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 group or generalized orthogonal group. 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.