Identity component SO0(p,q)

For integers $p,q\ge0$ with $p+q\ge1$ , the connected special orthogonal group

\operatorname{SO}_0(p,q):=\operatorname{SO}(p,q)^\circ

is the identity component of the real matrix group $\operatorname{SO}(p,q)$ . Explicitly, it consists of matrices joined to the identity by a continuous path inside

\{A\in\operatorname{GL}(p+q,\mathbb R): A^{\mathsf T}I_{p,q}A=I_{p,q},\ \det A=1\}, \qquad I_{p,q}=\operatorname{diag}(-I_p,I_q).

The parameter $p$ counts negative directions. Multiplication and inversion are the inherited matrix operations; $\operatorname{SO}^{+}(p,q)$ is another notation for this same identity component.

Dimension and compactness

The identity component is an open normal Lie subgroup, so it has the same Lie algebra and dimension as the full group:

\mathfrak{so}(p,q)=\{X:X^{\mathsf T}I_{p,q}+I_{p,q}X=0\}, \qquad \dim_{\mathbb R}\operatorname{SO}_0(p,q)=\frac{(p+q)(p+q-1)}2.

Indeed, $X\mapsto I_{p,q}X$ identifies these tangent matrices with skew-symmetric matrices, whose entries above the diagonal are independent.

If $p=0$ or $q=0$ , the group is compact $\operatorname{SO}(p+q)$ . If $p,q>0$ , the group is noncompact: on one negative and one positive coordinate, the matrices

\begin{pmatrix}\cosh t&\sinh t\\\sinh t&\cosh t\end{pmatrix},\qquad t\in\mathbb R,

extended by the identity on the remaining coordinates, form an unbounded continuous path through the identity.

Split orthogonal examples

The Lie algebra of $\operatorname{SO}_0(r,r+1)$ is split of type $B_r$ , and that of $\operatorname{SO}_0(r,r)$ is split of type $D_r$ . The visual table uses $r\ge2$ for $B_r$ and $r\ge4$ for $D_r$ , with real dimensions $r(2r+1)$ and $r(2r-1)$ , respectively. These entries specify connected matrix groups. Their Lie-algebra types alone do not specify a global covering or a central quotient.

The fixed specialization $p=1,q=3$ is the proper orthochronous Lorentz group.

References

Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §2.4, Proposition 2.6, pp. 21–22 (identity components); §6.1, p. 38 (orthogonal matrix groups); §41.2, pp. 219–220 (split orthogonal Lie algebras). The signature convention here counts negative directions first; exchanging the two entries gives an isomorphic group.