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orthogonal group

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Contents

Definition

For nn \in \mathbb{N} the orthogonal group is the group of isometries of a real nn-dimensional Hilbert space. This is naturally a Lie group. This is canonically isomorphic to the group of n×nn \times n orthogonal matrices.

More generally there is a notion of orthogonal group of an inner product space.

The analog for complex Hilbert spaces is the unitary group.

Properties

Homotopy groups

The homotopy groups of O=O(n)O = O(n) are for kk \in \mathbb{N} and for n>k+1n\gt k+1 (the “stable range”) are

π 8k+0(O) = 2 π 8k+1(O) = 2 π 8k+2(O) =0 π 8k+3(O) = π 8k+4(O) =0 π 8k+5(O) =0 π 8k+6(O) =0 π 8k+7(O) =. \array{ \pi_{8k+0}(O) & = \mathbb{Z}_2 \\ \pi_{8k+1}(O) & = \mathbb{Z}_2 \\ \pi_{8k+2}(O) & = 0 \\ \pi_{8k+3}(O) & = \mathbb{Z} \\ \pi_{8k+4}(O) & = 0 \\ \pi_{8k+5}(O) & = 0 \\ \pi_{8k+6}(O) & = 0 \\ \pi_{8k+7}(O) & = \mathbb{Z} } \,.

In the unstable range for low nn they instead start out as follows

GGπ 1\pi_1π 2\pi_2π 3\pi_3π 4\pi_4π 5\pi_5π 6\pi_6π 7\pi_7π 8\pi_8π 9\pi_9π 10\pi_10π 11\pi_11π 12\pi_12
SO(2)SO(2)\mathbb{Z}00000000000
SO(3)SO(3) 2\mathbb{Z}_20\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2 12\mathbb{Z}_{12} 2\mathbb{Z}_{2} 2\mathbb{Z}_{2} 3\mathbb{Z}_{3} 15\mathbb{Z}_{15} 2\mathbb{Z}_{2} 2 2\mathbb{Z}_{2} \oplus \mathbb{Z}_{2}
SO(4)SO(4)0\mathbb{Z} \oplus \mathbb{Z} 2 2\mathbb{Z}_{2} \oplus \mathbb{Z}_{2} 2 2\mathbb{Z}_{2} \oplus \mathbb{Z}_{2} 12 12\mathbb{Z}_{12} \oplus \mathbb{Z}_{12} 2 2\mathbb{Z}_{2} \oplus \mathbb{Z}_{2} 2 2\mathbb{Z}_{2} \oplus \mathbb{Z}_{2} 3 3\mathbb{Z}_{3} \oplus \mathbb{Z}_{3} 15 15\mathbb{Z}_{15}\oplus \mathbb{Z}_{15} 2 2\mathbb{Z}_{2}\oplus \mathbb{Z}_{2} 2 4\mathbb{Z}_{2}^{\oplus 4}
SO(5)SO(5)\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}00 120\mathbb{Z}_{120} 2\mathbb{Z}_{2} 2 2\mathbb{Z}_{2} \oplus \mathbb{Z}_{2}
SO(6)SO(6)0\mathbb{Z}0\mathbb{Z}00 120 2\mathbb{Z}_{120}\oplus\mathbb{Z}_2 4\mathbb{Z}_{4} 60\mathbb{Z}_{60}
SO(7)SO(7)00\mathbb{Z}00 8\mathbb{Z}_{8} 12\mathbb{Z}\oplus\mathbb{Z}_{12}0
SO(8)SO(8)0\mathbb{Z} \oplus \mathbb{Z} 2 3\mathbb{Z}_{2}^{\oplus 3} 2 3\mathbb{Z}_{2}^{\oplus 3} 24 8\mathbb{Z}_{24} \oplus \mathbb{Z}_{8} 2\mathbb{Z} \oplus \mathbb{Z}_{2}0
SO(9)SO(9)\mathbb{Z} 2 2\mathbb{Z}_{2}\oplus \mathbb{Z}_{2} 2 2\mathbb{Z}_{2}\oplus \mathbb{Z}_{2} 8\mathbb{Z}_{8} 2\mathbb{Z}\oplus \mathbb{Z}_{2}0
SO(10)SO(10) 2\mathbb{Z}_{2} 2\mathbb{Z}\oplus \mathbb{Z}_{2} 4\mathbb{Z}_{4}\mathbb{Z} 12\mathbb{Z}_{12}
SO(11)SO(11) 2\mathbb{Z}_{2} 2\mathbb{Z}_{2}\mathbb{Z} 2\mathbb{Z}_{2}
SO(12)SO(12)0\mathbb{Z} \oplus \mathbb{Z} 2\mathbb{Z}\oplus \mathbb{Z}_{2}

The SO(6)SO(6) row can be found using Mimura-Toda 63, using Spin(6)=SU(4)Spin(6) = SU(4), and that Spin(6)Spin(6) is a 2\mathbb{Z}_2-covering space of SO(6)SO(6). The SO(7)SO(7) row can be derived from the homotopy groups of Spin(7)Spin(7) as found in Mimura 67. Otherwise the table is given in columns π i\pi_i, i=10,11,12i=10,11,12, and in rows SO(n)SO(n), n=8,,12n=8,\ldots,12, by the Encyclopedic Dictionary of Mathematics?, Table 6.VII in Appendix A.

Homology and cohomology

(Pittie 91)

Whitehead tower and higher orientation structures

The Whitehead tower of the orthogonal group plays an important role in applications related to quantum physics.

The first steps are

Fivebrane(n)String(n)Spin(n)SO(n)O(n). \cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n) \,.

Fivebrane group to String group to Spin group to special orthogonal group to orthogonal group.

Given a manifold XX, lifts of the structure map XO(n)X \to \mathcal{B}O(n) of the O(n)O(n)-principal bundle to which the tangent bundle is associated through this tower define, respectively

on XX.

\cdots\to fivebrane group \to string group \to spin group \to special orthogonal group \to orthogonal group

groupsymboluniversal coversymbolhigher coversymbol
orthogonal groupO(n)\mathrm{O}(n)Pin groupPin(n)Pin(n)Tring groupTring(n)Tring(n)
special orthogonal groupSO(n)SO(n)Spin groupSpin(n)Spin(n)String groupString(n)String(n)
Lorentz groupO(n,1)\mathrm{O}(n,1)\,Spin(n,1)Spin(n,1)\,\,
anti de Sitter groupO(n,2)\mathrm{O}(n,2)\,Spin(n,2)Spin(n,2)\,\,
Narain groupO(n,n)O(n,n)
Poincaré groupISO(n,1)ISO(n,1)Poincaré spin groupISO^(n,1)\widehat {ISO}(n,1)\,\,
super Poincaré groupsISO(n,1)sISO(n,1)\,\,\,\,

References

Examples of sporadic (exceptional) isogenies from spin groups onto orthogonal groups are discussed in

The homotopy groups of O(n)O(n) are listed for instance in

  • Alexander Abanov, Homotopy groups of Lie groups 2009 (pdf)

  • M. Mimura and H. Toda, Homotopy Groups of SU(3)SU(3), SU(4)SU(4) and Sp(2)Sp(2), J. Math. Kyoto Univ. Volume 3, Number 2 (1963), 217-250. (Euclid)

  • M. Mimura, The Homotopy groups of Lie groups of low rank, Math. Kyoto Univ. Volume 6, Number 2 (1967), 131-176. (Euclid)

The ordinary cohomology and ordinary homotopy? of the manifolds SO(n)SO(n) is discussed in

  • Harsh V. Pittie, The integral homology and cohomology rings of SO(n) and Spin(n), Journal of Pure and Applied Algebra Volume 73, Issue 2, 19 August 1991, Pages 105–153 (web)

Revised on November 27, 2014 02:02:04 by David Roberts (129.127.210.209)