Indefinite orthogonal group

<h2 id="examples">Examples</h2>

The basic example is the <a href="/facts/Squeeze_mapping/CYxzaaKR">squeeze mappings</a>, which is the group SO+(1, 1) of (the identity component of) linear transforms preserving the <a href="/facts/Unit_hyperbola/AvwG6gJ9">unit hyperbola</a>. Concretely, these are the <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> 
 
 
 
 
 [
 
 
 
 
 
 cosh
 ⁡
 (
 α
 )
 
 
 sinh
 ⁡
 (
 α
 )
 
 
 
 
 sinh
 ⁡
 (
 α
 )
 
 
 cosh
 ⁡
 (
 α
 )
 
 
 
 
 
 ]
 
 ,
 
 
 {\displaystyle \left[{\begin{smallmatrix}\cosh(\alpha )&\sinh(\alpha )\\\sinh(\alpha )&\cosh(\alpha )\end{smallmatrix}}\right],}
 
 and can be interpreted as hyperbolic rotations, just as the group SO(2) can be interpreted as circular rotations.
In <a href="/facts/Physics/a7aH2y08">physics</a>, the <a href="/facts/Lorentz_group/t1APc2Ye">Lorentz group</a> O(1,3) is of central importance, being the setting for <a href="/facts/Electromagnetism/oAkmALHT">electromagnetism</a> and <a href="/facts/Special_relativity/vsFFc63E">special relativity</a>. (Some texts use O(3,1) for the Lorentz group; however, O(1,3) is prevalent in <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a> because the geometric properties of the <a href="/facts/Dirac_equation/DzFaXS4p">Dirac equation</a> are more natural in O(1,3).)

<h2 id="matrix-definition">Matrix definition</h2>
One can define O(p, q) as a group of matrices, just as for the classical <a href="/facts/Orthogonal_group/jKWM1KDM">orthogonal group</a> O(n). Consider the 
 
 
 
 (
 p
 +
 q
 )
 ×
 (
 p
 +
 q
 )
 
 
 {\displaystyle (p+q)\times (p+q)}
 
 <a href="/facts/Diagonal_matrix/tjIAHrE6">diagonal matrix</a> 
 
 
 
 g
 
 
 {\displaystyle g}
 
 given by

g
        =
        
          d
          i
          a
          g
        
        (
        
          
            
              
                1
                ,
                …
                ,
                1
              
              ⏟
            
          
          
            p
          
        
        ,
        
          
            
              
                −
                1
                ,
                …
                ,
                −
                1
              
              ⏟
            
          
          
            q
          
        
        )
        .
      
    
    {\displaystyle g=\mathrm {diag} (\underbrace {1,\ldots ,1} _{p},\underbrace {-1,\ldots ,-1} _{q}).}

Then we may define a <a href="/facts/Symmetric_bilinear_form/UI9b6RK7">symmetric bilinear form</a> 
 
 
 
 [
 ⋅
 ,
 ⋅
 
 ]
 
 p
 ,
 q
 
 
 
 
 {\displaystyle [\cdot ,\cdot ]_{p,q}}
 
 on 
 
 
 
 
 
 R
 
 
 p
 +
 q
 
 
 
 
 {\displaystyle \mathbb {R} ^{p+q}}
 
 by the formula

[
 x
 ,
 y
 
 ]
 
 p
 ,
 q
 
 
 =
 ⟨
 x
 ,
 g
 y
 ⟩
 =
 
 x
 
 1
 
 
 
 y
 
 1
 
 
 +
 ⋯
 +
 
 x
 
 p
 
 
 
 y
 
 p
 
 
 −
 
 x
 
 p
 +
 1
 
 
 
 y
 
 p
 +
 1
 
 
 −
 ⋯
 −
 
 x
 
 p
 +
 q
 
 
 
 y
 
 p
 +
 q
 
 
 
 
 {\displaystyle [x,y]_{p,q}=\langle x,gy\rangle =x_{1}y_{1}+\cdots +x_{p}y_{p}-x_{p+1}y_{p+1}-\cdots -x_{p+q}y_{p+q}}
 
,
where 
 
 
 
 ⟨
 ⋅
 ,
 ⋅
 ⟩
 
 
 {\displaystyle \langle \cdot ,\cdot \rangle }
 
 is the standard <a href="/facts/Inner_product/HoyjElEy">inner product</a> on 
 
 
 
 
 
 R
 
 
 p
 +
 q
 
 
 
 
 {\displaystyle \mathbb {R} ^{p+q}}
 
.
We then define 
 
 
 
 
 O
 
 (
 p
 ,
 q
 )
 
 
 {\displaystyle \mathrm {O} (p,q)}
 
 to be the group of 
 
 
 
 (
 p
 +
 q
 )
 ×
 (
 p
 +
 q
 )
 
 
 {\displaystyle (p+q)\times (p+q)}
 
 matrices that preserve this bilinear form:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

O
 
 (
 p
 ,
 q
 )
 =
 {
 A
 ∈
 
 M
 
 p
 +
 q
 
 
 (
 
 R
 
 )
 
 |
 
 [
 A
 x
 ,
 A
 y
 
 ]
 
 p
 ,
 q
 
 
 =
 [
 x
 ,
 y
 
 ]
 
 p
 ,
 q
 
 
 
 ∀
 x
 ,
 y
 ∈
 
 
 R
 
 
 p
 +
 q
 
 
 }
 
 
 {\displaystyle \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}\}}
 
.
More explicitly, 
 
 
 
 
 O
 
 (
 p
 ,
 q
 )
 
 
 {\displaystyle \mathrm {O} (p,q)}
 
 consists of matrices 
 
 
 
 A
 
 
 {\displaystyle A}
 
 such that<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

g
 
 A
 
 
 t
 r
 
 
 
 g
 =
 
 A
 
 −
 1
 
 
 
 
 {\displaystyle gA^{\mathrm {tr} }g=A^{-1}}
 
,
where 
 
 
 
 
 A
 
 
 t
 r
 
 
 
 
 
 {\displaystyle A^{\mathrm {tr} }}
 
 is the <a href="/facts/Transpose/8wmsagGS">transpose</a> of 
 
 
 
 A
 
 
 {\displaystyle A}
 
.
One obtains an isomorphic group (indeed, a conjugate subgroup of GL(p + q)) by replacing g with any <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric matrix</a> with p positive <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> and q negative ones. <a href="/facts/Diagonalizable_matrix/y3MkyTCy">Diagonalizing</a> this matrix gives a conjugation of this group with the standard group O(p, q).

<h3>Subgroups</h3>
The group SO+(p, q) and related subgroups of O(p, q) can be described algebraically. Partition a matrix L in O(p, q) as a <a href="/facts/Block_matrix/fPI8HX9f">block matrix</a>:

L
        =
        
          
            (
            
              
                
                  A
                
                
                  B
                
              
              
                
                  C
                
                
                  D
                
              
            
            )
          
        
      
    
    {\displaystyle L={\begin{pmatrix}A&B\\C&D\end{pmatrix}}}

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 O(p, q) whose upper-left p×p block A has positive determinant is a subgroup. Or, to put it another way, if

L
        =
        
          
            (
            
              
                
                  A
                
                
                  B
                
              
              
                
                  C
                
                
                  D
                
              
            
            )
          
        
        
        
          a
          n
          d
        
        
        M
        =
        
          
            (
            
              
                
                  W
                
                
                  X
                
              
              
                
                  Y
                
                
                  Z
                
              
            
            )
          
        
      
    
    {\displaystyle L={\begin{pmatrix}A&B\\C&D\end{pmatrix}}\;\mathrm {and} \;M={\begin{pmatrix}W&X\\Y&Z\end{pmatrix}}}

are in O(p, q), then

(
        sgn
        ⁡
        det
        A
        )
        (
        sgn
        ⁡
        det
        W
        )
        =
        sgn
        ⁡
        det
        (
        A
        W
        +
        B
        Y
        )
        .
      
    
    {\displaystyle (\operatorname {sgn} \det A)(\operatorname {sgn} \det W)=\operatorname {sgn} \det(AW+BY).}

The analogous result for the bottom-right q×q block also holds. The subgroup SO+(p, q) consists of matrices L such that det A and det D are both positive.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
For all matrices L in O(p, q), the determinants of A and D have the property that 
 
 
 
 
 
 
 det
 A
 
 
 det
 D
 
 
 
 =
 det
 L
 
 
 {\textstyle {\frac {\det A}{\det D}}=\det L}
 
 and that 
 
 
 
 
 |
 
 
 det
 A
 
 
 |
 
 =
 
 |
 
 
 det
 D
 
 
 |
 
 ≥
 1.
 
 
 {\displaystyle |{\det A}|=|{\det D}|\geq 1.}
 
<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> In particular, the subgroup SO(p, q) consists of matrices L such that det A and det D have the same sign.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<h2 id="topology">Topology</h2>
Assuming both p and q are positive, neither of the groups O(p, q) nor SO(p, q) are <a href="/facts/Connected_space/5CUTxfoA">connected</a>, having 4 and 2 components respectively.
π0(O(p, q)) ≅ C2 × C2 is the <a href="/facts/Klein_four-group/X2oQCYBy">Klein four-group</a>, 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 π0(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.
The <a href="/facts/Identity_component/rFCx5kro">identity component</a> of O(p, q) is often denoted SO+(p, q) and can be identified with the set of elements in SO(p, q) that preserve both orientations. This notation is related to the notation O+(1, 3) for the <a href="/facts/Orthochronous_Lorentz_group/t1APc2Ye">orthochronous Lorentz group</a>, where the + refers to preserving the orientation on the first (temporal) dimension.
The group O(p, q) is also not <a href="/facts/Compact_group/Itl2HJaR">compact</a>, but contains the compact subgroups O(p) and O(q) acting on the subspaces on which the form is definite. In fact, O(p) × O(q) is a <a href="/facts/Maximal_compact_subgroup/8EA7XRcu">maximal compact subgroup</a> of O(p, q), while S(O(p) × O(q)) is a maximal compact subgroup of SO(p, q).
Likewise, SO(p) × SO(q) is a maximal compact subgroup of SO+(p, q).
Thus, the spaces are homotopy equivalent to products of (special) orthogonal groups, from which algebro-topological invariants can be computed. (See <a href="/facts/Maximal_compact_subgroup/8EA7XRcu">Maximal compact subgroup</a>.)
In particular, the <a href="/facts/Fundamental_group/VyYtLqSP">fundamental group</a> of SO+(p, q) is the product of the fundamental groups of the components, π1(SO+(p, q)) = π1(SO(p)) × π1(SO(q)), and is given by:

<table><tbody><tr><th>π1(SO+(p, q))</th><th>p = 1</th><th>p = 2</th><th>p ≥ 3</th></tr><tr><th>q = 1</th><td>C1</td><td>Z</td><td>C2</td></tr><tr><th>q = 2</th><td>Z</td><td>Z × Z</td><td>Z × C2</td></tr><tr><th>q ≥ 3</th><td>C2</td><td>C2 × Z</td><td>C2 × C2</td></tr></tbody></table>
<h2 id="split-orthogonal-group">Split orthogonal group</h2>
In even dimensions, the middle group O(n, n) is known as the split orthogonal group, and is of particular interest, as it occurs as the group of <a href="/facts/T-duality/54taLJa1">T-duality</a> transformations in <a href="/facts/String_theory/HUJFIPYH">string theory</a>, for example. It is the <a href="/facts/Split_Lie_group/59D5Khvs">split Lie group</a> corresponding to the complex <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> so2n (the Lie group of the <a href="/facts/Split_real_form/7Frha9gL">split real form</a> 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 O(n) := O(n, 0) = O(0, n), which is the <a href="/facts/Compact_real_form/7Frha9gL">compact real form</a> of the complex Lie algebra.
The group SO(1, 1) may be identified with the group of unit <a href="/facts/Split-complex_number/DtavqChA">split-complex numbers</a>.
In terms of being a <a href="/facts/Group_of_Lie_type/gmboug78">group of Lie type</a> – i.e., construction of an algebraic group from a Lie algebra – split orthogonal groups are <a href="/facts/Chevalley_group/gmboug78">Chevalley groups</a>, while the non-split orthogonal groups require a slightly more complicated construction, and are <a href="/facts/Steinberg_group_(Lie_theory)/gmboug78">Steinberg groups</a>.
Split orthogonal groups are used to construct the <a href="/facts/Generalized_flag_variety/rdTKdfBl">generalized flag variety</a> over non-<a href="/facts/Algebraically_closed_field/HuVQClok">algebraically closed fields</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Orthogonal_group/jKWM1KDM">Orthogonal group</a></li>
<li><a href="/facts/Lorentz_group/t1APc2Ye">Lorentz group</a></li>
<li><a href="/facts/Poincar%C3%A9_group/3IerRYG1">Poincaré group</a></li>
<li><a href="/facts/Symmetric_bilinear_form/UI9b6RK7">Symmetric bilinear form</a></li></ul>

<h2 id="sources">Sources</h2>

<ul><li>Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3319134666</li>
<li><a href="/facts/Anthony_W._Knapp/Jal9nngl">Anthony Knapp</a>, Lie Groups Beyond an Introduction, Second Edition, Progress in Mathematics, vol. 140, Birkhäuser, Boston, 2002. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8176-4259-5 – see page 372 for a description of the indefinite orthogonal group</li>
<li><a href="/facts/Vladimir_L._Popov/U9VGpfRW">Popov, V. L.</a> (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Orthogonal_group">"Orthogonal group"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li>Shirokov, D. S. (2012). <a href="https://www.mathnet.ru/links/856008704d1b4844a21d3d20f25f3fdc/book1373.pdf">Lectures on Clifford algebras and spinors Лекции по алгебрам клиффорда и спинорам</a> (PDF) (in Russian). <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4213%2Fbook1373">10.4213/book1373</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1291.15063">1291.15063</a>.</li>
<li><a href="/facts/Joseph_A._Wolf/P12DVYGG">Joseph A. Wolf</a>, Spaces of constant curvature, (1967) page. 335.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Popov 2001 - Popov, V. L. (2001) [1994], "Orthogonal group", Encyclopedia of Mathematics, EMS Press <a href="https://www.encyclopediaofmath.org/index.php?title=Orthogonal_group" target="_blank">https://www.encyclopediaofmath.org/index.php?title=Orthogonal_group</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Hall 2015, p. 8, Section 1.2 - Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Hall 2015 Section 1.2.3 - Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Hall 2015 Chapter 1, Exercise 1 - Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Lester, J. A. (1993). "Orthochronous subgroups of O(p,q)". Linear and Multilinear Algebra. 36 (2): 111–113. doi:10.1080/03081089308818280. Zbl 0799.20041. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Shirokov 2012, pp. 88–96, Section 7.1 - Shirokov, D. S. (2012). Lectures on Clifford algebras and spinors Лекции по алгебрам клиффорда и спинорам (PDF) (in Russian). doi:10.4213/book1373. Zbl 1291.15063. <a href="https://www.mathnet.ru/links/856008704d1b4844a21d3d20f25f3fdc/book1373.pdf" target="_blank">https://www.mathnet.ru/links/856008704d1b4844a21d3d20f25f3fdc/book1373.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Shirokov 2012, pp. 89–91, Lemmas 7.1 and 7.2 - Shirokov, D. S. (2012). Lectures on Clifford algebras and spinors Лекции по алгебрам клиффорда и спинорам (PDF) (in Russian). doi:10.4213/book1373. Zbl 1291.15063. <a href="https://www.mathnet.ru/links/856008704d1b4844a21d3d20f25f3fdc/book1373.pdf" target="_blank">https://www.mathnet.ru/links/856008704d1b4844a21d3d20f25f3fdc/book1373.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Lester, J. A. (1993). "Orthochronous subgroups of O(p,q)". Linear and Multilinear Algebra. 36 (2): 111–113. doi:10.1080/03081089308818280. Zbl 0799.20041. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Indefinite orthogonal group open-in-new

Indefinite orthogonal group