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
Famous quotes containing the words lie and/or groups:
“... when one reflects on the books one never has written, and never may, though their schedules lie in the beautiful chirography which marks the inception of an unexpressed thought upon the pages of one’s notebook, one is aware, of any given idea, that the chances are against its ever being offered to one’s dearest readers.”
—Elizabeth Stuart Phelps (1844–1911)
“Under weak government, in a wide, thinly populated country, in the struggle against the raw natural environment and with the free play of economic forces, unified social groups become the transmitters of culture.”
—Johan Huizinga (1872–1945)