Schedule of the final presentation

Topics with brief abstracts

Baker-Campbell-Hausdorff Formula. We have seen in class that exp(X)exp(Y) is in general not equal to exp(Y)exp(X). BCH formula explicitly computes the error terms in this equality.

Poisson Lie Groups and Lie bialgebras. A Poisson-Lie group is a Lie group which has a compatible structure of a Poisson manifold. The the Lie algebra inherits a 'cobracket', making it into a Lie bialgebra. Thus Lie's correspondence restricts to a correspondence between Poisson-Lie groups and Lie bialgebras.

Belavin-Drinfeld classification. A certain family of Lie bialgebra structures on g come from an element of tensor-square of g, called r-matrix. The axioms of LBA can be translated to r, and become the classical Yang-Baxter equation. Belavin and Drinfeld classified solutions of this for a simple Lie algebra g.

PBW theorem For a Lie algebra g, PBW theorem asserts that the associated graded of the universal enveloping algebra U(g) is isomorphic to the symmetric algebra S(g).

Haar measure Let G be a locally compact topological group. A Haar measure is a left-invariant measure on the Borel sigma algbera of measurable subsets of G. It is unique up to normalization (relative to a fixed compact subset K of G).

Stone von Neumann theorem This is a famous result regarding decomposition of an action of Heisenberg group on Hilbert spaces (involves Peter-Weyl theorem).

Weyl Character formula Using the construction of Verma modules we can prove the Weyl character formula purely algebraically.

Lie group of type G2 The exceptional Lie group G of type G2 arises as automorphisms of octanions.

Bruhat decomposition B orbits on G/B, via left action, are parametrized by the Weyl group.