nLab
SL(2,O)

Contents

Contents

Idea

The concept of a special linear group SL(2,K)SL(2,K) with K=𝕆K = \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,𝕆)SL(2,\mathbb{O}), after all, and that it should essentially be the spin group Spin(9,1)Spin(9,1):

exceptional spinors and real normed division algebras

Lorentzian
spacetime
dimension
AA\phantom{AA}spin groupnormed division algebra\,\, brane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})A\phantom{A} \mathbb{R} the real numberssuper 1-brane in 3d
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})A\phantom{A} \mathbb{C} the complex numberssuper 2-brane in 4d
6=5+16 = 5+1Spin(5,1)Spin(5,1) \simeq SL(2,H)A\phantom{A} \mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1) {\simeq}SL(2,O)A\phantom{A} 𝕆\mathbb{O} the octonionsheterotic/type II string

Realizations

By Manogue–Schray

Manogue and Schray have shown how to interpret Spin(9,1)Spin(9,1) as SL(2,𝕆)SL(2,\mathbb{O}) as follows. For a clear summary of results see also Dray and Manogue.

First, a warmup with the complex numbers. The space 𝔥 2()\mathfrak{h}_2(\mathbb{C}) of 2×22 \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)(1,3). Any element gSL(2,)g \in \mathrm{SL}(2,\mathbb{C}) acts as a linear transformation of 𝔥 2()\mathfrak{h}_2(\mathbb{C}) as follows:

XgXg *,X𝔥 2() X \mapsto g X g^\ast , \qquad X \in \mathfrak{h}_2(\mathbb{C})

and this action preserves the determinant on 𝔥 2()\mathfrak{h}_2(\mathbb{C}), so we obtain a group homomorphism from SL(2,)\mathrm{SL}(2,\mathbb{C}) to O(1,3)\mathrm{O}(1,3). In fact this is a 2-1 homomorphism from SL(2,)\mathrm{SL}(2,\mathbb{C}) onto SO 0(1,3)SO_0(1,3). Since SL(2,)\mathrm{SL}(2,\mathbb{C}) is simply connected, this allows us to identify SL(2,)\mathrm{SL}(2,\mathbb{C}) with Spin(1,3)Spin(3,1)\mathrm{Spin}(1,3) \cong \mathrm{Spin}(3,1).

Following this pattern, the space 𝔥 2(𝕆)\mathfrak{h}_2(\mathbb{O}) of 2×22 \times 2 hermitian matrices with entries in the octonions 𝕆\mathbb{O} may be identified with 10-dimensional Minkowski spacetime, since the determinant is a quadratic form of signature (1,9)(1,9). (The determinant of an element of 𝔥 2(𝕆)\mathfrak{h}_2(\mathbb{O}) is well-defined using the usual formula for the determinant of a 2×22 \times 2 matrix, because any octonion commutes with its conjugate.) Let 𝕄 2(𝕆)\mathbb{M}_2(\mathbb{O}) be the set of 2×22 \times 2 octonionic matrices all of whose entries lie in an arbitrary subalgebra of 𝕆\mathbb{O} isomorphic to \mathbb{C}. Then the determinant of g𝕄 2(𝕆)g \in \mathbb{M}_2(\mathbb{O}) is well-defined by the usual formula, and furthermore (gX)g *=g(Xg *)(g X) g^\ast = g ( X g^\ast) for all X𝔥 2(𝕆)X \in \mathfrak{h}_2(\mathbb{O}).

Let then SL(2,𝕆)\mathrm{SL}(2,\mathbb{O}) be the subgroup of linear transformations of 𝕆 2\mathbb{O}^2 generated by those of the form

ψgψ \psi \mapsto g \psi

where g𝕄 2(𝕆)g \in \mathbb{M}_2(\mathbb{O}) has det(g)=1det(g) = 1. This group acts by linear transformations of 𝔥 2(𝕆)\mathfrak{h}_2(\mathbb{O}) in a unique way such that

X(gX)g *,X𝔥 2(𝕆) X \mapsto (g X) g^\ast , \qquad X \in \mathfrak{h}_2(\mathbb{O})

when g𝕄 2(𝕆)g \in \mathbb{M}_2(\mathbb{O}) has det(g)=1det(g) = 1. This action preserves the determinant on 𝔥 2(𝕆)\mathfrak{h}_2(\mathbb{O}). Thus, there is a group homomorphism from SL(2,𝕆)\mathrm{SL}(2,\mathbb{O}) to O(1,9)\mathrm{O}(1,9), and in fact it maps to SO 0(1,9)\mathrm{SO}_0(1,9) in a 2-1 and onto way, which allows us to identify SL(2,𝕆)\mathrm{SL}(2,\mathbb{O}) with Spin(1,9)Spin(9,1)\mathrm{Spin}(1,9) \cong \mathrm{Spin}(9,1).

By Hitchin

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

References

  • John Baez, The octonions, Bull. Amer. Math. Soc. 39 (2002), 145–205. Section 3.3: 𝕆P 1\mathbb{O}\mathrm{P}^1 and Lorentzian geometry. (html)

  • Tevian Dray, Corinne Manogue, Octonionic Cayley spinors and E 6\mathrm{E}_6, Comment. Math. Univ. Carolin. 51 (2010), 193–207. (arXiv:0911.2255)

  • Tevian Dray, John Huerta, Joshua Kincaid, The magic square of Lie groups: the 2×22 \times 2 case, Lett. Math. Phys. 104 (2014), 1445–68. (arXiv:2009.00390)

  • Nigel Hitchin, SL(2)SL(2) over the octonions, Mathematical Proceedings of the Royal Irish Academy. Vol. 118. No. 1. Royal Irish Academy, 2018. (arXiv:1805.02224)

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

Last revised on December 7, 2020 at 12:37:22. See the history of this page for a list of all contributions to it.