MAST90132

Mon, Mar 1, 2021

This is the webpage for MAST90132: Lie Algebras. The lectures (Evan Williams Tues 3:15-4:15pm, Wed 10-11am and Fri 10-11am) will be delivered in person but also through Zoom (see the LMS page for the link). Office hours are Wed 11am-12pm and Friday 11am-1pm on the same Zoom meeting ID. There will be four assignments due at regular intervals during semester amounting to a total of 40%, and a 3-hour written examination in the examination period worth 60%.

2021 Lectures

The ground truth of this subject is to be found in the lecture notes, which contain all details and which will be posted before each lecture (generally the night before). For Lie algebras and their representations we will generally follow David Ridout’s notes but at the beginning of semester my approach will be different. For background on topology and Hilbert spaces you may wish to consult my MAST30026 page

  • Lecture 1: What is real? (notes)
  • Lecture 2: Wigner’s theorem (notes)
  • Lecture 3: Symmetries of Hilbert space (notes)
  • Lecture 4: Infinitesimal symmetries (notes)
  • Lecture 5: Angular momentum (notes)
  • Background 1: Matrix exponentials (notes)
  • Lecture 6: Lie groups and Lie algebras (notes)
  • Lecture 7: Representation theory of Lie algebras (notes)
  • Background 2: Category theory (notes)

Assignment 1 due by 4pm on 22-3:

  • Exercise L2-2 (inj not sur), Exercise L2-4 (qed).

Assignment 2 due by 4pm on 23-4:

  • Exercise L3-6 (everything rotates), Exercise L3B-2 (sheaf condition), Exercise L3-13 (Laplacian vs Laplacian)
  • Exercise L4-4 (families of operators), Exercise L4-6 (suspicious formula), Exercise L4-8 (E pur si muove)

Assignment 3 due by 4pm on 14-5:

  • Exercise B1-2 (exponentials acting), Exercise B1-5 (trace vs det), Exercise B1-6 (Heisenberg)
  • Exercise L5-3 (skew vs unitary)

Assignment 4 due by 4pm on 28-5:

  • Exercise L6-4 (commutator), Exercise L6-5 (fullness)

Exam

The bulk of the exam will be questions that are either exactly the same as exercises assigned in lectures, or which are very similar. The questions will be at a level of difficulty below the one-star exercises (there will be one part of one question at the level of the easiest one-star problem). To be clear, exercises which appear on the exam may not have appeared on any homework assignment, and may not have solutions provided in the lecture notes.