5/6/16: Equivariant line bundles over G/B and the integral weight
lattice. The Borel-Weil-Bott (BWB) theorem: formulation. Proof of the
Borel-Weil part, which deals with H0 or global
sections of equivariant line bundles over G/B. [Lecture notes, Chapter
27 (The Bruhat Decomposition) of Bump's text
on Lie
Groups, Lurie's
notes]
5/4/16: The equivariant Picard group of G/B is isomorphic to the usual
Picard group of G/B. [Lecture
notes, Lurie's
notes]
5/2/16: Equivariant vector bundles over G/B. [Lecture notes]
4/29/16: Homework is due at the beginning of the class. Generalized
flag varieties G/B. Examples: flag varieties for classical simple Lie
groups. [Text: Sections 8.3-7; lecture notes]
4/27/16: The adjoint representation of sl(n,C). Characters of
representations of sl(n,C), the Vandermonde determinant and the
Schur functions. The Borel-Weil-Bott theorem: motivation. [Text:
Section 8.7 through the end and lecture notes]
4/25/16: Representations of sl(n,C) and Young diagrams. [Text:
Section 8.7 through 8.44]
4/22/16: The positive Weyl chamber and the fundamental domain for the
action of W on E. Dominant weights and the fundamental domain for the
action of W on the integral weight lattice P. Multiplicities and a
basis of the W-invariant subalgebra of the group
algebra C[P]. [Text: Exercise 7.11.8, Lemma 8.22, and Section
8.6]
4/20/16: The q-dimension and the usual dimension of a
finite-dimensional irreducible representation. [Text: Section 8.5
through the end]
4/18/16: The Weyl character formula: the character of a
finite-dimensional irreducible representation. The q-dimension. [Text:
Section 8.5 through Definition 8.38]
4/15/16: Homework is due at the beginning of the class. The Weyl
character formula: the completed group algebra of the weight lattice
and the character of a Verma module. [Text: Sections 7.9 and 8.1-2;
8.5 through Lemma 8.33]
4/13/16: Dominant weights and classification of irreducible
finite-dimensional representations (skipping part of the proof). The
BGG resolution. [Sections 8.3-4]
4/11/16: The weight decomposition of Verma modules, with a detailed
proof of Theorem 8.14, which is less immediate than presented in the
textbook, as it deals with studying the weight decomposition of the
universal enveloping algebra Un- as a
subrepresentation of the Cartan subalgebra h via the adjoint action
on Ug. [Section 8.2 through the end]
4/8/16: More on the characters for sl(2,C). Verma
modules. [Section 8.2 through 8.13]
4/6/16: Properties of characters. The invariance of weights, weight
spaces, and the character under the Weyl group action. Example:
sl(2,C). [Section 8.1 through the end]
4/4/16: Representations of a semisimple Lie algebra. The weight
lattice P of a simisimple Lie algebra and the
weights P(V) of a representation V. The character of a
representation. Example: sl(2,C). [Section 8.1 through 8.6]
4/1/16: Homework is due at the beginning of the class. Discussion of
homework (7.15). Verification of the Chevalley-Serre relations. [Text:
Sections 7.6-8, 10; 7.9 through the end]
3/30/16: The Chevalley-Serre relations and the Serre theorem (skipping
the proofs). Classification of simple Lie algebras. [Text: Section
7.9, skipping the proofs]
3/28/16: Proof of the classification theorem in the simply laced
case. [Text: Section 7.10]
3/25/16: The Cartan matrix and the Dynkin diagram of a root
system. Classification of Dynkin diagrams. [Text: Section 7.8]
3/21/16 - 3/23/16: Classes canceled: I will be at
a conference
on Higher Structures at the Max Planck Intitute in Bonn,
Germany.
3/14/16 - 3/18/16: No classes: Spring Break.
3/11/16: Applications of the notion of the length: the freeness of the
action of the Weyl group on the sets of Weyl chambers and
polarizations, the longest Weyl group element. Irreducible root
systems. [Text: Section 7.7 through the end]
3/9/16: The length of an element of the Weyl group as the number of
separating hyperplanes and as the minimal number of simple
reflections. [Text: Section 7.7 through Theorem 3.37]
3/7/16: The action of the Weyl group on the set of
polarizations. Simple reflections. The orbits of simple roots under
the Weyl group action. [Text: Sections 7.6 through the end and 7.7
through the proof of theorem 7.30]
3/4/16: Homework is due at the beginning of the class. Transitivity of
the action of the Weyl group on the set of Weyl chambers. [Text:
Sections 6.6 after Example 6.40, 7.1-4, 7.6 through Definition 7.22;
7.6 through Corollary 7.28]
3/2/16: Polarizations, positive and simple roots, Weyl
chambers. [Text: Sections 7.4, 7.6 through Definition 7.22]
2/29/16: The Weyl group. Seven options for relative position of two
linearly independent roots. Classification of rank two root systems:
A1 x A1, A2, B2, and
G2. [Text: Sections 7.2-3]
2/26/16: The imaginary part of the Cartan subalgebra and the compact
real form. Reduced root systems. Examples: the root system of
sl(n,C) and the root system An-1. [Text: Section 7.1
through the end]
2/24/16: Remarks about the real span of the roots in a semisimple Lie
algebra. Definition of an abstract root system. [Text: Sections 6.6
through the end and 7.1 through Theorem 7.3]
2/22/16: The structure of the root system for a semisimple Lie
algebra. [Text: Sections 6.6 through Theorem 6.44]
2/19/16: Homework is due at the beginning of the class. Discusson of
homework: Problems 6.5 and 6.7, as well as Theorem 6.39. Further
properties of root systems. [Text: Sections 4.8, 6.4-6.6 through 6.40;
6.6 through Lamma 6.43]
2/17/16: Fishing out copies of sl(2,C) in a semisimple Lie
algebra. [Text: Section 6.6 through Lemma 6.42]
2/15/16: A maximal toral subalgebra in a semisimple Lie algebra is
Cartan. The root decomposition and the root system for a semisimple
Lie algebra. Example: sl(n,C). [Text: Sections 6.5
through the end and 6.6 through 6.40]
2/12/16: Toral and maximal toral (Cartan) subalgebras. Root
decomposition with respect to a toral subalgebra. The diagonal Li
subalgebra in sl(n,C) is Cartan. [Text: Sections 6.4
through the end, 6.5 through Example 6.34]
2/10/16: Semisimple and nilpotent elements in a Lie algebra. Toral
subalgebras. [Text: Section 6.4 through 6.29]
2/8/16: Representations of sl(2,C), continued: Verma modules
and classification of irreducible representations. [Text: Section 4.8]
2/5/16: Homework is due at the beginning of the class: Problems 4 and
5. [Class notes. Text: Sections 5.6, 5.8, 6.1-3]
2/3/16: Representations of sl(2,C). [Text: Section 4.8 through
Lemma 4.57]
2/1/16: Complete reducibility of representations of a semisimple Lie
algebra: Lie algebra cohomology, Whitehead's lemma, proof of the main
theorem. [Text: Section 6.3 through the end]
1/29/16: Complete reducibility of complex representations of a
semisimple Lie algebra: action of Casimirs on irreducible
representations, short exact sequences, extensions, the Ext functor.
[Text: Section 6.3 up to Lemma 6.21]
1/27/16: The compact real form of a complex semisimple Lie
algebra. The only Lie algebra with positive definite Killing form is
0. Complete reducibility of representations of a semisimple Lie
algebra: proof using Weyl's unitary trick and compact real forms,
Casimir elements, main theorem (formulation). [Text: Sections 6.2
through the end and 6.3 through 6.15]
1/25/16: Recollection: the Levi decomposition, the Killing form,
Cartan's criteria for semisimplicity and solvability, decomposition of
a semisimple Lie algebra into the direct sum of simple ones; Der g = g
for a semisimple Lie algebra g. Relation between semisimple Lie
algebras and compact groups. [Text: Sections 5.6, 5.8, 6.1, and 6.2
through Remark 6.11]
1/22/16: Maurer-Cartan forms: relation to the Haar measure,
principal G-bundles and connections, the MC form as a
principal G-connection, the Maurer-Cartan equation and the
flatness of the connection. [Class
notes. Connection
(principal bundle) on Wikipedia]
1/20/16: Introduction. Syllabus handed out. Maurer-Cartan forms:
definition, left invariance, and examples.
[Syllabus.
Class
notes. Maurer-Cartan
form on Wikipedia]
Last modified: (2016-09-08 20:37:23 CDT)