Home Page of Alistair Savage | Department of Mathematics and Statistics | Blackboard Learn | Sessional Dates |

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 R^{n}, quaternions and rotations of R^{4} |
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) |
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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) |
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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) |
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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 |
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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) |
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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) |
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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) |
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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 |