sp(1,2) — indefinite quaternionic symplectic real Lie algebra

The indefinite quaternionic symplectic real Lie algebra $\mathfrak{sp}(1,2)$ is defined by

\mathfrak{sp}(1,2)=\{X\in M_{3}(\mathbb H):X^*\eta+\eta X=0\},\qquad \eta=\operatorname{diag}(I_{1},-I_{2}).

Its bracket is $[X,Y]=XY-YX$ . The star is conjugate transpose using quaternionic conjugation.

Signature, dimension and scalars

The signature convention is $p$ positive and $q$ negative directions. Negating the entire form leaves the matrix Lie algebra unchanged, and exchanging the two coordinate blocks identifies signatures $(p,q)$ and $(q,p)$ . The matrix $\eta X$ is skew-Hermitian over the quaternions. Counting three real coordinates on each diagonal entry and four on each off-diagonal pair gives $(p+q)(2(p+q)+1)$ real dimensions. The parameters count quaternionic coordinates, not complex coordinates.

The scalar field of this Lie algebra is $\mathbb R$ .

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.