MAT 4144/5158 (Winter 2015)

Introduction to Lie Groups

Syllabus

The syllabus will be updated as the course progresses. Recommended exercises are in addition to the exercises in the notes. In the columns Text Sections and Recommended Exercises, 'S' refers to the textbook by Stillwell and 'H' refers to the textbook by Hall (see here).

Date Notes Sections Text Sections Material Recommended Exercises
Jan 12 1.1, 1.2 S: 1.1 Introduction, a categorical definition of Lie groups, rotations and the circle. S: 1.1.1–1.1.5
Jan 14 1.3, 1.4 S: 1.2–1.4 Matrix representations of complex numbers, quaternions. S: 1.3.1–1.3.4, 1.4.1–1.4.4
Jan 19 1.5 S: 1.5, 2.3 Quaternions and space rotations S: 2.3.1–2.3.5
Jan 21 1.6, 1.7 S: 2.4–2.6 Isometries of Rn, quaternions and rotations of R4 S: 2.5.1–2.5.4, 2.6.1–2.6.3, 2.6.5
Jan 26 1.8, 2.1–2.6 S: 2.7, 3.1–3.3;
H: 1.1, 1.2		 SU(2)xSU(2) and SO(4), matrix Lie groups: definitions and examples S: 3.1.3, 3.2.1–3.2.3, 3.3.2–3.3.6
H: §1.9 (16)
Jan 28 2.7–2.11 S: 3.4; H: 1.2 More examples of linear Lie groups, homomorphisms of Lie groups S: 3.4.1–3.4.3
H: §1.9 (4, 17, 20)
Feb 2 3.1 H: 1.4 Connectedness
Feb 4 3.2, 3.3 H: 1.3, 1.7 Polar decompositions, compactness H: §1.9 (14,15)
Feb 9 4.1 S: 3.5, 3.6 Maximal tori S: 3.5.1, 3.5.2, 3.6.1, 3.6.2
Feb 11 4.2, 5.1 S: 3.5, 3.7, 3.8, 4.1 Centres and discrete subgroups, the exponential map S: 3.7.1–3.7.3, 3.8.1–3.8.5
Feb 16–20
Reading break
Feb 23 5.2–5.4 S: 4.2–4.4
H: 2.1, 2.2		 The Lie algebra of SU(2) S: 4.2.1–4.2.3, 4.3.2–4.3.4, 4.4.1, 4.4.2
H: §2.10 (1–4)
Feb 25 5.5, 5.6 S: 4.5, 5.1, 5.2 The exponential map, tangent spaces to O(n), U(n), Sp(n), SO(n) S: 4.5.1–4.5.6
Mar 2 Midterm exam
Mar 4 5.6, 5.7 S: 5.3–5.5
H: 2.5–2.6, 2.8		 Tangent spaces to SU(n), GL(n,C), SL(n,C); the tangent space as a Lie algebra S: 5.3.1–5.3.7; 5.4.1–5.4.5, 5.5.1, 5.5.2
H: §2.10 (9, 12, 15, 16, 22, 26)
Mar 9 5.8, 5.9 S: 5.6, 7.1
H: 2.7, 2.9		 Complex Lie groups and complexification, the matrix logarithm S: 5.6.1, 5.6.2, 5.6.4–5.6.6, 5.6.8
H: §2.10 (10, 11, 17, 23, 27)
Mar 11 5.10, 5.11 S: 7.2–7.4
H: 2.7		 The exponential mapping and the logarithm into the tangent space S: 7.2.1–7.2.6, 7.3.1, 7.3.2
Mar 16 5.11, 5.12 S: 7.4
H: 2.4, 2.6, 2.7		 The logarithm into the tangent space, other properties of the exponential S: 7.4.1, 7.4.2
H: §2.10 (25)
Mar 18 5.13 H: 2.6 The functor from Lie groups to Lie algebras, the adjoint mapping H: §2.10 (18,19,11,28)
Mar 23 5.14, 5.15 S: 7.6, 7.7
H: 3.1–3.5		 Course evaluations. Lie algebras and normal subgroups, the Baker-Campbell-Hausdorff Formula S: 7.5.2, 7.5.4, 7.6.1, 7.6.2, 7.7.1, 7.7.2
H: §3.9 (8)
Mar 25 5.15, 6.1 S: 7.7, 8.7
H: 1.5, 3.7		 The Baker-Campbell-Hausdorff Formula, simple connectedness H: §3.9 (1)
Mar 30 6.2–6.4 S: 9.1, 9.2 Simply connected Lie groups, nonisomorphic Lie groups with isomorphic Lie algebras S: 9.1.2–9.1.4
Apr 1 6.5, 6.6 S: 9.3–9.6
H: 3.6, 3.7		 The Lie group-Lie algebra correspondence, covering groups H: §3.9 (10,11,13)
Apr 8 6.6, 6.7 H: 3.7, 3.8 Covering groups, subalgebras and subgroups H: §3.9 (15,16)
Apr 13 7 Concluding remarks, directions of further study

Lecture Notes

Below are lecture notes for the course. They are a single file that will be updated as the course progresses. Please keep in mind that these are notes that I write for myself when preparing the lecture and should be complemented by a student's own class notes. I will often say more in class than is in the notes.