Editing SO(8)

{{Short description|Rotation group in 8-dimensional Euclidean space}}
[[Image:Dynkin diagram D4.png|thumb|[[D4 (root system)|D<sub>4</sub>]], [[Dynkin diagram]] of SO(8)]]
{{Group theory sidebar |Topological}}

In [[mathematics]], '''SO(8)''' is the [[special orthogonal group]] acting on eight-dimensional [[Euclidean space]]. It could be either a real or complex [[simple Lie group]] of rank 4 and dimension 28.

==Spin(8)==
Like all special orthogonal groups SO(n) with n ≥ 2, SO(8) is not [[simply connected]]. And like all SO(n) with n > 2, the [[fundamental group]] of SO(8) is [[isomorphic]] to [[cyclic group|'''Z'''<sub>2</sub>]]. The [[universal cover]] of SO(8) is the [[spin group]] '''Spin(8)'''.

==Center==
The [[center of a group|center]] of SO(8) is '''Z'''<sub>2</sub>, the diagonal matrices {±I} (as for all SO(2''n'') with 2''n'' ≥ 4), while the center of Spin(8) is '''Z'''<sub>2</sub>&times;'''Z'''<sub>2</sub> (as for all Spin(4''n''), 4''n'' ≥ 4).

==Triality==
{{main|Triality}}

SO(8) is unique among the [[simple Lie group]]s in that its [[Dynkin diagram]], {{CDD|node|3||node|split1|nodes}} ([[D4 (root system)|D<sub>4</sub>]] under the Dynkin classification), possesses a three-fold [[symmetry]]. This gives rise to peculiar feature of Spin(8) known as [[triality]]. Related to this is the fact that the two [[spinor]] [[group representation|representations]], as well as the [[fundamental representation|fundamental]] vector representation, of Spin(8) are all eight-dimensional (for all other spin groups the spinor representation is either smaller or larger than the vector representation). The triality [[automorphism]] of Spin(8) lives in the [[outer automorphism group]] of Spin(8) which is isomorphic to the [[symmetric group]] S<sub>3</sub> that permutes these three representations. The automorphism group acts on the center '''Z'''<sub>2</sub> x '''Z'''<sub>2</sub> (which also has automorphism group isomorphic to ''S''<sub>3</sub> which may also be considered as the [[general linear group]] over the finite field with two elements,  ''S''<sub>3</sub> ≅GL(2,2)). When one quotients Spin(8) by one central '''Z'''<sub>2</sub>, breaking this symmetry and obtaining SO(8), the remaining [[outer automorphism group]] is only '''Z'''<sub>2</sub>. The triality symmetry acts again on the further quotient SO(8)/'''Z'''<sub>2</sub>.

Sometimes Spin(8) appears naturally in an "enlarged" form, as the automorphism group of Spin(8), which breaks up as a [[semidirect product]]: Aut(Spin(8)) ≅ PSO (8) ⋊ ''S''<sub>3</sub>.

==Unit octonions==
Elements of SO(8) can be described with unit [[octonions]], analogously to how elements of SO(2) can be described with [[Circle group|unit complex numbers]] and elements of [[SO(4)]] can be described with [[versor|unit quaternions]]. However the relationship is more complicated, partly due to the [[non-associativity]] of the octonions. A general element in SO(8) can be described as the product of 7 left-multiplications, 7 right-multiplications and also 7 bimultiplications by unit octonions (a bimultiplication being the composition of a left-multiplication and a right-multiplication by the same octonion and is unambiguously defined due to octonions obeying the [[Moufang identities]]).

It can be shown that an element of SO(8) can be constructed with bimultiplications, by first showing that pairs of reflections through the origin in 8-dimensional space correspond to pairs of bimultiplications by unit octonions. The [[triality]] automorphism of Spin(8) described below provides similar constructions with left multiplications and right multiplications.<ref name="ConwaySmith2003">{{cite book|author1=John H. Conway|author2=Derek A. Smith|title=On Quaternions and Octonions|url=https://books.google.com/books?id=E_HCwwxMbfMC|date=23 January 2003|publisher=Taylor & Francis|isbn=978-1-56881-134-5}}</ref>

===Octonions and triality===
If <math>x,y,z\in\mathbb{O}</math> and <math>(xy)z=1</math>, it can be shown that this is equivalent to <math>x(yz)=1</math>, meaning that <math>xyz=1</math> without ambiguity. A triple of maps <math>(\alpha,\beta,\gamma)</math> that preserve this identity, so that <math>x^\alpha y^\beta z^\gamma=1</math> is called an [[Isotopy of loops|isotopy]]. If the three maps of an isotopy are in <math>\operatorname{SO(8)}</math>, the isotopy is called an orthogonal isotopy. If <math>\gamma\in \operatorname{SO(8)}</math>, then following the above <math>\gamma</math> can be described as the product of bimultiplications of unit octonions, say <math>\gamma=B_{u_1}...B_{u_n}</math>. Let <math>\alpha,\beta \in \operatorname{SO(8)}</math> be the corresponding products of left and right multiplications by the conjugates (i.e., the multiplicative inverses) of the same unit octonions, so <math>\alpha=L_{\overline{u_1}}...L_{\overline{u_n}}</math>, <math>\beta=R_{\overline{u_1}}...R_{\overline{u_n}}</math>. A simple calculation shows that <math>(\alpha,\beta,\gamma)</math> is an isotopy. As a result of the non-associativity of the octonions, the only other orthogonal isotopy for <math>\gamma</math> is <math>(-\alpha,-\beta,\gamma)</math>. As the set of orthogonal isotopies produce a 2-to-1 cover of <math>\operatorname{SO}(8)</math>, they must in fact be <math>\operatorname{Spin}(8)</math>.

Multiplicative inverses of octonions are two-sided, which means that <math>xyz=1</math> is equivalent to <math>yzx=1</math>. This means that a given isotopy <math>(\alpha,\beta,\gamma)</math> can be permuted cyclically to give two further isotopies <math>(\beta,\gamma,\alpha)</math> and <math>(\gamma,\alpha,\beta)</math>. This produces an order 3 [[outer automorphism]] of <math>\operatorname{Spin}(8)</math>. This "triality" automorphism is exceptional among [[spin group]]s. There is no triality automorphism of <math>\operatorname{SO}(8)</math>, as for a given <math>\gamma</math> the corresponding maps <math>\alpha,\beta</math> are only uniquely determined up to sign.<ref name="ConwaySmith2003"/>

==[[Root system]]==

: <math>(\pm 1,\pm 1,0,0)</math>
: <math>(\pm 1,0,\pm 1,0)</math>
: <math>(\pm 1,0,0,\pm 1)</math>
: <math>(0,\pm 1,\pm 1,0)</math>
: <math>(0,\pm 1,0,\pm 1)</math>
: <math>(0,0,\pm 1,\pm 1)</math>

==[[Weyl group]]==
Its [[Weyl group|Weyl]]/[[Coxeter group]] has 4!&nbsp;&times;&nbsp;8&nbsp;=&nbsp;192 elements.

==[[Cartan matrix]]==

: <math>
\begin{pmatrix}
2  & -1 & -1 & -1\\
-1 &  2 &  0 &  0\\
-1 &  0 &  2 &  0\\
-1 &  0 &  0 &  2
\end{pmatrix}
</math>

==See also==
* [[Octonions]]
* [[Clifford algebra]]
* [[G2 (mathematics)|''G''<sub>2</sub>]]

==References==
{{Reflist}}
*{{citation|last=Adams|first=J.F.|authorlink=Frank Adams|title=Lectures on exceptional Lie groups|series=Chicago Lectures in Mathematics|publisher= [[University of Chicago Press]]|year= 1996|isbn= 0-226-00526-7}}
*{{citation|last=Chevalley|first=Claude|authorlink=Claude Chevalley|title=The algebraic theory of spinors and Clifford algebras|series=Collected works|volume=2|publisher=Springer-Verlag|year=1997|isbn= 3-540-57063-2}} (originally published in 1954 by [[Columbia University Press]]) 
*{{citation|last=Porteous|first= Ian R.|authorlink=Ian R. Porteous|title=Clifford algebras and the classical groups|series=
Cambridge Studies in Advanced Mathematics|volume= 50|publisher= [[Cambridge University Press]]|year= 1995|isbn= 0-521-55177-3}}

[[Category:Lie groups]]
[[Category:Octonions]]