M206: Lie Groups and Lie algebras
This is the course website for the course M206 Fall 2017 with material relevant to the course.
The class meets Monday 5-7PM and Tuesday 1-2PM and 3-4PM at 706. Office hours Tuesday 2-3PM.
In the first part of the course we focus on Lie groups.
Part I. Lie Groups:
- Lie's Integrability Theorem
- Unitary Representations and Haar measure
- Fourier Decomposition and Peter-Weyl Theorem
- Coadjoint orbits and Borel-Weil-Bott
In the second part of the course we discuss Lie algebras.
Part II. Lie algebras:
- Structure theory of Lie algebras
- Representations of sl(2,C) and root systems
- Classification of simple Lie algebras
Textbook References:
- Lectures on Lie Groups and Lie Algebras, R.W. Carter, I.G. MacDonald, G.B. Segal, CUP (2012).
- Lectures on Lie groups and geometry, S.K. Donaldson, Imperial College London (2011).
- Introduction to Lie groups and Lie algebras, A. A. Kirillov, CUP (2008).
- Lectures on the Orbit Method, A. A. Kirillov, GSM 64 (2004).
Research articles:
- Lie algebra theory without algebra, S.K. Donaldson, Algebra, Arithmetic, and Geometry - PM Vol 269.
- Moment maps and Diffeomorphisms, S.K. Donaldson, Survery in Diff. Geom. (2000).
- Unitary Representations of nilpotent Lie groups, A. A. Kirillov, Russ. Math. Surv. (1962).
- Merits and demerits of the orbit method, A. A. Kirillov, Bull. AMS (1999).
- The Moment Map of a Lie Group Representation, N. J. Wildberger, Trans. AMS (1992).
Problem Sets :
- Problem Set 1: Due Oct 23.
- Problem Set 2: Due Nov 30.
Lecture Diaries
- Sep 29: Introduction to M206. (2h)
Symmetries of polygons. Representation of finite groups.
Continuous groups of symmetries. Lie groups of dimension 2.
The rotation group SO(3). The unitary group SU(2) and SL(2,R).
Quaternions as 3-dimensional rotations. The map SU(2) to SO(3).
- Oct 2: Constructions of Lie groups.
Definition of a Lie group. Classical groups.
Connected component of the identity. Universal cover.
Closed subgroup theorem. Irrational one-parameter groups.
Lie subgroups and images of Lie group morphisms.
- Oct 2: Lie group actions.
The coset space G/H. Examples for SO(n) and SU(n).
Orbits and stabilizers. Left multiplication and conjugation.
The tangent space g of G. Adjoint action of a Lie group.
- Oct 3: Classical Lie groups.
Exponential map for matrix groups. Smooth structure from g.
List of classical groups and their Lie algebras. Dimension counts.
The exponential map for a general G. One-parameter subgroups.
- Oct 3: The Lie algebra (g,[·,·]).
The commutator [·,·] via the exponential map. Matrix group examples.
Skew-symmetry and the Jacobi identity. (so(3),[·,·]) and cross product.
Group morphisms induce algebra morphisms. Example SU(2) to SO(3).
- Oct 9: Infinitesimal adjoint.
Adjoint representation of SU(2). Pauli matrices and orbits.
Infinitesimal adjoint ad: g → End(g). Description for sl(2,C).
The identity ad(x,y)=[x,y]. Campbell-Hausdorff formula.
- Oct 9: Reconstruction Statements.
Lie subalgebras and ideals. Counter-examples to lifting Lie algebra morphisms.
Non-uniqueness of integrating Lie group. The map SU(2) to SO(3) revisited.
Integral submanifolds. The three fundamental questions and discussion.
- Oct 10: The Lie group Diff(M).
Lie algebra (Vect(M),[·,·]). Adjoint representation of Diff(M).
Representation Diff(M) in C^{∞}(M). Infinitesimal adjoint of Diff(M).
Maurer-Cartan Equation and Integrability of PDEs. First Lie Theorem.
- Oct 10: The Second and Third Lie Theorems.
Equivalence of categories. The Second Lie Theorem.
Simply-connectedness. Ado's Theorem.
The Third Lie Theorem. Consequences.
- Oct 23: Representation of Lie Groups.
Finite groups. Representations of cyclic groups.
Tensor and dual operations. Irreps and complete reducibility.
Examples. Intertwining operators. Schur's Lemma.
- Oct 23: Weyl's Unitary Trick.
G-invariant inner product. Finite group case. Haar Measure.
Representation of compact groups are completely reducible.
Polynomial Representation of SU(2). Tensor intrigue.
- Oct 24: Character Theory.
Matrix coefficients. Orthogonality of characters.
Irreps Decomposition. Peter-Weyl Theorem. Fourier series.
- Oct 24: Symmetric Representations of SU(2).
Symmetric Representations. SU(2) Characters.
Classification of irreducibles. Clebsch-Gordan decomposition.
Classification of irreps for SO(3), SO(4) and U(2).
- Oct 30: Representations of sl(2,C).
Weight decomposition. Verma modules.
Classification of irreps. Integrality of weights.
Correspondence SU(2) symmetric representations.
- Nov 13: Universal Enveloping Algebra.
Enveloping Algebra. Universal property.
Filtered algebras. Ordered monomials generate.
Heisenberg Lie algebra and U(sl(2,C)). PBW Theorem.
Centers and Casimir operators. Harish-Chandra Principle.
- Nov 13: Nilpotency and Solvability.
Commutant ideal. Nilpotent and solvable Lie algebras.
Derived series and lower central series. Flag examples.
Lie's Theorem for solvable representations. Engel's Theorem.
- Nov 14: Semisimplicity.
Radical ideal. Semisimple Lie algebras.
Levi decomposition. Reductive algebras.
Example gl(n,C). Semisimplicity for sl(2,C).
- Nov 14: Cartan's criteria.
Bilinear invariants forms for representations. Killing form.
Example sl(2,C) for weights 2 and 3. Jordan decomposition.
Cartan Solvability Criterion. Cartan Semisimplicity criterion.
- Nov 20: Semisimplicity and complete reducibility I.
Complete reducibility for compact groups revisited.
Theorem: g semisimple iff reps are completely reducible.
Steps of Proof: g semisimple have completely reducible adjoint.
All reps of g completely reducible implies semisimple.
- Nov 20: Semisimplicity and complete reducibility II.
Complementary subreps to V' in V and trivial subreps of Hom(V/V',V).
Short exact sequences (ses). Complete reducibility from splitting ses.
Casimir elements revisited. Non-trivial irreps admit Casimirs with scalar action.
- Nov 21: Semisimplicity and complete reducibility III.
Ses of reps split if core rep is irreducible.
Inductive argument that every ses of reps splits.
Conclusion main proof. Reps for complex tori and sl(2,C) revisited.
Toral subalgebra and Cartan subalgebra of semisimples.
- Nov 21: Cartan subalgebras.
g decomposes as simultaneous eigenspaces for a toral subalgebra.
[g_{α}, g_{ β}] is contained in g_{ α+ β}. Cartan subalgebras are self-centralising.
Cartan algebras are conjugate. Rank of Lie algebra. Root system and root decomposition.
The case sl(n, C). Orthogonality of g_{ α} and non-degenerate pairing between g_{ α} and g_{ -α}.