## A Gentle Introduction to Lie algebras and Orbits |

**David Vella**

September 29, 2005

1:00 pm

Bailey Hall 102

I will start from scratch - that is, with the definition of a Lie algebra, and then, starting with a very simple example (g = sl_2(C), the complex 2x2 matrices of trace 0), outline the theory of orbits of elements. We will see the difference between orbits of semisimple elements and orbits of nilpotent elements, which will lead us to ask 4 basic questions about (nilpotent) orbits. These 4 questions will be the guideposts for the entire semester. Our aim is to understand the answers to these questions for more and more complicated Lie algebras over the complex numbers. At the end of the talk, I will outline the structure theory of a complex semisimple Lie algebra (root systems, etc.) For those unfamiliar with root systems, it will be possible to continue to attend the seminar for at least the first half of the semester, and still learn something, always thinking in terms of the canonical example of g = sl_n(C), the special linear Lie algebras.

In subsequent weeks, we will continue this study, although it is not clear if we will have a formal lecture every week or if the format will relax into a casual discussion of the proofs of certain key theorems.

For additional information, send e-mail to math@union.edu or call (518) 388-6246.

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