nLab SL(2,O)

Idea

The concept of a special linear group $SL(2,K)$ with $K = \mathbb{O}$ the octonions does not literally make sense, due to the failure of the octonions to be an associative algebra.

Nevertheless, the pattern of the special linear groups of the other three real normed division algebras suggests that for some suitably adjusted concept of “special linear group”, it should make sense to speak of $SL(2,\mathbb{O})$, after all, and that it should essentially be the spin group $Spin(9,1)$:

exceptional spinors and real normed division algebras

Lorentzian spacetime dimension	$\phantom{AA}$spin group	normed division algebra	$\,\,$ brane scan entry
$3 = 2+1$	$Spin(2,1) \simeq SL(2,\mathbb{R})$	$\phantom{A}$ $\mathbb{R}$ the real numbers	super 1-brane in 3d
$4 = 3+1$	$Spin(3,1) \simeq SL(2, \mathbb{C})$	$\phantom{A}$ $\mathbb{C}$ the complex numbers	super 2-brane in 4d
$6 = 5+1$	$Spin(5,1) \simeq$ SL(2,H)	$\phantom{A}$ $\mathbb{H}$ the quaternions	little string
$10 = 9+1$	Spin(9,1) ${\simeq}$ “SL(2,O)”	$\phantom{A}$ $\mathbb{O}$ the octonions	heterotic/type II string

Realizations

By Manogue, Schray & Dray

Manogue & Schray 1993 have shown how to interpret $Spin(9,1)$ as $SL(2,\mathbb{O})$ as follows. A clear summary of these results is given by Dray & Manogue 2010, p. 4, which we follow now.

First, a warmup with the complex numbers. The space $\mathfrak{h}_2(\mathbb{C})$ of $2 \times 2$ hermitian matrices with entries in $\mathbb{C}$ may be identified with 4-dimensional Minkowski spacetime, since the determinant is a quadratic form of signature $(1,3)$. (For details on this and the following identification see at geometry of physics – supersymmetry the section Spacetime in dimensions 3, 4, 6 and 10).

Any element $g \in \mathrm{SL}(2,\mathbb{C})$ acts as a linear transformation of $\mathfrak{h}_2(\mathbb{C})$ by conjugation:

$$X \mapsto g X g^\ast , \qquad X \in \mathfrak{h}_2(\mathbb{C})$$

and this action preserves the determinant on $\mathfrak{h}_2(\mathbb{C})$, so that we obtain a group homomorphism from $\mathrm{SL}(2,\mathbb{C})$ to $\mathrm{O}(1,3)$. In fact this is a double cover from $\mathrm{SL}(2,\mathbb{C})$ onto $SO_0(1,3)$. Since $\mathrm{SL}(2,\mathbb{C})$ is simply connected, this allows us to identify $\mathrm{SL}(2,\mathbb{C})$ with $\mathrm{Spin}(1,3) \cong \mathrm{Spin}(3,1)$.

Following this pattern, the space $\mathfrak{h}_2(\mathbb{O})$ of $2 \times 2$ hermitian matrices with entries in the octonions $\mathbb{O}$ may be identified with 10-dimensional Minkowski spacetime, since the determinant is a quadratic form of signature $(1,9)$.

(To note here that the usual formula for the determinant of a $2\times 2$ matrix is indeed well-defined on elements of $\mathfrak{h}_2(\mathbb{O})$, because any octonion commutes with its conjugate.)

Let $\mathbb{M}_2(\mathbb{O})$ be the set of those $2 \times 2$ octonionic matrices all of whose entries lie in some subalgebra of $\mathbb{O}$ isomorphic to $\mathbb{C}$. Then the determinant of $g \in \mathbb{M}_2(\mathbb{O})$ is well-defined by the usual formula, and furthermore $(g X) g^\ast = g (X g^\ast)$ for all $X \in \mathfrak{h}_2(\mathbb{O})$.

Now define $\mathrm{SL}(2,\mathbb{O})$ to be the subgroup of linear transformations of $\mathbb{O}^2$ generated by those of the form

$$\psi \mapsto g \psi \,,$$

where $g \in \mathbb{M}_2(\mathbb{O})$ has $det(g) = \pm 1$ [Dray & Manogue 2010 (18)(].

This group acts by well-defined conjugation action on $\mathfrak{h}_2(\mathbb{O})$ via

$$X \mapsto (g X) g^\ast = g (X g^\ast) , \qquad X \in \mathfrak{h}_2(\mathbb{O}) \,,$$

and this action preserves the determinant on $\mathfrak{h}_2(\mathbb{O})$. This construction gives a group homomorphism from $\mathrm{SL}(2,\mathbb{O})$ to $\mathrm{O}(1,9)$ which turns out to be a double cover. This shows that $\mathrm{SL}(2,\mathbb{O})$ as defined above is isomorphic $\mathrm{Spin}(1,9) \cong \mathrm{Spin}(9,1)$.

By Hitchin

Another proposal for making sense of $SL(2,\mathbb{O})$ is due to Hitchin. In this approach, “$SL(2,\mathbb{O})$” is a submanifold of “$GL(2,\mathbb{O})$”, which is an open orbit of $Spin(9,1) \times GL(2,\mathbb{R})$ on ${S} \otimes \mathbb{R}^2$. Here ${S}$ is the 16-dimensional spin representation of $Spin(9,1)$; this may be identified with $\mathbb{O}^2$.

References

• Corinne Manogue, Jörg Schray: Finite Lorentz transformations, automorphisms, and division algebras, Section 5 of: Lorentz transformations, J. Math. Phys. 34 (1993) 3746-3767 [arXiv:hep-th/9302044, doi:10.1063/1.530056

  https://doi.org/10.1063/1.530056)]

• John Baez: $\mathbb{O}\mathrm{P}^1$ and Lorentzian geometry, Section 3.3 of: The octonions, Bull. Amer. Math. Soc. 39 (2002) 145–205 [web, arXiv:math/0105155]

• Tevian Dray, Corinne Manogue, Octonionic Cayley spinors and $\mathrm{E}_6$, Comment. Math. Univ. Carolin. 51 (2010), 193–207 [arXiv:0911.2255, eudml:37752]

• Tevian Dray, John Huerta, Joshua Kincaid: The magic square of Lie groups: the $2 \times 2$ case, Lett. Math. Phys. 104 (2014) 1445–68. [arXiv:2009.00390, doi:10.1007/s11005-014-0720-3]

• Nigel Hitchin: $SL(2)$ over the octonions, Mathematical Proceedings of the Royal Irish Academy. 118 1, Royal Irish Academy, (2018) [arXiv:1805.02224, doi:10.3318/pria.2018.118.04]