Editing Hopf algebra (section)

===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>