Math 261B: Lie Groups—Fall 2020

Announcements

Professor

Time and place

Subject matter

Textbooks

Homework and grades

Problem sets

List of lecture topics

Announcements

(12/1) Posted Problem Set 2. All homework due on the last day of final exam week, which is Friday, Dec. 18.

(10/6) Posted a revised version of Problem Set 1, fixing a mistake in problem 3(d).

(9/24) Problem Set 1 is posted, see below.

(8/27) Under "Textbooks" I've posted links to references that may be helpful for topics in the first part of the course. I'm posting slides of the lectures alongside the lecture topics. Links to the Zoom recordings are on the bCourses page.

(8/13) Welcome to Math 261B! This class will be held on Zoom. The Zoom link is posted on bCourses. If you have any difficulty accessing the link, please let me know.

Professor

Mark Haiman,
Office hours by appointment

Time and place

Tues-Thurs 12:30-2pm, online

Subject matter

This is the second semester of the Math 261A-B Lie Groups sequence.

I will assume familiarity with basic Lie theory, as covered in Math 261A. This couse will largely be a seminar on more advanced topics, although I will probably review some of the basics as needed.

Some topics I hope to cover are: (1) Algebraic groups and their representations; (2) Construction of complex reductive Lie groups as algebraic groups; (3) Reductive algebraic groups over fields of characteristic p; (4) Chevalley groups; (5) Hopf algebras, especially the enveloping algebra of a Lie algebra and the algebra of functions on a linear algebraic group; (6) Quantum groups and crystal/canonical bases.

Textbooks

There is no textbook for this course. I will give a self-contained presentation of the material in the lectures. You may find some of the following reference books helpful. The online access links below use the UC Berkeley library proxy server, so they should work from off campus with CalNet authentication.

• R. W. Carter, I. G. Macdonald and G. B. Segal, Lectures on Lie Groups and Lie Algebras, Cambridge Univ. Press (1995). The third part, by Macdonald, is on algebraic groups. Online access.
• T. A. Springer, Linear Algebraic Groups, 2nd Ed., Birkhauser Boston (1998). Online access

Homework and grades

I will assign occasional problems on the topics discussed. You can pick and choose to do whichever of them interest you. For grading purposes, I only require that you turn in some reasonable number of problems.

Problem sets

• Problem Set 1. Here is the LaTeX source in case you want to use it as a template for writing solutions.
• Problem Set 2, LaTeX source.

List of lecture topics

I will update this list as the course goes on.

1. (8/27) Overview of the general structure of algebraic groups and comparison with Lie groups over ℂ. Slides
2. (9/1) Hopf algebra structure of the algebra A of functions on an algebraic group G. Representations of G interpreted as comodules for A. Slides
3. (9/3) Local finiteness of comodules; why affine algebraic groups are linear. Examples of G acting on 𝒪(G) for 𝔾_a and 𝔾_m. General geometric lemmas. Closed subgroups: G/H is a linear algebraic group with coordinate ring 𝒪(G)^H if H is normal, or a smooth but generally not affine algebraic variety if H is not normal. Slides
4. (9/8) All algebraic irreducible representations of G occur in 𝒪(G). Characterizations of unipotent groups: the only irreducible is the trivial representation; unitriangularity of the action on all representations; existence of a faithful unitriangular representation. Unipotent radical. Reynolds operator R∈A^*: example for 𝔾_m, action by projection on V^G. Complete reducibility via Reynolds operator. Slides
5. (9/10) Reductive and unipotent decompostion of linear algebraic groups in general, example of Borel in GL₂. Tori, characters and cocharacters. Cartan data: getting started on GL_n as an example. Slides
6. (9/15) Smooth varieties, tangent spaces, regular local coordinates, vector fields as derivations on 𝒪(X), Kahler 1-forms, tangent and cotangent bundles. For A = 𝒪(G): Lie algebra 𝔤 = T_eG as a subspace of A^*, Lie bracket as commutator in A^*, and adjoint action by conjugation in A^*. Aside on the algebra 𝒰 ⊂ A^* of left invariant differential operators and comparison with the enveloping algebra 𝒰(𝔤)^* in characteristic 0 versus p. Computation of the Lie bracket and adjoint action for G = GL_n. Slides
7. (9/17) Borels and maximal tori. Action of the torus on the Lie algebra for GL_n. Weight lattice, roots, positive roots, simple roots. Compare with SL_n. Root homomorphisms SL₂ → G, co-weight lattice, co-roots. Cartan matrix. Quick look at PGL_n and Langlands duals. Slides
8. (9/22) More details on the root systems of SL_n, GL_n, and PGL_n. The center of a reductive algebraic group. Center of SL_n as a group scheme in any characteristic. Canonical homomorphism SL_p → PGL_p is bijective but not an isomorphism in char p. Representation theory and complete reducibility for SL₂ in characteristic zero. Slides
9. (9/24) Clarification about characteristic p phenomena—everything works except representations of reductive groups aren't completely reducible. Example: the irreducible and standard representations of SL₂(K) when char(K) = 3, for highest weights up to 4. Root subgroups U_α and their action on weight spaces (started). Slides
10. (9/29) Root subgroups continued. The Weyl group and its action on weights. Axioms for Cartan data, and what they classify: (i) reductive algebraic groups over any algebraic closed field, (ii) complex reductive (= semisimple × complex torus) Lie groups, (iii) compact real Lie groups. Start on examples: the classical groups SO_n and Sp_2n. Slides
11. (10/1) SO_n as group preserving a bilinear form for char(K) ≠ 2. Roots, coroots, Weyl group, Cartan matrix and Dynkin diagram in detail for SO_2n+1. Isomorphism PSL₂ → SO₃ giving the root subgroups for the short roots. Slides
12. (10/6) Review SO_2n+1 from last time. Groups with finite center and this root system: Spin_2n+1 (simply connected) and SO_2n+1 (adjoint). Spin₃ ≅ SL₂. Compact real forms and a physical demonstration of Spin₃(ℝ) in action. Slides
13. (10/8) SO_2n, Cartan matrix and Weyl group of type D_n. Spin groups. Isomorphisms of SO₄ and its cousins with other groups. Symplectic group Sp_2n. Langlands dualities between classical groups. Slides
14. (10/13) Fixing characteristic 2 problems by defining O_N as the group that preserves a standard quadratic form. Examples for N=2 (because SO₂ as a subgroup of O₂ already involves a subtle point) and N=3, with computation of the Lie algebra. Introduction to reductive group schemes over ℤ and the functors they represent, with basic examples. Slides
15. (10/15) Start on representations of reductive groups. Highest weight vectors. Standard modules from line bundles on the flag variety. Embedding V^* → H⁰(G/B, ℒ_−λ) for irreducible V with highest weight λ. Slides
16. (10/20) Representation theory continued. Examples of standard highest weight modules: SL₂ modules revisited; adjoint action of SL₃ on its Lie algebra; exterior powers of the defining representation of GL_n and corresponding line bundles on the flag variety. Slides
17. (10/22) Highest weight module examples continued. Parabolic subgroups and which G/P a line budle ℒ_λ comes from. Symmetric powers of the defining representation of GL_n. Introductory remarks and classical examples on Chevalley ℤ forms. Slides
18. (10/27) Definition of the Chevalley ℤ form of V^λ. Further examples. Slides
19. (10/29) More examples of V^λ_ℤ. Construction of 𝒪_ℤ(G) and 𝒰_ℤ(𝔤). Slides
20. (11/3) Examples. (I) Showing that 𝒰_ℤ(𝔤) is a subring of 𝒰(𝔤) by straightening with respect to the PBW decomposition: (a) to move 𝒰_ℤ(𝔱), use its Hopf algebra structure and adjoint action on weight vectors; (b) to switch 𝒰_ℤ(𝔲_±), reduce to a nice computation in SL₂. (II) Isomorphism PSL₂ ≅ SO₃ over ℤ (started). Slides
21. Finished discussing the isomorphism PSL₂ ≅ SO₃ over ℤ. Finite Chevalley groups and finite groups of Lie type. Slides
22. Examples of finite groups of Lie type. Start on quantum groups. Hopf pairing between 𝒪(B) and 𝒰(𝔟) for the Borel B in SL₂ as initial motivation. Slides
23. Quantum tori and Hopf pairing between 𝒪_q(T) and 𝒪_q(T^∨). Quantum 𝒪_v(B) and 𝒰_v(𝔟) in two steps: (1) replace 𝒰(𝔱) with 𝒪_q(T^∨); (2) deform the relations of 𝒪(B) and the coproduct in 𝒰(𝔟) to get isomorphic self-dual Hopf algebras which are neither commutative nor co-commutative. Definition of 𝒰_q(𝔰𝔩₂). Slides
24. More details on 𝒰_q(𝔰𝔩₂). Standard highest weight modules V^m. Tensor product of modules for Hopf algebras. Example: V¹⊗V¹ ≅ V³ ⊕ V⁰. Slides
25. Representation theory of 𝒰_q(𝔰𝔩₂) for modules with standard weight spaces. Quantum Casimir operator and complete reducibility. Computation of V^m ⊗ Vⁿ using characters. Definition of 𝒪_q(SL₂) as the subalgebra of 𝒰_q(𝔰𝔩₂)^* spanned by matrix entries of the standard irreducibles. Generation by matrix entries a, b, c, d of V¹ (relations to be discussed next lecture). Expression for the coproduct. Slides
26. Working out defining relations of 𝒪_q(SL₂). Definition of 𝒰_q(𝔤)^*, given Cartan data of a general G and symmetrizing numbers d_i for the Cartan matrix. Consistency of the basic relations with the co-product. Triangular decomposition. Quantum Serre relations to be discussed next time. Slides
27. Quantum Serre relations defined and explained: (1) as kernel of the Hopf pairing between upper and lower halves; (2) as precisely the additional relations that hold in integrable modules. Slides
28. Representation theory of 𝒰_q(𝔤) compared to classical 𝒰(𝔤). Quantum function algebra 𝒪_q(G). Canonical/crystal bases of the irreducible representations V^λ: examples, defining properties, tensor product rule for the crystal graphs. Slides

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