Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 102 / 3

After ordering a symplectic basis in two blocks, writeMatrices in the Symplectic Lie algebra have block formThe root-space decomposition isFor example, these one-dimensional spaces are spanned respectively byThus this is the Cn root systemThe upper-triangular choice givesIts simple roots, highest root, fundamental weights, and half-sum of positive roots are

Using the notation requested in the paper, the root lattice $P$ and weight lattice $Q$ areso $Q/P\cong \mathbb{Z}/2\mathbb{Z}$. This reverses the common notation in which the root lattice is called $Q$ and the weight lattice is called $P$.

Since a multiple-edge arrow in a Dynkin diagram points toward the shorter root, the finite and extended diagrams areand

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The Weyl reflection formula $s_{\alpha }(x)=x-⟨x,{\alpha }^{\vee }⟩\alpha$ becomes especially concrete on $x=(x_1,\dots ,x_n)$:with all unlisted coordinates fixed. The Weyl group is therefore the group of signed permutations

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Withthe equation $AJ+J{A}^T=0$ reduces to $p+s=0$. Henceas matrix Lie algebras, with the same commutator bracket.

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