Knowlpedia

Definition

O(3,R)

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

O⁡(3,R)\operatorname{O}(3,\mathbb R) is the

O⁡(3,R)={A∈M3(R):ATA=I},\operatorname{O}(3,\mathbb R)=\{A\in M_{3}(\mathbb R):A^{\mathsf T}A=I\},

with matrix multiplication. The preserved form is the positive-definite real Euclidean form.

Dimensions and structure

This group has real dimension 33. Its tangent algebra consists of skew-symmetric matrices, XT+X=0X^{\mathsf T}+X=0. It has two distinguished by determinant.

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