View source for Indefinite orthogonal group

{{short description|Orthogonal group of an indefinite quadratic form}}
In [[mathematics]], the '''indefinite orthogonal group''', {{nowrap|O(''p'', ''q'')}} is the [[Lie group]] of all [[linear transformation]]s of an ''n''-[[dimension (vector space)|dimensional]] [[real number|real]] [[vector space]] that leave invariant a [[nondegenerate form|nondegenerate]], [[symmetric bilinear form]] of [[signature of a quadratic form|signature]] {{nowrap|(''p'', ''q'')}}, where {{nowrap|1=''n'' = ''p'' + ''q''}}. It is also called the '''pseudo-orthogonal group'''<ref>{{harvnb|Popov|2001}}</ref> or '''generalized orthogonal group'''.<ref>{{harnvb|Hall|2015|loc=Section 1.2|p=8}}</ref> The dimension of the group is {{nowrap|''n''(''n'' − 1)/2}}.

The '''indefinite special orthogonal group''', {{nowrap|SO(''p'', ''q'')}} is the [[subgroup]] of {{nowrap|O(''p'', ''q'')}} consisting of all elements with [[determinant]] 1. Unlike in the definite case, {{nowrap|SO(''p'', ''q'')}} is not [[connected space|connected]] – it has 2 [[connected component (topology)|components]] – and there are two additional finite [[index of a subgroup|index]] subgroups, namely the connected {{nowrap|SO<sup>+</sup>(''p'', ''q'')}} and {{nowrap|O<sup>+</sup>(''p'', ''q'')}}, which has 2 components – see ''{{slink||Topology}}'' for definition and discussion.

The signature of the form determines the group up to [[isomorphism]]; interchanging ''p'' with ''q'' amounts to replacing the metric by its negative, and so gives the same group. If either ''p'' or ''q'' equals zero, then the group is isomorphic to the ordinary [[orthogonal group]] O(''n''). We assume in what follows that both ''p'' and ''q'' are positive.

The group {{nowrap|O(''p'', ''q'')}} is defined for vector spaces over the reals. For [[complex number|complex]] spaces, all groups {{nowrap|O(''p'', ''q''; '''C''')}} are isomorphic to the usual [[orthogonal group]] {{nowrap|O(''p'' + ''q''; '''C''')}}, since the transform <math>z_j \mapsto iz_j</math> changes the signature of a form. This should not be confused with the [[indefinite unitary group]] {{nowrap|U(''p'', ''q'')}} which preserves a [[sesquilinear form]] of signature {{nowrap|(''p'', ''q'')}}.

In even dimension {{nowrap|1=''n'' = 2''p''}}, {{nowrap|O(''p'', ''p'')}} is known as the [[#Split orthogonal group|split orthogonal group]].

== Examples ==
[[File:Squeeze r=1.5.svg|thumb|[[Squeeze mapping]]s, here {{nowrap|1=''r'' = 3/2}}, are the basic hyperbolic symmetries.]]
The basic example is the [[squeeze mapping]]s, which is the group {{nowrap|SO<sup>+</sup>(1, 1)}} of (the identity component of) linear transforms preserving the [[unit hyperbola]]. Concretely, these are the [[matrix (mathematics)|matrices]] <math>\left[\begin{smallmatrix} \cosh(\alpha) & \sinh(\alpha)  \\ \sinh(\alpha)  & \cosh(\alpha) \end{smallmatrix}\right],</math> and can be interpreted as ''hyperbolic rotations,'' just as the group SO(2) can be interpreted as ''circular rotations.''

In [[physics]], the [[Lorentz group]] {{nowrap|O(1,3)}} is of central importance, being the setting for [[electromagnetism]] and [[special relativity]]. (Some texts use {{nowrap|O(3,1)}} for the Lorentz group; however, {{nowrap|O(1,3)}} is prevalent in [[quantum field theory]] because the geometric properties of the [[Dirac equation]] are more natural in {{nowrap|O(1,3)}}.)

==Matrix definition==
One can define {{nowrap|O(''p'', ''q'')}} as a group of matrices, just as for the classical [[orthogonal group]] O(''n''). Consider the <math>(p+q)\times(p+q)</math> [[diagonal matrix]] <math>g</math> given by
:<math>g = \mathrm{diag}(\underbrace{1,\ldots,1}_{p},\underbrace{-1,\ldots,-1}_{q}) .</math>
Then we may define a [[symmetric bilinear form]] <math>[\cdot,\cdot]_{p,q}</math> on <math>\mathbb R^{p+q}</math> by the formula
:<math>[x,y]_{p,q}=\langle x,gy\rangle=x_1y_1+\cdots +x_py_p-x_{p+1}y_{p+1}-\cdots -x_{p+q}y_{p+q}</math>,
where <math>\langle\cdot,\cdot\rangle</math> is the standard [[inner product]] on <math>\mathbb R^{p+q}</math>.

We then define <math>\mathrm{O}(p,q)</math> to be the group of <math>(p+q)\times(p+q)</math> matrices that preserve this bilinear form:<ref>{{harvnb|Hall|2015}} Section 1.2.3</ref>
:<math>\mathrm{O}(p,q)=\{A\in M_{p+q}(\mathbb R)|[Ax,Ay]_{p,q}=[x,y]_{p,q}\,\forall x,y\in\mathbb R^{p+q}\}</math>.

More explicitly, <math>\mathrm{O}(p,q)</math> consists of matrices <math>A</math> such that<ref>{{harvnb|Hall|2015}} Chapter 1, Exercise 1</ref>
:<math>gA^{\mathrm{tr}}g = A^{-1}</math>,
where <math>A^{\mathrm{tr}}</math> is the [[transpose]] of <math>A</math>.

One obtains an isomorphic group (indeed, a conjugate subgroup of {{nowrap|GL(''p'' + ''q'')}}) by replacing ''g'' with any [[symmetric matrix]] with ''p'' positive [[eigenvalue]]s and ''q'' negative ones. [[Diagonalizable matrix|Diagonalizing]] this matrix gives a conjugation of this group with the standard group {{nowrap|O(''p'', ''q'')}}.

===Subgroups===

The group {{nowrap|SO<sup>+</sup>(''p'', ''q'')}} and related subgroups of {{nowrap|O(''p'', ''q'')}} can be described algebraically. Partition a matrix ''L'' in {{nowrap|O(''p'', ''q'')}} as a [[block matrix]]:
:<math>L = \begin{pmatrix}
A & B \\
C & D
\end{pmatrix}
</math>
where ''A'', ''B'', ''C'', and ''D'' are ''p''×''p'', ''p''×''q'', ''q''×''p'', and ''q''×''q'' blocks, respectively. It can be shown that the set of matrices in {{nowrap|O(''p'', ''q'')}} whose upper-left ''p''×''p'' block ''A'' has positive determinant is a subgroup. Or, to put it another way, if
:<math>L = \begin{pmatrix}
A & B \\
C & D
\end{pmatrix}
\;\mathrm{and}\;
M = \begin{pmatrix}
W & X \\
Y & Z
\end{pmatrix}</math>
are in {{nowrap|O(''p'', ''q'')}}, then
:<math>(\sgn \det A)(\sgn \det W) = \sgn \det (AW+BY).</math>
The analogous result for the bottom-right ''q''×''q'' block also holds. The subgroup {{nowrap|SO<sup>+</sup>(''p'', ''q'')}} consists of matrices ''L'' such that det&thinsp;''A'' and det&thinsp;''D'' are both positive.<ref name="lester">{{Cite journal |last=Lester |first=J. A. |title=Orthochronous subgroups of O(p,q) |journal=Linear and Multilinear Algebra |volume=36 |issue=2 |pages=111–113 |date=1993 |doi=10.1080/03081089308818280 |zbl=0799.20041}}</ref><ref>{{harvnb|Shirokov|2012|loc=Section 7.1|pp=88–96}}</ref>

For all matrices ''L'' in {{nowrap|O(''p'', ''q'')}}, the determinants of ''A'' and ''D'' have the property that <math display="inline">\frac{\det A}{\det D} = \det L</math> and that <math>|{\det A}| = |{\det D}| \ge 1.</math><ref>{{harvnb|Shirokov|2012|loc=Lemmas 7.1 and 7.2|pp=89–91}}</ref> In particular, the subgroup {{nowrap|SO(''p'', ''q'')}} consists of matrices ''L'' such that det&thinsp;''A'' and det&thinsp;''D'' have the same sign.<ref name="lester" />

==Topology==
Assuming both ''p'' and ''q'' are positive, neither of the groups {{nowrap|O(''p'', ''q'')}} nor {{nowrap|SO(''p'', ''q'')}} are [[connected space|connected]], having 4 and 2 components respectively.
{{nowrap|1=''π''<sub>0</sub>(O(''p'', ''q'')) ≅ C<sub>2</sub> × C<sub>2</sub>}} is the [[Klein four-group]], with each factor being whether an element preserves or reverses the respective orientations on the ''p'' and ''q'' dimensional subspaces on which the form is definite; note that reversing orientation on only one of these subspaces reverses orientation on the whole space. The special orthogonal group has components {{nowrap|1=''π''<sub>0</sub>(SO(''p'', ''q'')) = {(1, 1), (−1, −1)}}}, each of which either preserves both orientations or reverses both orientations, in either case preserving the overall orientation.{{clarify|date=December 2020|reason=Usually, the word ''orientation'' refers to the sign on the [[volume form]], and the sign on that flips or not, depending on even or odd dimensions. This paragraph seems to be talking about two different ''parity transformations'' (or parity and time reversal) and ''not'' orientation. Also, it should be clarified whether these parity transformations are [[inner automorphism]]s or not. I think they are(?), but I'm not sure. Maybe they're only inner in some dimensions and not others? }}

The [[identity component]] of {{nowrap|O(''p'', ''q'')}} is often denoted {{nowrap|SO<sup>+</sup>(''p'', ''q'')}} and can be identified with the set of elements in {{nowrap|SO(''p'', ''q'')}} that preserve both orientations. This notation is related to the notation {{nowrap|O<sup>+</sup>(1, 3)}} for the [[orthochronous Lorentz group]], where the + refers to preserving the orientation on the first (temporal) dimension.

The group {{nowrap|O(''p'', ''q'')}} is also not [[compact group|compact]], but contains the compact subgroups O(''p'') and O(''q'') acting on the subspaces on which the form is definite. In fact, {{nowrap|O(''p'') × O(''q'')}} is a [[maximal compact subgroup]] of {{nowrap|O(''p'', ''q'')}}, while {{nowrap|S(O(''p'') × O(''q''))}} is a maximal compact subgroup of {{nowrap|SO(''p'', ''q'')}}.
Likewise, {{nowrap|SO(''p'') × SO(''q'')}} is a maximal compact subgroup of {{nowrap|SO<sup>+</sup>(''p'', ''q'')}}.
Thus, the spaces are homotopy equivalent to products of (special) orthogonal groups, from which algebro-topological invariants can be computed. (See [[Maximal compact subgroup#Topology|Maximal compact subgroup]].)

In particular, the [[fundamental group]] of {{nowrap|SO<sup>+</sup>(''p'', ''q'')}} is the product of the fundamental groups of the components, {{nowrap|1=''π''<sub>1</sub>(SO<sup>+</sup>(''p'', ''q'')) = ''π''<sub>1</sub>(SO(''p'')) × ''π''<sub>1</sub>(SO(''q''))}}, and is given by:
:{| border="1" cellpadding="11" style="border-collapse: collapse; border: 1px #aaa solid;"
!style="background:#efefef;"| ''&pi;''<sub>1</sub>(SO<sup>+</sup>(''p'', ''q''))
!style="background:#efefef;"| ''p'' = 1
!style="background:#efefef;"| ''p'' = 2
!style="background:#efefef;"| ''p'' ≥ 3
|-
!style="background:#efefef;"| ''q'' = 1
| C<sub>1</sub> || '''Z''' || C<sub>2</sub>
|-
!style="background:#efefef;"| ''q'' = 2
| '''Z''' || '''Z''' × '''Z'''
 || '''Z''' × C<sub>2</sub>
|-
!style="background:#efefef;"| ''q'' ≥ 3
| C<sub>2</sub> || C<sub>2</sub> × '''Z'''
 || C<sub>2</sub> × C<sub>2</sub>
|}

==Split orthogonal group==
In even dimensions, the middle group {{nowrap|O(''n'', ''n'')}} is known as the '''split orthogonal group''', and is of particular interest, as it occurs as the group of [[T-duality]] transformations in [[string theory]], for example. It is the [[split Lie group]] corresponding to the complex [[Lie algebra]] so<sub>2''n''</sub> (the Lie group of the [[split real form]] of the Lie algebra); more precisely, the identity component is the split Lie group, as non-identity components cannot be reconstructed from the Lie algebra. In this sense it is opposite to the definite orthogonal group {{nowrap|1=O(''n'') := O(''n'', 0) = O(0, ''n'')}}, which is the [[compact real form|''compact'' real form]] of the [[complex Lie algebra]].

The group {{nowrap|SO(1, 1)}} may be identified with the group of unit [[split-complex number]]s.

In terms of being a [[group of Lie type]] – i.e., construction of an [[algebraic group]] from a Lie algebra – split orthogonal groups are [[Chevalley group]]s, while the non-split orthogonal groups require a slightly more complicated construction, and are [[Steinberg group (Lie theory)|Steinberg groups]].

Split orthogonal groups are used to construct the [[generalized flag variety]] over non-[[algebraically closed field]]s.

{{Expand section|date=March 2011}}

==See also==
*[[Orthogonal group]]
*[[Lorentz group]]
*[[Poincaré group]]
*[[Symmetric bilinear form]]

==References==
{{reflist}}

==Sources==
{{sfn whitelist |CITEREFPopov2001}}
{{refbegin}}
* {{citation|first=Brian C.|last=Hall|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|edition= 2nd|series=Graduate Texts in Mathematics|volume=222 |publisher=Springer|year=2015|isbn=978-3319134666}}
*[[Anthony W. Knapp|Anthony Knapp]], ''Lie Groups Beyond an Introduction'', Second Edition, Progress in Mathematics, vol. 140, Birkhäuser, Boston, 2002. {{ISBN|0-8176-4259-5}} – see page 372 for a description of the indefinite orthogonal group
*{{springer|id=O/o070300|title=Orthogonal group|author-link=Vladimir L. Popov|first=V. L.|last=Popov}}
*{{Cite journal |last=Shirokov |first=D. S. |script-title=ru:Лекции по алгебрам клиффорда и спинорам |title=Lectures on Clifford algebras and spinors |journal=Лекционные Курсы Ноц |date=2012 |volume=19 |language=ru |doi=10.4213/book1373 |zbl=1291.15063 |url=http://www.mathnet.ru/links/856008704d1b4844a21d3d20f25f3fdc/book1373.pdf}}
*[[Joseph A. Wolf]], ''Spaces of constant curvature'', (1967) page. 335.
{{refend}}

[[Category:Lie groups]]