{{short description|Orthogonal group of an indefinite quadratic form}} In mathematics, the '''indefinite orthogonal group''', <math>\operatorname{O}(p,q)</math> is the Lie group of all linear transformations of an <math>n</math>-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature <math>(p,q)</math>, where <math>n=p+q</math>. 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 <math>n(n-1)/2</math>.
The '''indefinite special orthogonal group''', <math>\operatorname{SO}(p,q)</math> is the subgroup of <math>\operatorname{O}(p,q)</math> consisting of all elements with determinant <math>1</math>. Unlike in the definite case, <math>\operatorname{SO}(p,q)</math> is not connected – it has 2 components – and there are two additional finite index subgroups, namely the connected <math>\operatorname{SO}^+(p,q)</math> and <math>\operatorname{O}^+(p,q)</math>, which has 2 components – see ''{{slink||Topology}}'' for definition and discussion.
The signature of the form determines the group up to isomorphism; interchanging <math>p</math> with <math>q</math> amounts to replacing the metric by its negative, and so gives the same group. If either <math>p</math> or <math>q</math> equals zero, then the group is isomorphic to the ordinary orthogonal group <math>\operatorname{O}(n)</math>. We assume in what follows that both <math>p</math> and <math>q</math> are positive.
The group <math>\operatorname{O}(p,q)</math> is defined for vector spaces over the reals. On complex spaces, all nondegenerate symmetric bilinear forms are the same up to change of coordinates; however, one can define the indefinite unitary group <math>\operatorname{U}(p,q)</math> which preserves a sesquilinear form of signature <math>(p,q)</math>.
In even dimension <math>n=2p</math>, <math>\operatorname{O}(p,p)</math> is known as the split orthogonal group.
== Examples == [[File:Squeeze r=1.5.svg|thumb|Squeeze mappings, here <math>r=3/2</math>, are the basic hyperbolic symmetries.]] The basic example is the squeeze mappings, which is the group <math>\operatorname{SO}^+(1,1)</math> of (the identity component of) linear transforms preserving the unit hyperbola. Concretely, these are the 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 <math>\operatorname{SO}(2)</math> can be interpreted as ''circular rotations.''
In physics, the Lorentz group <math>\operatorname{O}(1,3)</math> is of central importance, being the setting for electromagnetism and special relativity. (Some texts use <math>\operatorname{O}(3,1)</math> for the Lorentz group; however, <math>\operatorname{O}(1,3)</math> is prevalent in quantum field theory because the geometric properties of the Dirac equation are more natural in <math>\operatorname{O}(1,3)</math>.)
==Matrix definition== One can define <math>\operatorname{O}(p,q)</math> as a group of matrices, just as for the classical orthogonal group <math>\operatorname{O}(n)</math>. Consider the <math>(p+q)\times(p+q)</math> diagonal matrix <math>g</math> given by <math display="block">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 display="block">[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 display="block">\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 display="block">gA^Tg = A^{-1},</math> where <math>A^T</math> is the transpose of <math>A</math>.
One obtains an isomorphic group (indeed, a conjugate subgroup of <math>\operatorname{GL}(p+q)</math>) by replacing <math>g</math> with any symmetric matrix with <math>p</math> positive eigenvalues and <math>q</math> negative ones. Diagonalizing this matrix gives a conjugation of this group with the standard group <math>\operatorname{O}(p,q)</math>.
===Subgroups=== The group <math>\operatorname{SO}^+(p,q)</math> and related subgroups of <math>\operatorname{O}(p,q)</math> can be described algebraically. Partition a matrix <math>L</math> in <math>\operatorname{O}(p,q)</math> as a block matrix: <math display="block">L = \begin{pmatrix} A & B \\ C & D \end{pmatrix} </math> where <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math> are <math>p\times p</math>, <math>p\times q</math>, <math>q\times p</math>, and <math>q\times q</math> blocks, respectively. It can be shown that the set of matrices in <math>\operatorname{O}(p,q)</math> whose upper-left <math>p\times p</math> block <math>A</math> has positive determinant is a subgroup. Or, to put it another way, if <math display="block">L = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \;\mathrm{and}\; M = \begin{pmatrix} W & X \\ Y & Z \end{pmatrix}</math> are in <math>\operatorname{O}(p,q)</math>, then <math display="block">(\sgn \det A)(\sgn \det W) = \sgn \det (AW+BY).</math>
The analogous result for the bottom-right <math>q\times q</math> block also holds. The subgroup <math>\operatorname{SO}^+(p,q)</math> consists of matrices <math>L</math> such that <math>\det A</math> and <math>\det D</math> 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 <math>L</math> in <math>\operatorname{O}(p,q)</math>, the determinants of <math>A</math> and <math>D</math> 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 <math>\operatorname{SO}(p,q)</math> consists of matrices <math>L</math> such that <math>\det A</math> and <math>\det D</math> have the same sign.<ref name="lester" />
==Topology== Assuming both <math>p</math> and <math>q</math> are positive, neither of the groups <math>\operatorname{O}(p,q)</math> nor <math>\operatorname{SO}(p,q)</math> are connected, having <math>4</math> and <math>2</math> components respectively. <math>\pi_0(\operatorname{O}(p,q))\cong C_2\times C_2</math> is the Klein four-group, with each factor being whether an element preserves or reverses the respective orientations on the <math>p</math> and <math>q</math> 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 <math>\pi_0(\operatorname{SO}(p,q))=\{(1,1),(-1,-1)\}</math>, 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 automorphisms 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 <math>\operatorname{O}(p,q)</math> is often denoted <math>\operatorname{SO}^+(p,q)</math> and can be identified with the set of elements in <math>\operatorname{SO}(p,q)</math> that preserve both orientations. This notation is related to the notation <math>\operatorname{O}^+(1,3)</math> for the orthochronous Lorentz group, where the <math>+</math> refers to preserving the orientation on the first (temporal) dimension.
The group <math>\operatorname{O}(p,q)</math> is also not compact, but contains the compact subgroups <math>\operatorname{O}(p)</math> and <math>\operatorname{O}(q)</math> acting on the subspaces on which the form is definite. In fact, <math>\operatorname{O}(p)\times\operatorname{O}(q)</math> is a maximal compact subgroup of <math>\operatorname{O}(p,q)</math>, while <math>\operatorname{S}(\operatorname{O}(p)\times\operatorname{O}(q))</math> is a maximal compact subgroup of <math>\operatorname{SO}(p,q)</math>. Likewise, <math>\operatorname{SO}(p)\times\operatorname{SO}(q)</math> is a maximal compact subgroup of <math>\operatorname{SO}^+(p,q)</math>. Thus, the spaces are homotopy equivalent to products of (special) orthogonal groups, from which algebro-topological invariants can be computed. (See Maximal compact subgroup.)
In particular, the fundamental group of <math>\operatorname{SO}^+(p,q)</math> is the product of the fundamental groups of the components, <math>\pi_1(\operatorname{SO}^+(p,q))=\pi_1(\operatorname{SO}(p))\times\pi_1(\operatorname{SO}(q))</math>, and is given by: :{| border="1" cellpadding="11" style="border-collapse: collapse; border: 1px #aaa solid;" !style="background:#efefef;"| <math>\pi_1(\operatorname{SO}^+(p,q))</math> !style="background:#efefef;"| <math>p=1</math> !style="background:#efefef;"| <math>p=2</math> !style="background:#efefef;"| <math>p\geq 3</math> |- !style="background:#efefef;"| <math>q=1</math> | <math>C_1</math> || <math>\mathbb{Z}</math> || <math>C_2</math> |- !style="background:#efefef;"| <math>q=2</math> | <math>\mathbb{Z}</math> || <math>\mathbb{Z}\times\mathbb{Z}</math> || <math>\mathbb{Z}\times C_2</math> |- !style="background:#efefef;"| <math>q\geq 3</math> | <math>C_2</math> || <math>C_2\times\mathbb{Z}</math> || <math>C_2\times C_2</math> |}
==Split orthogonal group== In even dimensions, the middle group <math>\operatorname{O}(n,n)</math> 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 <math>\mathfrak{so}_{2n}</math> (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 <math>\operatorname{O}(n):=\operatorname{O}(n,0)=\operatorname{O}(0,n)</math>, which is the ''compact'' real form of the complex Lie algebra.
The group <math>\operatorname{SO}(1,1)</math> may be identified with the unit hyperbola group, a subgroup of the group of units in split-complex numbers.
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 groups, while the non-split orthogonal groups require a slightly more complicated construction, and are Steinberg groups.
Split orthogonal groups are used to construct the generalized flag variety over non-algebraically closed fields.
==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 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