SO(8)

Group theory

Group theory
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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.

1 Spin(8)
2 Center
3 Triality
4 Root system
5 Weyl group
6 Cartan matrix
7 See also
8 References

Spin(8)

Like all special orthogonal groups of $n > 2,$ SO(8) is not simply connected, having a fundamental group isomorphic to Z₂. The universal cover of SO(8) is the spin group Spin(8).

Center

The center of SO(8) is Z₂, the diagonal matrices $\{\pm I\},$ (as for all SO(2n) for $2n > 2$ ), while the center of Spin(8) is Z₂×Z₂ (as for all Spin(4n), $4n > 0$ ).

Triality

Main article: Triality

SO(8) is unique among the simple Lie groups in that its Dynkin diagram (shown right) (D₄ 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 representations, as well as the 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 S₃ that permutes these three representations. The automorphism group acts on the center Z₂ x Z₂ (which also has automorphism group isomorphic to $S_3,$ which may also be considered as the general linear group over the finite field with two elements, $S_3 \cong \operatorname{GL}(2,2)$ ). When one quotients Spin(8) by one central Z₂, breaking this symmetry and obtaining SO(8), the remaining outer automorphism group is only Z₂. The triality symmetry acts again on the further quotient SO(8)/Z₂.

Sometimes Spin(8) appears naturally in an "enlarged" form, as the automorphism group of Spin(8), which breaks up as a semidirect product: $\operatorname{Aut}(\operatorname{Spin}(8)) \cong \operatorname{Spin}(8) \rtimes S_3$ .

Root system

$(\pm 1,\pm 1,0,0)$

$(\pm 1,0,\pm 1,0)$

$(\pm 1,0,0,\pm 1)$

$(0,\pm 1,\pm 1,0)$

$(0,\pm 1,0,\pm 1)$

$(0,0,\pm 1,\pm 1)$

Weyl group

Its Weyl/Coxeter group has 4!×8=192 elements.

Cartan matrix

$\begin{pmatrix} 2 & -1 & -1 & -1\\ -1 & 2 & 0 & 0\\ -1 & 0 & 2 & 0\\ -1 & 0 & 0 & 2 \end{pmatrix}$

References

Adams, J.F. (1996), Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University of Chicago Press, ISBN 0-226-00526-7
Chevalley, Claude (1997), The algebraic theory of spinors and Clifford algebras, Collected works, 2, Springer-Verlag, ISBN 3-540-57063-2 (originally published in 1954 by Columbia University Press)
Porteous, Ian R. (1995), Clifford algebras and the classical groups, Cambridge Studies in Advanced Mathematics, 50, Cambridge University Press, ISBN 0-521-55177-3