Editing Hopf algebra (section)

===Hopf subalgebras===
A subalgebra ''A'' of a Hopf algebra ''H'' is a Hopf subalgebra if it is a subcoalgebra of ''H'' and the antipode ''S'' maps ''A'' into ''A''. In other words, a Hopf subalgebra A is a Hopf algebra in its own right when the multiplication, comultiplication, counit and antipode of ''H'' are restricted to ''A'' (and additionally the identity 1 of ''H'' is  required to be in A). The Nichols–Zoeller freeness theorem of Warren Nichols and [[Bettina Richmond|Bettina Zoeller]] (1989) established that the natural ''A''-module ''H'' is free of finite rank if ''H'' is finite-dimensional: a generalization of [[Lagrange's theorem (group theory)|Lagrange's theorem for subgroups]].<ref>{{citation
 | last1 = Nichols | first1 = Warren D.
 | last2 = Zoeller | first2 = M. Bettina | author2-link = Bettina Richmond
 | doi = 10.2307/2374514
 | issue = 2
 | journal = [[American Journal of Mathematics]]
 | mr = 987762
 | pages = 381–385
 | title = A Hopf algebra freeness theorem
 | volume = 111
 | year = 1989| jstor = 2374514
 }}</ref> As a corollary of this and integral theory, a Hopf subalgebra of a semisimple finite-dimensional Hopf algebra is automatically semisimple.

A Hopf subalgebra ''A'' is said to be right normal in a Hopf algebra ''H'' if it satisfies the condition of stability, ''ad<sub>r</sub>''(''h'')(''A'') ⊆ ''A'' for all ''h'' in ''H'', where the right adjoint mapping ''ad<sub>r</sub>'' is defined by ''ad<sub>r</sub>''(''h'')(''a'') = ''S''(''h''<sub>(1)</sub>)''ah''<sub>(2)</sub> for all ''a'' in ''A'', ''h'' in ''H''. Similarly, a Hopf subalgebra ''A'' is left normal in ''H'' if it is stable under the left adjoint mapping defined by ''ad<sub>l</sub>''(''h'')(''a'') = ''h''<sub>(1)</sub>''aS''(''h''<sub>(2)</sub>). The two conditions of normality are equivalent if the antipode ''S'' is bijective, in which case ''A'' is said to be a normal Hopf subalgebra.

A normal Hopf subalgebra ''A'' in ''H'' satisfies the condition (of equality of subsets of H): ''HA''<sup>+</sup> = ''A''<sup>+</sup>''H'' where ''A''<sup>+</sup> denotes the kernel of the counit on ''A''. This normality condition implies that ''HA''<sup>+</sup> is a Hopf ideal of ''H'' (i.e. an algebra ideal in the kernel of the counit, a coalgebra coideal and stable under the antipode). As a consequence one has a quotient Hopf algebra ''H''/''HA''<sup>+</sup> and epimorphism ''H'' → ''H''/''A''<sup>+</sup>''H'', a theory analogous to that of normal subgroups and quotient groups in [[group theory]].<ref>{{harvnb|Montgomery|1993|p=36}}</ref>