Knowlpedia

Definition

so(3,R) — real orthogonal Lie algebra

so(3,R) — real orthogonal Lie algebra.

The real orthogonal so(3,R)\mathfrak{so}(3,\mathbb R) consists of skew-symmetric 3×33\times3 matrices, with commutator bracket:

so(3,R)={X∈M3(R):XT+X=0},[X,Y]=XY−YX.\mathfrak{so}(3,\mathbb R)=\{X\in M_{3}(\mathbb R):X^T+X=0\},\qquad [X,Y]=XY-YX.

The form preserved here is the positive-definite real form x12+⋯+x32x_1^2+\cdots+x_3^2.

Dimension and structure

The independent entries above the diagonal give real dimension 33; a basis is Eij−Eji (1≤i<j≤3)E_{ij}-E_{ji}\ (1\leq i<j\leq 3).

explains its general construction.

Scalar convention

This record retains R\mathbb R as its defining field. Its complexification is a different catalogue object; a matrix presentation over complex numbers does not change this field automatically.