'')ε(''b'') for some ''a,b'' in ''H''. Instead one requires the following:

::<math> (\Delta(1) \otimes 1)(1 \otimes \Delta(1)) = (1 \otimes \Delta(1))(\Delta(1) \otimes 1) = (\Delta \otimes \mbox{Id})\Delta(1)</math>
::<math> \epsilon(abc) = \sum \epsilon(ab_{(1)})\epsilon(b_{(2)}c) = \sum \epsilon(ab_{(2)})\epsilon(b_{(1)}c)</math>

:for all ''a'', ''b'', and ''c'' in ''H''.
* ''H'' has a weakened antipode ''S'': ''H'' → ''H'' satisfying the axioms:

#<math>S(a_{(1)})a_{(2)} = 1_{(1)} \epsilon(a 1_{(2)})</math> for all ''a'' in ''H'' (the right-hand side is the interesting projection usually denoted by Π<sup>''R''</sup>(''a'') or ε<sub>''s''</sub>(''a'') with image a separable subalgebra denoted by ''H<sup>R</sup>'' or ''H<sub>s</sub>''); 
#<math>a_{(1)}S(a_{(2)}) =  \epsilon(1_{(1)}a)1_{(2)}</math> for all ''a'' in ''H'' (another interesting projection usually denoted by Π<sup>''R''</sup>(''a'') or ε<sub>''t''</sub>(''a'') with image a separable algebra ''H<sup>L</sup>'' or ''H<sub>t</sub>'', anti-isomorphic to ''H<sup>L</sup>'' via ''S'');
#<math>S(a_{(1)})a_{(2)}S(a_{(3)}) = S(a) </math> for all ''a'' in ''H''.

:Note that if Δ(1) = 1 ⊗ 1, these conditions reduce to the two usual conditions on the antipode of a Hopf algebra.

The axioms are partly chosen so that the category of ''H''-modules is a [[rigid category|rigid monoidal category]]. The unit ''H''-module is the separable algebra ''H<sup>L</sup>'' mentioned above. 
 
For example, a finite [[groupoid]] algebra is a weak Hopf algebra. In particular, the groupoid algebra on [n] with one pair of invertible arrows ''e<sub>ij</sub>'' and ''e<sub>ji</sub>''  between ''i'' and ''j'' in [''n''] is isomorphic to the algebra ''H'' of ''n'' x ''n'' matrices. The weak Hopf algebra structure on this particular ''H'' is given by coproduct Δ(''e<sub>ij</sub>'') = ''e<sub>ij</sub>'' ⊗ ''e<sub>ij</sub>'', counit ε(''e<sub>ij</sub>'') = 1 and antipode ''S''(''e<sub>ij</sub>'') = ''e<sub>ji</sub>''. The separable subalgebras ''H<sup>L</sup>'' and ''H<sup>R</sup>'' coincide and are non-central commutative algebras in this particular case (the subalgebra of diagonal matrices).

Early theoretical contributions to weak Hopf algebras are to be found in<ref>{{cite journal | last1 = Böhm | first1 = Gabriella | last2 = Nill | first2 = Florian | last3 = Szlachanyi | first3 = Kornel | year = 1999 | title =  Weak Hopf Algebras| journal = J. Algebra | volume = 221 | issue = 2| pages = 385–438 | doi=10.1006/jabr.1999.7984| arxiv = math/9805116 | s2cid = 14889155 }}</ref> as well as<ref>{{cite book |first1=Dmitri |last1=Nikshych |first2=Leonid |last2=Vainerman |chapter=Finite groupoids and their applications |chapter-url=https://books.google.com/books?id=I3IK9U5Co_0C&pg=PA211 |editor-first=S. |editor-last=Montgomery |editor2-first=H.-J. |editor2-last=Schneider |title=New directions in Hopf algebras |publisher=M.S.R.I. Publications |location=Cambridge |date=2002 |isbn=9780521815123   |pages=211–262 |url= |volume=43}}</ref>