 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-55/1/solution

Use the Lie bracket convention

[X, Y] = X Y - Y X

. If

E_{ij}

denotes a matrix unit, then

E_{ij} E_{k l} = δ_{jk} E_{i l}

. Here this gives

[X_{0}, X_{1}] = X_{1}, [X_{0}, X_{2}] = X_{2}, [X_{1}, X_{2}] = 0.

[X_{j}, X_{k}] = f^{i}_{jk} X_{i}

, the nonzero structure constants of a Lie algebra are

f^{1}_{01} = f^{2}_{02} = 1, f^{1}_{10} = f^{2}_{20} = - 1.

All other entries vanish. This is the Lie algebra of the translation-dilation group of the plane: its two-dimensional translation ideal is abelian, and

X_{0}

acts on that ideal as the identity. In group coordinates the multiplication and inverse are

(ρ, x) (ρ^{'}, x^{'}) = (ρ ρ^{'}, x + ρ x^{'}), (ρ, x)^{- 1} = (ρ^{- 1}, - ρ^{- 1} x) .

These formulas also make the semidirect product structure explicit.

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