2.3 Adjoint

In Lie groups, it is often necessary to transform a tangent vector from the tangent space around one element to the tangent space of another. The adjoint performs this transformation. One very nice property of Lie groups in general is that this transformation is linear. For an element $X$ of a Lie group, the adjoint is written $\mathrm{Adj}_{X}$:

$$\boldsymbol{\omega} \in \mathrm{so(3)},\;\mathbf{R\in}\mathrm{SO(3)}$$ (19)

$$\mathbf{R}\cdot\exp\left(\boldsymbol{\omega}\right) = \mathrm{\exp\left(Adj_{\mathbf{R}}\cdot\boldsymbol{\omega}\right)}\cdot\mathbf{R}$$ (20)

The adjoint can be computed from the generators of the Lie algebra:

$$\mathrm{\exp\left(Adj_{\mathbf{R}}\cdot\boldsymbol{\omega}\right)} = \mathbf{R}\cdot\exp\left(\boldsymbol{\omega}\right)\cdot\mathbf{R}^{-1}$$ (21)

$$\mathrm{Adj}_{\mathbf{R}}\cdot\boldsymbol{\omega} = \mathbf{R}\cdot\left(\sum_{i=1}^{3}\boldsymbol{\omega}_{i}G_{i}\right)\cdot\mathbf{R}^{-1}$$ (22)

$$= \mathbf{R}\cdot\boldsymbol{\omega}_{\times}\cdot\mathbf{R}^{-1}$$ (23)

$$= \left(\mathbf{R}\boldsymbol{\omega}\right)_{\times}$$ (24)

$$\implies\mathrm{Adj}_{\mathbf{R}} = \mathbf{R}$$ (25)

In the case of $\mathrm{SO(3)}$, the adjoint transformation for an element is particularly simple: it is the same rotation matrix used to represent the element. Rotating a tangent vector by an element ``moves'' it from the tangent space on the right side of the element to the tangent space on the left.