Knowlpedia

Definition

SO(p,q)

SO(p,q) as a separately identified real Lie group, with its defining operations and dimension.

For nonnegative integers p,qp,q with p+q≥1p+q\ge1, SO⁡(p,q)\operatorname{SO}(p,q) is the real

SO⁡(p,q)={A∈GL(p+q,R):ATIp,qA=Ip,q, det⁡A=1},Ip,q=diag⁡(−Ip,Iq).\operatorname{SO}(p,q)=\{A\in\mathrm{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 first parameter counts negative directions. The form is real symmetric bilinear.

Dimensions and structure

This group has real dimension (p+q)(p+q−1)/2(p+q)(p+q-1)/2. The definition specifies the full matrix group. When both signature entries are positive it has two components; its is written SO⁡0(p,q)\operatorname{SO}_0(p,q) or SO⁡+(p,q)\operatorname{SO}^{+}(p,q). Determinant one alone does not impose .

References
  1. Pavel Etingof, Lie Groups and Lie Algebras (MIT lecture notes), §6.1, pp. 38–39; Proposition 6.3 and Exercise 6.5.