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orthogonal group

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Group Theory

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

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

Compactness

Proposition

The orthogonal group O(n)O(n) is compact topological space, hence in particular a compact Lie group.

Homotopy groups

Proposition

For n,kn,k \in \mathbb{N}, nkn \leq k, then the canonical inclusion of orthogonal groups

O(n)O(k) O(n) \hookrightarrow O(k)

is an (n-1)-equivalence?, hence induces an isomorphism on homotopy groups in degrees <n1\lt n-1 and a surjection in degree n1n-1.

Proof

Consider the coset quotient projection

O(n)O(n+1)O(n+1)/O(n). O(n) \longrightarrow O(n+1) \longrightarrow O(n+1)/O(n) \,.

By prop. 1 and by this corolary, the projection O(n+1)O(n+1)/O(n)O(n+1)\to O(n+1)/O(n) is a Serre fibration. Furthermore, example 1 identifies the coset with the n-sphere

S nO(n+1)/O(n). S^{n}\simeq O(n+1)/O(n) \,.

Therefore the long exact sequence of homotopy groups of the fiber sequence O(n)O(n+1)S nO(n)\to O(n+1)\to S^n looks like

π +1(S n)π (O(n))π (O(n+1))π (S n) \cdots \to \pi_{\bullet+1}(S^n) \longrightarrow \pi_\bullet(O(n)) \longrightarrow \pi_\bullet(O(n+1)) \longrightarrow \pi_\bullet(S^n) \to \cdots

Since π <n(S n)=0\pi_{\lt n}(S^n) = 0, this implies that

π <n1(O(n))π <n1(O(n+1)) \pi_{\lt n-1}(O(n)) \overset{\simeq}{\longrightarrow} \pi_{\lt n-1}(O(n+1))

is an isomorphism and that

π n1(O(n))π n1(O(n+1)) \pi_{n-1}(O(n)) \overset{\simeq}{\longrightarrow} \pi_{n-1}(O(n+1))

is surjective. Hence now the statement follows by induction over knk-n.

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} 24\mathbb{Z}_{24} 2\mathbb{Z}_2 120 2\mathbb{Z}_{120}\oplus\mathbb{Z}_2 4\mathbb{Z}_{4} 60\mathbb{Z}_{60}
SO(7)SO(7)00\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(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 2\mathbb{Z}_{2}\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.

Beware that the maps

π 3(SO(3))π 3(SO(4))π 3(SO(5)) \array{ \pi_3(SO(3)) \longrightarrow \pi_3(SO(4)) \longrightarrow \pi_3(SO(5)) \\ \mathbb{Z}\longrightarrow \mathbb{Z}\oplus \mathbb{Z} \longrightarrow\mathbb{Z} }

are inclusion of the first summand followed by projection onto the second. So even though π 3(SO(3))\pi_3(SO(3)) \simeq \mathbb{Z} superficially looks like it already stabilized, in fact these elements disappear in the stabilization and another copy of \mathbb{Z} appears (e.g. Tamura 57). The same is also true for π 7(SO(7))π 7(SO(8))π 7(SO(9))\pi_7(SO(7)) \to \pi_7(SO(8)) \to \pi_7(SO(9)).

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.

Coset spaces

Example

The n-spheres are coset spaces of orthogonal groups

S nO(n+1)/O(n). S^n \simeq O(n+1)/O(n) \,.

For fix a unit vector in n+1\mathbb{R}^{n+1}. Then its orbit under the defining O(n+1)O(n+1)-action on n+1\mathbb{R}^{n+1} is clearly the canonical embedding S n n+1S^n \hookrightarrow \mathbb{R}^{n+1}. But precisely the subgroup of O(n+1)O(n+1) that consists of rotations around the axis formed by that unit vector stabilizes it, and that subgroup is isomorphic to O(n)O(n), hence S nO(n+1)/O(n)S^n \simeq O(n+1)/O(n).

Example

The coset

V n(k)O(k)/O(n) V_n(k) \coloneqq O(k)/O(n)

is called the nnth Stiefel manifold of k\mathbb{R}^k.

Proposition

The Stiefel manifold V n(k)V_n(k) is (n-1)-connected.

Proof

Consider the coset quotient projection

O(n)O(k)O(k)/O(n)=V n(k). O(n) \longrightarrow O(k) \longrightarrow O(k)/O(n) = V_n(k) \,.

By prop. 1 and by this corolarry the projection O(k)O(k)/O(n)O(k)\to O(k)/O(n) is a Serre fibration. Therefore there is the long exact sequence of homotopy groups of this fiber sequence and by prop. 2 it has the following structure in degrees bounded by nn:

π n1(O(k))epiπ n1(O(n))0π n1(V n(k))0π 1<n1(O(k))π 1<n1(O(n)). \cdots \to \pi_{\bullet \leq n-1}(O(k)) \overset{epi}{\longrightarrow} \pi_{\bullet \leq n-1}(O(n)) \overset{0}{\longrightarrow} \pi_{\bullet \leq n-1}(V_n(k)) \overset{0}{\longrightarrow} \pi_{\bullet-1 \lt n-1}(O(k)) \overset{\simeq}{\longrightarrow} \pi_{\bullet-1 \lt n-1}(O(n)) \to \cdots \,.

This implies the claim. (Exactness of the sequence says that every element in π n1(V n(k))\pi_{\bullet \leq n-1}(V_n(k)) is in the kernel of zero, hence in the image of 0, hence is 0 itself.)

\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)\,\,
conformal groupO(n+1,t+1)\mathrm{O}(n+1,t+1)\,
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)\,\,\,\,
superconformal group

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 homology 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)

See also

  • Itiro Tamura, On Pontrjagon classes of homotopy types of manifolds, Journal of the mathematical society of Japan, Vol. 9 No. 2 , 1957 pdf

Revised on May 3, 2016 04:35:33 by Urs Schreiber (131.220.184.222)