SO(2) and Its Lie Algebra

Special Orthogonal Group SO(2)

The special orthogonal group $S O (2)$ consists of all $2 \times 2$ orthogonal matrices with determinant 1. It represents rotations in the plane and is given by:

S O (2) = {R (θ) = [\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}] | θ \in R}

This group is compact, connected, and abelian, with the Lie group structure of a one-dimensional torus ( $S^{1}$ ).

Lie Algebra of SO(2)

The Lie algebra of $S O (2)$ , denoted $so (2)$ , consists of all skew-symmetric $2 \times 2$ matrices:

so (2) = {X = [\begin{matrix} 0 & - ω \\ ω & 0 \end{matrix}] | ω \in R}

A common basis for $so (2)$ is:

J = [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}]

Since $so (2)$ is one-dimensional, any element can be written as $ω J$ for $ω \in R$ . The Lie bracket is trivial:

[J, J] = 0

indicating that $so (2)$ is an abelian Lie algebra.

Exponential Map

The matrix exponential relates the Lie algebra to the Lie group:

e^{θ J} = \sum_{n = 0}^{\infty} \frac{(θ J)^{n}}{n!} = [\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}] = R (θ)

which shows that exponentiation recovers the rotation matrices in $S O (2)$ .