Hopf Algebras - Cohomology 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 multiplication G × GG. 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 Algebras

Famous quotes containing the words lie and/or groups:

    It is comforting when one has a sorrow to lie in the warmth of one’s bed and there, abandoning all effort and all resistance, to bury even one’s head under the cover, giving one’s self up to it completely, moaning like branches in the autumn wind. But there is still a better bed, full of divine odors. It is our sweet, our profound, our impenetrable friendship.
    Marcel Proust (1871–1922)

    Trees appeared in groups and singly, revolving coolly and blandly, displaying the latest fashions. The blue dampness of a ravine. A memory of love, disguised as a meadow. Wispy clouds—the greyhounds of heaven.
    Vladimir Nabokov (1899–1977)