Schedule of the final presentation
- Day 1 (April 24th Monday 12.40-1.35PM, JR 221) Han Zhang, Yuancheng Xie
- Day 2 (April 25th Tuesday 10AM-12.30PM MW 317) Junjei Chen, Matthew Harper, Nick Bruno, Yilong Zhang
- Day 3 (May 2nd Tuesday 12-1.45PM, JR 221) Aniket Shah, Andreu Ferre Moragues, Andrew Castillo
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.