Indefinite orthogonal group

In mathematics, the indefinite orthogonal group, ${\displaystyle \operatorname {O} (p,q)}$ is the Lie group of all linear transformations of an ${\displaystyle n}$-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature ${\displaystyle (p,q)}$, where ${\displaystyle n=p+q}$. It is also called the pseudo-orthogonal group¹ or generalized orthogonal group.² The dimension of the group is ${\displaystyle n(n-1)/2}$.

The indefinite special orthogonal group, ${\displaystyle \operatorname {SO} (p,q)}$ is the subgroup of ${\displaystyle \operatorname {O} (p,q)}$ consisting of all elements with determinant ${\displaystyle 1}$. Unlike in the definite case, ${\displaystyle \operatorname {SO} (p,q)}$ is not connected – it has 2 components – and there are two additional finite index subgroups, namely the connected ${\displaystyle \operatorname {SO} ^{+}(p,q)}$ and ${\displaystyle \operatorname {O} ^{+}(p,q)}$, which has 2 components – see § Topology for definition and discussion.

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

The group ${\displaystyle \operatorname {O} (p,q)}$ 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 ${\displaystyle \operatorname {U} (p,q)}$ which preserves a sesquilinear form of signature ${\displaystyle (p,q)}$.

In even dimension ${\displaystyle n=2p}$, ${\displaystyle \operatorname {O} (p,p)}$ is known as the split orthogonal group.

The basic example is the squeeze mappings, which is the group ${\displaystyle \operatorname {SO} ^{+}(1,1)}$ of (the identity component of) linear transforms preserving the unit hyperbola. Concretely, these are the matrices ${\displaystyle \left[{\begin{smallmatrix}\cosh(\alpha )&\sinh(\alpha )\\\sinh(\alpha )&\cosh(\alpha )\end{smallmatrix}}\right],}$ and can be interpreted as hyperbolic rotations, just as the group ${\displaystyle \operatorname {SO} (2)}$ can be interpreted as circular rotations.

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

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

We then define ${\displaystyle \mathrm {O} (p,q)}$ to be the group of ${\displaystyle (p+q)\times (p+q)}$ matrices that preserve this bilinear form:³ $${\displaystyle \mathrm {O} (p,q)=\{A\in M_{p+q}(\mathbb {R} ):[Ax,Ay]_{p,q}=[x,y]_{p,q}\,\forall x,y\in \mathbb {R} ^{p+q}\}.}$$

More explicitly, ${\displaystyle \mathrm {O} (p,q)}$ consists of matrices ${\displaystyle A}$ such that⁴ $${\displaystyle gA^{T}g=A^{-1},}$$ where ${\displaystyle A^{T}}$ is the transpose of ${\displaystyle A}$.

One obtains an isomorphic group (indeed, a conjugate subgroup of ${\displaystyle \operatorname {GL} (p+q)}$) by replacing ${\displaystyle g}$ with any symmetric matrix with ${\displaystyle p}$ positive eigenvalues and ${\displaystyle q}$ negative ones. Diagonalizing this matrix gives a conjugation of this group with the standard group ${\displaystyle \operatorname {O} (p,q)}$.

The group ${\displaystyle \operatorname {SO} ^{+}(p,q)}$ and related subgroups of ${\displaystyle \operatorname {O} (p,q)}$ can be described algebraically. Partition a matrix ${\displaystyle L}$ in ${\displaystyle \operatorname {O} (p,q)}$ as a block matrix: $${\displaystyle L={\begin{pmatrix}A&B\\C&D\end{pmatrix}}}$$ where ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$, and ${\displaystyle D}$ are ${\displaystyle p\times p}$, ${\displaystyle p\times q}$, ${\displaystyle q\times p}$, and ${\displaystyle q\times q}$ blocks, respectively. It can be shown that the set of matrices in ${\displaystyle \operatorname {O} (p,q)}$ whose upper-left ${\displaystyle p\times p}$ block ${\displaystyle A}$ has positive determinant is a subgroup. Or, to put it another way, if $${\displaystyle L={\begin{pmatrix}A&B\\C&D\end{pmatrix}}\;\mathrm {and} \;M={\begin{pmatrix}W&X\\Y&Z\end{pmatrix}}}$$ are in ${\displaystyle \operatorname {O} (p,q)}$, then $${\displaystyle (\operatorname {sgn} \det A)(\operatorname {sgn} \det W)=\operatorname {sgn} \det(AW+BY).}$$

The analogous result for the bottom-right ${\displaystyle q\times q}$ block also holds. The subgroup ${\displaystyle \operatorname {SO} ^{+}(p,q)}$ consists of matrices ${\displaystyle L}$ such that ${\displaystyle \det A}$ and ${\displaystyle \det D}$ are both positive.⁵⁶

For all matrices ${\displaystyle L}$ in ${\displaystyle \operatorname {O} (p,q)}$, the determinants of ${\displaystyle A}$ and ${\displaystyle D}$ have the property that ${\textstyle {\frac {\det A}{\det D}}=\det L}$ and that ${\displaystyle |{\det A}|=|{\det D}|\geq 1}$.⁷ In particular, the subgroup ${\displaystyle \operatorname {SO} (p,q)}$ consists of matrices ${\displaystyle L}$ such that ${\displaystyle \det A}$ and ${\displaystyle \det D}$ have the same sign.⁵

Assuming both ${\displaystyle p}$ and ${\displaystyle q}$ are positive, neither of the groups ${\displaystyle \operatorname {O} (p,q)}$ nor ${\displaystyle \operatorname {SO} (p,q)}$ are connected, having ${\displaystyle 4}$ and ${\displaystyle 2}$ components respectively. ${\displaystyle \pi _{0}(\operatorname {O} (p,q))\cong C_{2}\times C_{2}}$ is the Klein four-group, with each factor being whether an element preserves or reverses the respective orientations on the ${\displaystyle p}$ and ${\displaystyle q}$ 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 ${\displaystyle \pi _{0}(\operatorname {SO} (p,q))=\{(1,1),(-1,-1)\}}$, each of which either preserves both orientations or reverses both orientations, in either case preserving the overall orientation.

The identity component of ${\displaystyle \operatorname {O} (p,q)}$ is often denoted ${\displaystyle \operatorname {SO} ^{+}(p,q)}$ and can be identified with the set of elements in ${\displaystyle \operatorname {SO} (p,q)}$ that preserve both orientations. This notation is related to the notation ${\displaystyle \operatorname {O} ^{+}(1,3)}$ for the orthochronous Lorentz group, where the ${\displaystyle +}$ refers to preserving the orientation on the first (temporal) dimension.

The group ${\displaystyle \operatorname {O} (p,q)}$ is also not compact, but contains the compact subgroups ${\displaystyle \operatorname {O} (p)}$ and ${\displaystyle \operatorname {O} (q)}$ acting on the subspaces on which the form is definite. In fact, ${\displaystyle \operatorname {O} (p)\times \operatorname {O} (q)}$ is a maximal compact subgroup of ${\displaystyle \operatorname {O} (p,q)}$, while ${\displaystyle \operatorname {S} (\operatorname {O} (p)\times \operatorname {O} (q))}$ is a maximal compact subgroup of ${\displaystyle \operatorname {SO} (p,q)}$. Likewise, ${\displaystyle \operatorname {SO} (p)\times \operatorname {SO} (q)}$ is a maximal compact subgroup of ${\displaystyle \operatorname {SO} ^{+}(p,q)}$. Thus, the spaces are homotopy equivalent to products of (special) orthogonal groups, from which algebro-topological invariants can be computed. (See Maximal compact subgroup.)

In particular, the fundamental group of ${\displaystyle \operatorname {SO} ^{+}(p,q)}$ is the product of the fundamental groups of the components, ${\displaystyle \pi _{1}(\operatorname {SO} ^{+}(p,q))=\pi _{1}(\operatorname {SO} (p))\times \pi _{1}(\operatorname {SO} (q))}$, and is given by:

${\displaystyle \pi _{1}(\operatorname {SO} ^{+}(p,q))}$	${\displaystyle p=1}$	${\displaystyle p=2}$	${\displaystyle p\geq 3}$
${\displaystyle q=1}$	${\displaystyle C_{1}}$	${\displaystyle \mathbb {Z} }$	${\displaystyle C_{2}}$
${\displaystyle q=2}$	${\displaystyle \mathbb {Z} }$	${\displaystyle \mathbb {Z} \times \mathbb {Z} }$	${\displaystyle \mathbb {Z} \times C_{2}}$
${\displaystyle q\geq 3}$	${\displaystyle C_{2}}$	${\displaystyle C_{2}\times \mathbb {Z} }$	${\displaystyle C_{2}\times C_{2}}$

In even dimensions, the middle group ${\displaystyle \operatorname {O} (n,n)}$ 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 ${\displaystyle {\mathfrak {so}}_{2n}}$ (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 ${\displaystyle \operatorname {O} (n):=\operatorname {O} (n,0)=\operatorname {O} (0,n)}$, which is the compact real form of the complex Lie algebra.

The group ${\displaystyle \operatorname {SO} (1,1)}$ 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.