## Math 261 - Lie groups and Lie algebras

**Instructor: ** Allen Knutson, allenk@math.berkeley.edu

**Lectures have moved: **MWF 1:00pm-2:00pm,
Room 70 Evans Hall

**Course Control Number:** 55248

**Office:** 1033
Evans Hall,
642-4319

**Prerequisites:**

**Text:** Fulton and Harris, "Representation Theory"

Also suggested: Fulton, "Young Tableaux"

We have a full-TA now:

**TA:** Nick Proudfoot, email address proudf at-sign math,berkeley,edu.

Check present enrolment
here. It says "limit of 30" but I don't think that means anything.

There are some notes available online in PostScript
and PDF.

###
Here's a survey for you to fill out and
bring to class, so I can get an idea of the general background.

The general plan is to spend the first semester on the general linear
group. You may think you know linear algebra now, but there is
always more linear algebra worth learning. We'll get enough
of the general definitions in place to
Aside from being a nearly painless place to see all the main strands
of Lie theory, GL_n is far and away the most relevant Lie group for
the rest of mathematics. I've put it front and center for this reason.

The principal way this course will differ from standard books on
the subject is in eschewing algebra (in the sense of Lie algebras)
in favor of geometry pretty much wherever possible. Virtually all
current research on representation theory is based on geometry at
some level, and it is misleading to present it otherwise, in addition
to making it less intuitive (for most people). However, since I don't
expect people to have algebraic geometry at their fingertips,
we will sometimes prove our results by some other means and then
figure out what the geometry was ex post facto.

Old homework is here.
Current homework: none over Thanksgiving.

Here is a planned list of topics for the first term, in order.
[a.b] refers to Fulton and Harris, Chapter a, Verse b.

Here is a record of what we've actually done so far.

I know less about what'll happen second term, but some topics are
fairly obvious:
Lie algebras, Killing form
Classification of complex simple Lie algebras via root systems
Engel's, Lie's, and Levi's theorems
Universal enveloping algebras and their centers
Representation theory of complex Lie groups
and the high-weight classification
the Borel-Weil theorem for other Lie groups
Verma modules
conjugacy classes in compact Lie groups
Less obvious:
the automorphism group of a complex simple Lie algebra
the fundamental representations of the other Lie groups
conjugacy classes in complex Lie groups
real forms
Demazure modules
Bott-Samelson manifolds
The Demazure and Littelmann character formulae