12/18/15, noon: No class, but homework is due in my mailbox. [Text:
Sections 5.6-9; 6.1]
12/18/15, noon: No class, but homework is due in my mailbox. [Text:
Sections 5.6-9; 6.1]
12/16/15: Corollaries: decomposition of a semisimple Lie algebra into
the direct sum of simple ones; semisimplicity of quotients; the
commutant; no outer derivations. [Text: Section 6.1]
12/14/15: The Killing form. Cartan's criteria for semisimplicity and
solvability. The Jordan decomposition. [Text: Sections 5.8-9]
12/11/15: Reductivity criterion: proof. Semisimplicity of classical
Lie algebras. [Text: Section 5.7]
12/9/15: Reductive Lie algebras. Invariant bilinear forms on a Lie
algebra. Reductivity criterion: main lemma (Theorem 5.43). [Text:
Sections 5.6-7 through the statement of Theorem 5.48]
12/7/15: Simple and semisimple Lie algebras. Example: sl(2). The Levi
decomposition. [Text: Section 5.6 through 5.42]
12/4/15: Homework is due at the beginning of the class. Corollaries of
Lie's theorem. The idea of proof of Engel's theorem. The radical and
semisimple Lie algebras. [Text: Sections 5.2-5; 5.5 (5.32 and the
proof of 5.34), 5.6 (5.35, 5.39)]
12/2/15: Lie's and Engel's theorems. [Text: Section 5.5, skipping 5.32
and the proof of 5.34]
11/30/15: More on solvable and nilpotent Lie algebras. Upper
triangular matrices. [Text: Section 5.4]
11/27/15: Thanksgiving break. Use the Lie Theory homework (to be
posted soon) as an excuse to leave that boring Thanksgiving dinner.
11/25/15: Corollaries of the PBW theorem. Ideals and the
commutant. Solvable Lie algebras. Nilpotent Lie algebras. [Text:
Sections 5.2-4 through 5.27]
11/23/15: The "computer science" proof of the PBW theorem. [Text:
Section 5.2]
11/20/15: Homework is due at the beginning of the class. Discussion of
homework. The idea of proof of the Poincaré-Birkhoff-Witt (PBW)
theorem. [Text: Sections 4.7, 5.1; 5.2 through 5.11]
11/18/15: Representations of a Lie algebra g and its universal
enveloping algebra Ug. The center of Ug. Example: the Casimir operator
for so(3,R). The Poincaré-Birkhoff-Witt theorem (without
proof so far). [Text: Sections 5.1 and 5.2]
11/16/15: Representations of S1 and Fourier series. The
universal enveloping algebra: definition and construction. [Text:
Sections 4.7 (the very end) and 5.1 through 5.4]
11/13/15: The Peter-Weyl theorem. [Text: Section 4.7 through Theorem
4.49 (the Peter-Weyl theorem)]
11/11/15: Characters: orthogonality. [Text: Section 4.7 through 4.46]
11/9/15: Matrix coefficients: orthogonality (proofs). [Text: Section
4.7 through the proofs of 4.41 and 4.42]
11/6/15: Homework is due at the beginning of the class. Discussion of
homework: Problem F (the centers of Sp(n,K) and sp(n,K)). Matrix
coefficients: orthogonality (main statements: 4.41 and 4.42). [Text:
Sections 4.2-6; 4.7 through the statement of Lemma 4.42]
11/4/15: Haar measure for compact Lie groups and complete
reducibility. [Text: Section 4.6]
11/2/15: Representations of finite groups. Haar measure for Lie
groups. Volume forms for compact Lie groups. [Text: Sections 4.5-6
through Theorem 4.34(2)]
10/30/15: Application of Schur's lemma: centers of classical Lie
groups and algebras. Irreducible representations of
S1. Unitary representations. Complete reducibility. [Text:
Sections 4.4-5 through Theorem 4.30]
10/28/15: Representations of R. Intertwining operators and
Schur's lemma. Application: irreducible representations of abelian Lie
groups and algebras. [Text: Sections 4.3-4]
10/26/15: Subrepresentations, direct sums, tensor products, duals,
endomorphisms. Irreducibility. Complete reducibility. [Text: Sections
4.2.1-2, and 4.3 through p. 57]
10/23/15: Homework is due at the beginning of the class. Discussion of
homework: Problem 3.12. [Text: Sections 3.2-5, 3.7-9, 4.1]
10/21/15: Ideals and normal subgroups. {Representations of G} =
{Representations of Lie(G)}. Examples: SO(3,R) and
SU(2). Complexification: sl(n,K) and su(n). [Text: Sections
3.4, 3.9, and 4.1]
10/19/15: Sketches of proofs of the existence of a Lie subgroup
theorem and the fullness of the Lie functor theorem. [Text: Section 3.8]
10/16/15: Morphisms of Lie groups and Lie algebras. Lie subgroups and
Lie subalgebras. Sketch of proof of Lie's Third Theorem. [Text:
Section 3.8]
10/14/15: Two Lie brackets on vector fields on a manifold: through
action on points and on functions. The Lie algebra of left-invariant
vector fields. The Baker-Campbell-Hausdorff formula. [Text: Sections
3.5 and 3.7]
10/12/15: The commutator of vector fields. Note: we used a more common
convention for the commutator of vector fields; under it, the
commutator we defined on T1G corresponds to the commutator
of left-invariant vector fields on G, contrary to Corollary 3.28 and
Exercise 3.4. [Text: Section 3.5]
10/9/15: Homework is due at the beginning of the class. [x,y] and
Ad(X) for matrix groups. Ad and ad. The Jacobi identity. Abstract Lie
algebras. The happy end: a functor {Lie groups} → {Lie algebras}
having a faithful restriction to the subcategory {Connected Lie
groups}. [Text: Example 3.14 and Section 3.3]
10/7/15: Morphisms of Lie groups are determined by the differential
maps at 1. The commutator and its properties. [Text: Prop. 3.9, Section 3.2]
10/5/15: Bootstrapping a locally defined one-parameter subgroup to a
globally defined one: g(t) := g(t/n)n. The
exponential and logarithmic maps for a general Lie group. Examples:
U(1) and SO(3). [Text: Section 3.1 through the end, skipping
Prop. 3.9]
10/2/15: Remarks on the topology of classical groups. One-parameter
subgroups in a Lie group. [Text: Sections 2.7 through the end and 3.1
through Definition 3.2]
9/30/15: Classical groups. [Text: Section 2.7 through the proof of
Theorem 2.30]
9/28/15: Recap on left-invariant vector fields. Classical
groups. [Text: Sections 2.6 and 2.7 before Theorem 2.30]
9/25/15: Homework is due at the beginning of the class. Discussion of
homework. Actions of a Lie group on itself. [Text: Sections 2.2-6]
9/23/15: Orbits and homogeneous spaces. [Text: Section 2.5]
9/21/15: Lie subgroups and homomorphism theorem. Actions and
representations. [Text: Sections 2.3-4]
9/18/15: Cosets. [Text: Section 2.2 after Corollary 2.10]
9/16/15: Subgroups. Connected components and the universal
cover. [Text: Section 2.2 through Corollary 2.10]
9/14/15: Introduction. Syllabus handed out. Lie groups: definitions
and examples.
[
Syllabus. Text (Kirillov): Section 2.2 through Example 2.5]
9/11/15: No class meeting: I am still out of town. [Read Section 2.1
of the text]
9/9/15: No class meeting: I am out of
town. [Read
Syllabus and Text (Kirillov): Chapter 1 (do not worry, this is
just the introduction)]
Last modified: (2016-01-23 13:00:32 CST)