Past exam of the Lie algebras and representation theory of symmetric groups course of the University of Oxford / 2008 / A / 1 / c

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- Solution c

Solution

Induct on

i

, proving the claim for all

j

simultaneously. The case

i = 1

is the definition

[L, L_{j}] = L_{j + 1}

. If it holds for

i

, Jacobi gives

[L_{i + 1}, L_{j}] = [[L, L_{i}], L_{j}] \subseteq [L, [L_{i}, L_{j}]] + [L_{i}, [L, L_{j}]] \subseteq [L, L_{i + j}] + [L_{i}, L_{j + 1}] \subseteq L_{i + j + 1} .

Both uses of the induction hypothesis have first index

i

, so the induction is valid. Thus

[L_{i}, L_{j}] \subseteq L_{i + j}

for all positive indices.

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