Lie Algebra¶
Reference
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Lie Groups, Lie Algebras, and Representations, Brian Hall.
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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
and satisfy two axioms: identity \(\mathcal{E}\cdot \mathcal{v}=\mathcal{v}\) and compatibility
Lie Algebra
Lie algebra \(\mathcal{m}\) of a Lie group \(\mathcal{M}\) is defined to be the tangent space at the identity.
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
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.
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