Editing Hopf algebra (section)

=== Cohomology of Lie groups ===
The cohomology algebra (over a field <math>K</math>) of a Lie group <math>G</math> is a Hopf algebra: the multiplication is provided by the [[cup product]], and the comultiplication 
:<math>H^*(G,K) \rightarrow H^*(G\times G,K) \cong H^*(G,K)\otimes H^*(G,K)</math>
by the group multiplication <math>G\times G\to G</math>. This observation was actually a source of the notion of Hopf algebra. Using this structure, Hopf proved a structure theorem for the cohomology algebra of Lie groups.

'''Theorem (Hopf)'''<ref name="Hopf, 1941">{{cite journal|last1=Hopf|first1=Heinz|title=Über die Topologie der Gruppen–Mannigfaltigkeiten und ihre Verallgemeinerungen|journal=Ann. of Math. |series= 2|date=1941|volume=42|issue=1|pages=22–52|doi=10.2307/1968985|language=de|jstor=1968985}}<!--|access-date=7 March 2016--></ref> Let <math>A</math> be a finite-dimensional, [[Graded-commutative|graded commutative]], graded cocommutative Hopf algebra over a field of characteristic 0. Then <math>A</math> (as an algebra) is a free exterior algebra with generators of odd degree.