## 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:

• confidence in linear algebra (dual spaces and tensor products would be nice, but we'll go over them just in case)
• differential topology (moreso the second term)
• 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
• prove a GL_n theorem
• state the theorem for general Lie groups
• but wait to prove it until the second term.
• 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.

• Getting started
• Definitions of representation theory [1.1]
• Representation theory of finite groups; complete reducibility [1.2]
• Character theory and the classification of representations [2]
• Definition of physics (!), and application of symmetry to normal modes of molecules
• The group algebra and the representation ring [3.4]
• The Peter-Weyl theorem for finite groups
• Our first Lie group: representation theory of tori (and the Fourier transform)
• Definitions of matrix Lie groups and Lie algebras
• Our second Lie group: representation theory of SL_2(C) via Lie algebras [11]
• Representation theory of GL_n
• Classification in terms of highest weights:
• Injection into the representation ring of T
• The fundamental representations
• Irreps correspond 1:1 with dominant weights
• Note: The following topic is just a reinterpretation of things we'll have already proved, so don't worry if you don't know any algebraic geometry.
• Geometry of GL_n's representation theory
• The complex flag manifold
• The Plucker, Veronese, and Segre embeddings
• Homogeneous coordinate rings for the flag manifold
• Statement of Borel-Weil theorem for GL_n
• Some applications of Lie algebras (to Lie groups)
• Raising and lowering operators
• Irreps of U(2)
• The flag manifold, and applications of Borel-Weil
• The Bruhat decomposition (just the dense orbit statement)
• Proof of Borel-Weil
• The Weyl character formula
• Applications of Weyl character formula
• The Kostant multiplicity formula
• The Steinberg tensor product formula
• Character theory for GL_n
• The branching rule from GL_n to GL_{n-1}
• The Gel'fand-Cetlin basis for representations of GL_n
• Tensor products of GL_n representations
• The Pieri formula
• The hive ring
• Associativity of the hive ring
• Hives compute GL_n tensor products
• Hives correspond 1:1 with honeycombs (statement)
• Puzzles give inequalities on hives
• Relation to cohomology of Grassmannians (statement)
• A hifalutin approach: representations of the category Vec
• Schur-Weyl duality
• Weyl's unitary trick [9.3]: U(n) vs. GL_n
• 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