Knowlpedia

Definition

SO(1,3)

SO(1,3) as a separately identified real Lie group, with its defining operations and dimension.

SO⁡(1,3)\operatorname{SO}(1,3) is the real

SO⁡(1,3)={A∈GL(4,R):ATI1,3A=I1,3, det⁡A=1},I1,3=diag⁡(−I1,I3).\operatorname{SO}(1,3)=\{A\in\mathrm{GL}(4,\mathbb R):A^{\mathsf T}I_{1,3}A=I_{1,3},\ \det A=1\},\qquad I_{1,3}=\operatorname{diag}(-I_{1},I_{3}).

The first parameter counts negative directions. The form is real symmetric bilinear.

Dimensions and structure

This group has real dimension 66. The definition specifies the full matrix group. When both signature entries are positive it has two components; its is written 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.