Lie Algebra

Reference

• Lie Groups, Lie Algebras, and Representations, Brian Hall.

• A micro Lie theory for state estimation in robotics, Joan Solà, Jeremie Deray, Dinesh Atchuthan

Lie groups

A Lie group \(\mathcal{G}\) is a smooth manifold whose elements satisfies group axioms.

Check the definition of groups in ODE general theory.

We could conclude as 4 axioms: closure under \(\circ\), identity, inverse and associativity.

The smoothness of the manifold implies the existence of a unique tangent space at each point

group actions

Given a Lie group \(\mathcal{G}\) and a set \(\mathcal{V}\), we denote action \(\mathcal{X}\cdot \mathcal{v}\) to be

\[ \cdot: \mathcal{G}\times \mathcal{V}\rightarrow \mathcal{V}, (\mathcal{X}, \mathcal{v})\mapsto \mathcal{X}\cdot \mathcal{v}. \]

and satisfy two axioms: identity \(\mathcal{E}\cdot \mathcal{v}=\mathcal{v}\) and compatibility

\[ (\mathcal{X}\circ \mathcal{Y})\cdot \mathcal{v}=\mathcal{X}\cdot(\mathcal{Y}\cdot \mathcal{v}) \]

Lie Algebra

Lie algebra \(\mathcal{m}\) of a Lie group \(\mathcal{M}\) is defined to be the tangent space at the identity.

\[ \mathcal{m}=T_{\mathcal{E}}\mathcal{M} \]

this is a linear space, and its elements can be identified by \(\mathbb{R}^m\) where \(m\) is the dimension of \(\mathcal{m}\). Denote the element of a Lie algebra to be \(\mathcal{v}^{\land}\).

The structure of the Lie algebra can be found by time-differentiating the group constraint. For multiplicative group, we have

\[ \mathcal{v}^\land= \mathcal{X}^{-1}\dot{\mathcal{X}}, \]

where \(\mathcal{X}\) is chosen at \(t=0\) or identity.

Exponential map

\(\mathfrak{n}\)

Gaussian Mixture Model

Use this model to estimate a relatively complex distribution of data, which could be multi-dimensional.

\[ \mathcal{N}_\mathcal{M} (\pmb{x}\mid \pmb{\mu} , \Sigma)=(2\pi |\Sigma|)^{-1/2} \exp\left(\frac{1}{2}\log_{\pmb{\mu}} (\pmb{x})\Sigma^{-1} \log_{\pmb{\mu}} (\pmb{x}) \right) \]

where \(\pmb{x}\in \mathcal{M}\), and \(\pmb{\mu} \in \mathcal{M}\) is the mean of the distribution, \(\Sigma\in \mathcal{T}_{\pmb{\mu}}\mathcal{M}\) is the covariance matrix defined on the tangent space.

A mixture model is defined by

\[ p(\pmb{x}\mid \theta)=\sum_{k=1}^K \pi_k \mathcal{N}_\mathcal{M}(\pmb{x}\mid \pmb{\mu}_k, \Sigma_k). \]