Editing Hopf algebra (section)

==Formal definition==
Formally, a Hopf algebra is an (associative and coassociative) [[bialgebra]] ''H'' over a [[field (mathematics)|field]] ''K'' together with a [[linear transformation|''K''-linear]] map ''S'': ''H'' → ''H'' (called the '''antipode''') such that the following diagram [[commutative diagram|commutes]]:
<div style="text-align: center;">
[[File:Hopf algebra.svg|250px|antipode commutative diagram]]
</div>
Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. In the sumless [[Sweedler notation]], this property can also be expressed as
:<math>S(c_{(1)})c_{(2)}=c_{(1)}S(c_{(2)})=\varepsilon(c)1\qquad\mbox{ for all }c\in H.</math>

As for [[associative algebra|algebra]]s, one can replace the underlying field ''K'' with a [[commutative ring]] ''R'' in the above definition.<ref name=Und55>{{harvnb|Underwood|2011|p=55}}</ref>

The definition of Hopf algebra is [[Dual (category theory)|self-dual]] (as reflected in the symmetry of the above diagram), so if one can define a [[Dual space|dual]] of ''H'' (which is always possible if ''H'' is finite-dimensional), then it is automatically a Hopf algebra.<ref name=Und62>{{harvnb|Underwood|2011|p=62}}</ref>

=== Structure constants ===
Fixing a basis <math>\{e_k\}</math> for the underlying vector space, one may define the algebra in terms of [[structure constant]]s for multiplication:
:<math>e_i\nabla e_j = \sum_k \mu^k_{\;ij} e_k</math>
for co-multiplication:
:<math>\Delta e_i = \sum_{j,k} \nu^{\;jk}_i e_j\otimes e_k</math>
and the antipode:
:<math>S e_i = \sum_j \tau_i^{\;j} e_j</math>
Associativity then requires that
:<math>\mu^k_{\;ij}\mu^m_{\;kn}=\mu^k_{\;jn}\mu^m_{\;ik}</math>
while co-associativity requires that
:<math>\nu_k^{\;ij}\nu_i^{\;mn}=\nu_k^{\;mi}\nu_i^{\;nj}</math>
The connecting axiom requires that
:<math>\nu_k^{\;ij}\tau_j^{\;m}\mu^n_{\;im}=\nu_k^{\;jm}\tau_j^{\,\;i}\mu^n_{\;im}</math>

===Properties of the antipode===
The antipode ''S'' is sometimes required to have a ''K''-linear inverse, which is automatic in the finite-dimensional case{{clarify|date=May 2018|reason=Either provide a reference or briefly sketch an explanation}}, or if ''H'' is [[commutative]] or [[cocommutative]] (or more generally [[Quasitriangular Hopf algebra|quasitriangular]]).

In general, ''S'' is an [[antihomomorphism]],<ref>{{cite book|author=Dăscălescu, Năstăsescu & Raianu |chapter=Prop. 4.2.6|title=Hopf Algebra: An Introduction |year=2001|url={{Google books|plainurl=y|id=pBJ6sbPHA0IC|page=153|text=is an antimorphism of algebras}}|page=153}}</ref> so ''S''<sup>2</sup> is a [[homomorphism]], which is therefore an automorphism if ''S'' was invertible (as may be required).

If ''S''<sup>2</sup> = id<sub>''H''</sub>, then the Hopf algebra is said to be '''involutive''' (and the underlying algebra with involution is a [[*-algebra]]). If ''H'' is finite-dimensional semisimple over a field of characteristic zero, commutative, or cocommutative, then it is involutive.

If a bialgebra ''B'' admits an antipode ''S'', then ''S'' is unique ("a bialgebra admits at most 1 Hopf algebra structure").<ref>{{cite book|author=Dăscălescu, Năstăsescu & Raianu |chapter=Remarks 4.2.3|title=Hopf Algebra: An Introduction |year=2001|url={{Google books|plainurl=y|id=pBJ6sbPHA0IC|page=151|text=the antipode is unique}}|page=151}}</ref> Thus, the antipode does not pose any extra structure which we can choose: Being a Hopf algebra is a property of a bialgebra.

The antipode is an analog to the inversion map on a group that sends ''g'' to ''g''<sup>−1</sup>.<ref>[http://www.mathematik.uni-muenchen.de/~pareigis/Vorlesungen/98SS/Quantum_Groups/LN2_1.PDF Quantum groups lecture notes]</ref>

===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>

===Hopf orders===
A '''Hopf order''' ''O'' over an [[integral domain]] ''R'' with [[field of fractions]] ''K'' is an [[Order (ring theory)|order]] in a Hopf algebra ''H'' over ''K'' which is closed under the algebra and coalgebra operations: in particular, the comultiplication Δ maps ''O'' to ''O''⊗''O''.<ref name=Und82>{{harvnb|Underwood|2011|p=82}}</ref>

===Group-like elements===
A '''group-like element''' is a nonzero element ''x'' such that Δ(''x'') = ''x''⊗''x''. The group-like elements form a group with inverse given by the antipode.<ref>{{cite book | page=149 | title=Algebras, Rings, and Modules: Lie Algebras and Hopf Algebras | volume=168 | series=Mathematical surveys and monographs | first1=Michiel | last1=Hazewinkel | first2=Nadezhda Mikhaĭlovna | last2=Gubareni | first3=Vladimir V. | last3=Kirichenko | publisher=[[American Mathematical Society]] | year=2010 | isbn=978-0-8218-7549-0 }}</ref> A '''[[primitive element (co-algebra)|primitive element]]''' ''x'' satisfies Δ(''x'') = ''x''⊗1 + 1⊗''x''.<ref>{{cite book | at=p. 307, C.42  | title=The Concise Handbook of Algebra | editor1-first=Aleksandr Vasilʹevich | editor1-last=Mikhalev | editor2-first=Günter | editor2-last=Pilz | publisher=[[Springer-Verlag]] | year=2002 | isbn=978-0792370727 }}</ref><ref>{{cite book | title=Hopf Algebras | volume=74 | series=Cambridge Tracts in Mathematics | first=Eiichi | last=Abe | publisher=[[Cambridge University Press]] | year=2004 | isbn=978-0-521-60489-5 | page=59 }}</ref>