sp(1) — compact symplectic Lie algebra

The compact symplectic Lie algebra $\mathfrak{sp}(1)$ is the real vector space

\mathfrak{sp}(1)=\{X\in M_{1}(\mathbb H):X^*+X=0\}.

with commutator bracket $[X,Y]=XY-YX$ . Here $X^*$ means conjugate transpose using quaternionic conjugation.

Dimension and scalar field

Each diagonal entry is an imaginary quaternion with three real coordinates; each pair of off-diagonal entries contributes four. The real dimension is $3$ . The size parameter counts quaternionic rows, not complex rows. In rank one this is $\operatorname{Im}\mathbb H$ with the quaternion commutator, isomorphic to $\mathfrak{su}(2)$ .

This object is a real Lie algebra. Multiplication by $i$ does not generally preserve the defining skew-adjoint condition; there is no inherited complex-linear category membership.

References

Barton and Sudbery, Magic squares and matrix models of Lie algebras, §2, pp. 5–8, equations (2.19)–(2.27); quaternionic and indefinite matrix conventions.