 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-4/4/a/solution

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

(L, *)

, the

p^{n}

-th power of

X

is exactly

p^{n} X

, and its unique

p^{n}

-th root in the appropriate layer is division by

p^{n}

. The intrinsic addition is therefore

lim_{n \to \infty} p^{- n} BCH (p^{n} X, p^{n} Y) = X + Y .

The linear term is exact; every non-linear term tends to zero by the convergence estimates above. The intrinsic Lie bracket is

lim_{n \to \infty} p^{- 2 n} lo g (exp (- p^{n} X) exp (- p^{n} Y) exp (p^{n} X) exp (p^{n} Y)) = [X, Y],

since the leading term is

p^{2 n} [X, Y]

and the terms of degree at least three vanish after rescaling. Finally the intrinsic scalar operation sends

(λ, X)

λ X

. All underlying sets are identical, and all three operations agree. Thus

L_{(L, *)} = L .

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