Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 140 / 2 / b

Solution

Use the decomposition into Lagrangian subspaces

V = V_{+} \oplus V_{-}

from part (a). The bilinear form

Ω

V_{+}

and

V_{-}

nondegenerately: a vector in either summand annihilating the other already annihilates its own summand and hence all of

V

, so is zero.

Choose a basis

e_{1}, \dots, e_{n}

V_{+}

V_{-} ⟶ V_{+}^{*}, f ⟼ (e ⟼ Ω (e, f))

f_{j}

correspond to the dual basis of the

e_{i}

. Then

Ω (e_{i}, f_{j}) = δ_{ij}

, and the pairings within each summand vanish. Thus

(e_{1}, \dots, e_{n}, f_{1}, \dots, f_{n}) is a symplectic basis, S e_{i} = e_{i}, S f_{i} = - f_{i} .

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