Knowlpedia

Definition

O(p,q)

O(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, O⁡(p,q)\operatorname{O}(p,q) is the real

O⁡(p,q)={A∈GL(p+q,R):ATIp,qA=Ip,q},Ip,q=diag⁡(−Ip,Iq).\operatorname{O}(p,q)=\{A\in\mathrm{GL}(p+q,\mathbb R):A^{\mathsf T}I_{p,q}A=I_{p,q}\},\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 four components.

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