### 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). 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. 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) updated 18-6-21
• Background 2: Category theory (notes)
• Lecture 6: Lie groups and Lie algebras (notes) updated 21-6-21
• Lecture 7: Representation theory of Lie algebras (notes) updated 21-6-21

There is a bonus lecture (not examinable):

• Lecture 8: Complex Lie groups and spin (notes)

Assignment 1 due by 4pm on 22-3:

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

The solutions are thanks to Jackson Godfrey.

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)

The solutions are thanks to Brian Chan.

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)

The solutions are thanks to Brian Chan.

Assignment 4 due by 4pm on 28-5:

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

The solutions are thanks to Liam Carroll.

#### Exam

All but one of the questions on the exam will be questions that are either exactly the same as exercises assigned in lectures, or which are very similar. 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. The question on the exam which is not an exercise from the notes will ask you to reproduce one of the proofs (an explicit list of those proofs which might appear will be given before the end of semester).

• The exam will not contain questions about anything in L1, L2 or Background 2 (so, no Wigner’s theorem or category theory on the exam).

• The following exercises relying heavily on material from MHS will not appear on the exam: L3-1, L3-2, L3-7, L3-11, L4-3, L4-5, B1-1.

• You will be asked on the exam to give the proof of one of the following results from lectures: Lemma L3-10, Lemma L3-11, Lemma L5-4, Lemma L6-8, Lemma L6-9, Lemma L6-10, Theorem L6-12, Theorem L6-13, Corollary L6-14, Lemma L6-15, Example L7-2, Lemma L7-8, Theorem L7-10,

#### Index of topics covered

The following topics appear in the lecture notes.

• Lecture 1: Poincare group, Cramer’s rule, formalisation of observers in terms of Lie groups and their representations.
• Lecture 2: Physical states and rays in quantum mechanics, unitary and antiunitary transformations, Wigner’s theorem.
• Lecture 3: spherical coordinates, the L2 space of the sphere, action of SO(n+1) on L^2(S^n,C), structure of three dimensional rotations and SO(3), harmonic polynomials and spherical harmonics, smooth functions and differential operators on the sphere, the Laplacian on R^3 and S^2, decomposion of L^2(S^2,C) into spherical harmonics.
• Lecture 4: the action of rotations on L^2(S^2,C) is given by exponentials of infinitesimal symmetries which are differential operators (eta operators).
• Lecture 5: infinitesimal symmetries as directional derivatives, Taylor expansion on the sphere, spectral theorem for unitary and self-adjoint operators, relationship between unitary and self-adjoint operators, skew self-adjointness of the infinitesimal generator of rotations.
• Lecture 6: rotation matrices are exponentials of the delta matrices, the action of SO(3) on L^2(S^2,C) is the exponentiation of a linear map from so(3) to endomorphisms, one-parameter subgroups, commutators, bijection between matrices and one-parameter subgroups, matrix Lie groups and their Lie algebras, the Lie algebra of a Lie group is a vector space closed under commutators, origin and meaning of the Lie bracket, the Lie functor from the category of matrix Lie groups to the category of Lie algebras, definition of representations of matrix Lie groups and Lie algebras.
• Lecture 7: subrepresentations, trivial representation, defining representation, structure of H2, discovery of ladder operators, irreducibility of Hk and its structure in terms of ladder operators.
• Background 1: all norms on finite-dimensional normed spaces are Lipschitz equivalent, transformations between finite-dimensional normed spaces are bounded, absolutely convergent series in normed spaces, exponential of bounded operators on a Banach space, basic properties of exponentials, continuity of the exponential function, definition and continuity of the logarithm of a bounded operator, Lie product formula.
• Background 2: definition of a category, examples of categories, definition of a functor, free abelian groups, the tensor product of abelian groups.