**Cohomology of Lie Groups**

The cohomology algebra of a Lie group is a Hopf algebra: the multiplication is provided by the cup-product, and the comultiplication

by the group multiplication *G* × *G* → *G*. 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)** Let *A* be a finite-dimensional, graded commutative, graded cocommutative Hopf algebra over a field of characteristic 0. Then *A* (as an algebra) is a free exterior algebra with generators of odd degree.

Read more about this topic: Hopf Algebra

### Other articles related to "cohomology of lie groups, cohomology, lie group, group, of lie groups":

**Cohomology of Lie Groups**

... The

**cohomology**algebra of a

**Lie group**is a Hopf algebra the multiplication is provided by the cup-product, and the comultiplication by the

**group**... Using this structure, Hopf proved a structure theorem for the

**cohomology**algebra

**of Lie groups**...

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