SO(8)

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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. Like all special orthogonal groups (except SO(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). The center of Spin(8) is Z2×Z2 while the center of SO(8) is Z2.

Dynkin diagram of so(8)

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. 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 semidirect product \mathrm{Spin}(8) \rtimes S_3.

[edit] 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)

[edit] Weyl group

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

[edit] Cartan matrix

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

See also: Octonions, Clifford algebra, G2

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