**6510.001, Fall 2000**

**Lie Algebras**

**Conley**

the University of North Texas. |

**INSTRUCTOR:** Charles Conley, GAB
475,
(940) 565 - 3326

**OFFICE HOURS:** Monday and Wednesday,
11:30-12:30 and 2:00-3:00

**CLASS MEETS:** Monday and Wednesday, 12:30 - 1:50,
GAB
318

**TEXT:** *Introduction to Lie Algebras and
Representation Theory,* J. Humphreys

This text is a little bit too concise for our purposes, so I will try to hand out notes as we go along. I am open to suggestions for different texts: I would like something that avoids Lie groups, assumes only a minimal knowledge of linear algebra, and does many examples in detail. There are many choices, for example the well-known books by Varadarajan (published by Springer), Jacobson (Dover), and Serre.

**PLAN:** My aim is to teach this class as concretely
as
possible, so as to make it accessible to those who are not completely
comfortable
with abstract algebra. For this reason, I will prove most of the
main
theorems only in the setting of one of the main examples to which they
are
applied. I will assume only elementary linear algebra, and will
do
everything over the fields C and R. We will begin with the
definitions
of Lie algebras and their representations, and give some
examples.
Then we will look at nilpotent and solvable Lie algebras, concentrating
on
the examples of the upper triangular nilpotent matrices, and the upper
triangular
matrices. We will prove the theorems of Lie and Engel for these
examples,
and discuss the Cartan criterion for semisimplicity.

From this point on we will study semisimple Lie algebras,
concentrating on sl(n,C), the complex n by n matrices of trace 0,
especially the cases n=2
and n=3*.* We will begin with the representations of sl(2,C)*,*
and then study the "fine structure" of sl(n,C), i.e., its Cartan
subalgebras, root system, and Weyl group (which in this case is the
permutation group on
n letters). Finally, we will study the representations of
sl(n,C), especially
those of sl(3,C). We will conclude with an algebraic proof of the
Weyl
character formula, certainly for sl(2,C), and if there is time, for
sl(3,C),
and maybe even sl(n,C).

I