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

$$\cdot :\mathcal{G}\times \mathcal{V}\to \mathcal{V},(\mathcal{X},\mathcal{v})\mapsto \mathcal{X}\cdot \mathcal{v}.$$

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 ${\mathcal{v}}^{\wedge }$.

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

$${\mathcal{v}}^{\wedge }={\mathcal{X}}^{-1}\dot{\mathcal{X}},$$

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.

$${\mathcal{N}}_{\mathcal{M}}(xx\mid \mu \mu ,\mathrm{\Sigma })=(2\pi |\mathrm{\Sigma }|{)}^{-1/2}\exp (\frac{1}{2}{\log }_{\mu \mu }(xx){\mathrm{\Sigma }}^{-1}{\log }_{\mu \mu }(xx))$$

where $xx\in \mathcal{M}$, and $\mu \mu \in \mathcal{M}$ is the mean of the distribution, $\mathrm{\Sigma }\in {\mathcal{T}}_{\mu \mu }\mathcal{M}$ is the covariance matrix defined on the tangent space.

A mixture model is defined by

$$p(xx\mid \theta )=\sum _{k=1}^K{\pi }_k{\mathcal{N}}_{\mathcal{M}}(xx\mid \mu {\mu }_k,{\mathrm{\Sigma }}_k).$$