Octonionic special linear group

Definition

Every traceless $2\times2$ octonionic matrix acts as a real-linear endomorphism of $\mathbb O^2\cong\mathbb R^{16}$ . The octonionic special linear Lie algebra $\mathfrak{sl}_2(\mathbb O)$ is the Lie subalgebra of $\mathfrak{gl}_{16}(\mathbb R)$ generated by all these endomorphisms. The connected closed subgroup of $GL(16,\mathbb R)$ with this Lie algebra is denoted $SL_2(\mathbb O)$ .

Identification with a spin group

There are Lie algebra and Lie group isomorphisms

\mathfrak{sl}_2(\mathbb O)\cong\mathfrak{so}(9,1), \qquad SL_2(\mathbb O)\cong\operatorname{Spin}(9,1).

Its maximal compact subgroup is $\operatorname{Spin}(9)$ . The action on $\mathbb O^2$ restricts to the real 16-dimensional spin representation of that compact subgroup.

Why this is not a naive matrix group

The real-linear maps represented by octonionic matrices are not closed under commutators if one tries to compute using ordinary octonionic matrix multiplication: nonassociativity creates additional endomorphisms. Thus $SL_2(\mathbb O)$ is defined through the generated real Lie algebra, not as the set of octonionic matrices with a naive determinant equal to one.

Geometric action

The group preserves the positive cone and determinant on $H_2(\mathbb O)$ . It also acts conformally on octonionic lines and preserves the class of octonionic plurisubharmonic functions.

References

Corinne A. Manogue and Jörg Schray, “Finite Lorentz transformations, automorphisms, and division algebras,” Journal of Mathematical Physics 34 (1993), 3746–3767. DOI record.
Semyon Alesker, “Plurisubharmonic functions on the octonionic plane and $\operatorname{Spin}(9)$ -invariant valuations on convex sets,” Journal of Geometric Analysis 18 (2008), 651–686. arXiv record. Relevant: §1.4.