 2023

Solutions to all three questions of the Lie algebras course of Part C of the mathematics course of the University of Oxford examination, sat on 6 June 2023. Source: original released PDF, bundle pages 307–309; paper pages 1–3. Questions 1 and 2 occupy paper page 2, and Question 3 occupies paper page 3. Module C2.1 and formal paper code A12189W1 identify this Part C paper. The original question numbering is retained.

Throughout, $k$ is an algebraically closed field of characteristic zero. All Lie algebras and vector spaces are finite-dimensional vector spaces except the Lie algebra representation explicitly considered in Question 2(c). We use the operator commutator $[A,B]=AB-BA$ and the standard sl2 triple convention $[H,E]=2E$, $[H,F]=-2F$, $[E,F]=H$. The printed operators in Question 2(c)(i) have a sign inconsistency with that convention; its solution explains the correction. The displayed definition of the symplectic group on paper page 3 also lacks a closing set brace; its intended meaning is the group of invertible symplectic linear maps.