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 or generalized orthogonal group. 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 ' 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
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:
<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
<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.
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>. 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.