 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-4/1/iv/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 4 / 1 / iv / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

C^{3}

take the matrix units

ρ (p) = E_{12}, ρ (q) = E_{23}, ρ (c) = E_{13} .

(1)

Their products give

[E_{12}, E_{23}] = E_{13}

, while

E_{13}

commutes with each of the other two. Thus this is a Lie algebra representation. The three images are linearly independent, so its kernel is zero: it is faithful and finite-dimensional. It is reducible, as expected from the classification in part ii; for example, the first coordinate line is invariant.

 Read the full article

 New to topics? Read the docs here!