Math 261B: Lie Groups—Fall 2020

Time and place
Subject matter
Homework and grades
Problem sets
List of lecture topics


(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.


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.


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.

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

Due Friday, Dec. 18 (the last day of final exam week). Before Sunday, Dec. 13, please take a few minutes to fill out the Course Evaluation.

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 𝔾a and 𝔾m. 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 𝔾m, action by projection on VG. 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 GLn 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 = TeG 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 = GLn. Slides
  7. (9/17) Borels and maximal tori. Action of the torus on the Lie algebra for GLn. Weight lattice, roots, positive roots, simple roots. Compare with SLn. Root homomorphisms SL2 → G, co-weight lattice, co-roots. Cartan matrix. Quick look at PGLn and Langlands duals. Slides
  8. (9/22) More details on the root systems of SLn, GLn, and PGLn. The center of a reductive algebraic group. Center of SLn as a group scheme in any characteristic. Canonical homomorphism SLp → PGLp 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 Uα 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 SOn and Sp2n. Slides
  11. (10/1) SOn as group preserving a bilinear form for char(K) ≠ 2. Roots, coroots, Weyl group, Cartan matrix and Dynkin diagram in detail for SO2n+1. Isomorphism PSL2 → SO3 giving the root subgroups for the short roots. Slides
  12. (10/6) Review SO2n+1 from last time. Groups with finite center and this root system: Spin2n+1 (simply connected) and SO2n+1 (adjoint). Spin3 ≅ SL2. Compact real forms and a physical demonstration of Spin3(ℝ) in action. Slides
  13. (10/8) SO2n, Cartan matrix and Weyl group of type Dn. Spin groups. Isomorphisms of SO4 and its cousins with other groups. Symplectic group Sp2n. Langlands dualities between classical groups. Slides
  14. (10/13) Fixing characteristic 2 problems by defining ON 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 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 GLn 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 λ comes from. Symmetric powers of the defining representation of GLn. Introductory remarks and classical examples on Chevalley forms. Slides
  18. (10/27) Definition of the Chevalley form of Vλ. Further examples. Slides
  19. (10/29) More examples of Vλ. Construction of 𝒪(G) and 𝒰(𝔤). Slides
  20. (11/3) Examples. (I) Showing that 𝒰(𝔤) is a subring of 𝒰(𝔤) by straightening with respect to the PBW decomposition: (a) to move 𝒰(𝔱), use its Hopf algebra structure and adjoint action on weight vectors; (b) to switch 𝒰(𝔲±), reduce to a nice computation in SL2. (II) Isomorphism PSL2 ≅ SO3 over (started). Slides
  21. Finished discussing the isomorphism PSL2 ≅ SO3 over . 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 𝒪q(T) and 𝒪q(T). Quantum 𝒪v(B) and 𝒰v(𝔟) in two steps: (1) replace 𝒰(𝔱) with 𝒪q(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 𝒰q(𝔰𝔩2). Slides
  24. More details on 𝒰q(𝔰𝔩2). Standard highest weight modules Vm. Tensor product of modules for Hopf algebras. Example: V1⊗V1 ≅ V3 ⊕ V0. Slides
  25. Representation theory of 𝒰q(𝔰𝔩2) for modules with standard weight spaces. Quantum Casimir operator and complete reducibility. Computation of Vm ⊗ Vn using characters. Definition of 𝒪q(SL2) as the subalgebra of 𝒰q(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 𝒪q(SL2). Definition of 𝒰q(𝔤)*, given Cartan data of a general G and symmetrizing numbers di 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 𝒰q(𝔤) compared to classical 𝒰(𝔤). Quantum function algebra 𝒪q(G). Canonical/crystal bases of the irreducible representations Vλ: examples, defining properties, tensor product rule for the crystal graphs. Slides

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