Hopf Algebra - 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 Algebra

Famous quotes containing the words lie and/or groups:

    In the hollow Lotos land to live and lie reclined
    On the hills like Gods together, careless of mankind.
    Alfred Tennyson (1809–1892)

    Women over fifty already form one of the largest groups in the population structure of the western world. As long as they like themselves, they will not be an oppressed minority. In order to like themselves they must reject trivialization by others of who and what they are. A grown woman should not have to masquerade as a girl in order to remain in the land of the living.
    Germaine Greer (b. 1939)