sl(n,H) — quaternionic special linear Lie algebra

For an integer $n\geq1$ , the quaternionic special linear Lie algebra $\mathfrak{sl}(n,\mathbb H)$ is the real vector space

\mathfrak{sl}(n,\mathbb H)=\{X\in M_{n}(\mathbb H):\operatorname{Re}\operatorname{tr}X=0\},\qquad [X,Y]=XY-YX.

Trace and scalar conventions

Quaternionic matrix multiplication is associative, so its commutator is a real Lie bracket. The real part of the trace is essential: the full quaternionic trace need not be cyclic, but its real part is. Thus the displayed subspace is closed under brackets and has real codimension one. Its conventional complex-matrix name is $\mathfrak{su}^*(2n)$ .

Quaternionic entries do not make this a Lie algebra over a commutative field $\mathbb H$ . All scalar-linear categories in this record are over $\mathbb R$ .

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.