Lie group, Lie algebra
Lie group
Parent:
Rotations / SO(3) group index
Source: http://www.seas.upenn.edu/~meam620/slides/kinematicsI.pdf
A group that is a differentiable (smooth) manifold is called a Lie group.
Lie algebra
Source: http://en.wikipedia.org/wiki/3D_rotation_group
Lie algebra $\mathfrak{so}(3)$
- Every Lie group has an associated Lie algebra
- Lie algebra: linear space with same dimension as the Lie group
- Consists of all skew-symmetric 3x3 matrices
- Elements of the Lie algebra $\mathfrak{so}(3)$ are elements of the tangent space of the manifold SO(3)/Lie group at the identity element .
Source: Forster 2017 IMU Preintegration
An Euler vector $\mathbf{\omega} = \left(x,y,z\right) \in \mathbb{R}^3$ can be represented by a skew symmetric matrix in the Lie algebra
$$
\hat{\mathbf{\omega}} = \left[\begin{array}{ccc}
0 & -z & y\
z & 0 & -x\
-y & x & 0
\end{array}\right] \in \mathfrak{so}(3)
$$
Mappings: