4.1 Basics of Lie Group and Lie Algebra

4.1.1 Group

A group is an algebraic structure of one set plus one operator. $$G = (A, \cdot)$$ Properties:

• Closure $$\ \forall a_1, a_2 \in A, \ a_1 \cdot a_2 \in A$$

• Combination $$\forall a_1, a_2, a_3 \in A, \ (a_1 \cdot a_2) \cdot a_3 = a_1 \cdot ( a_2 \cdot a_3)$$

• Unite Element $$\ \exists a_0 \in A, \ \mathrm{s.t.} \ \forall a \in A, \ a_0 \cdot a = a \cdot a_0 = a$$

• Inverse Element

$$\ \forall a \in A, \ \exists a^{-1} \in A, \ st \ a \cdot a^{-1} = a_0$$

4.1.2 Introduction of the Lie Algebra

Consider an arbitrary rotation matrix $\mathbf{R}$

It satisfies: $$\mathbf{R}(t) \mathbf{R}(t) ^T = \mathbf{I}$$ Deriving time on both sides yields: $$\dot{\mathbf{R}} (t) \mathbf{R} {(t)^T} + \mathbf{R} (t) \dot{\mathbf{R}} {(t) ^T} = 0.$$ Move the second term to the right and commute the matrices by using the transposed relation: $$\dot{\mathbf{R}} (t) \mathbf{R} {(t)^T} = - \left( \dot{\mathbf{R}} (t) \mathbf{R} {(t)^T} \right)^T$$ It can be seen that $\dot{\mathbf{R}} (t) \mathbf{R} {(t)^T}$ is a skew-symmetric matrix, which turns a vector into a skew-symmetric matrix: $${\mathbf{a}^ \wedge } = \mathbf{A} = \left[ {\begin{array}{*{20}{c}} 0&{ - {a_3}}&{ {a_2}}\\ { {a_3}}&0&{ - {a_1}}\\ { - {a_2}}&{ {a_1}}&0 \end{array}} \right], \quad { \mathbf{A}^ \vee } = \mathbf{a}$$ Since $\dot{\mathbf{R}} (t) \mathbf{R} {(t)^T}$ is a skew-symmetric matrix, we can find a 3-D vector $\pmb{\phi} (t) \in \mathbb{R}^3$: $$\dot{\mathbf{R}} (t) \mathbf{R}(t)^T = \pmb{\phi} (t) ^ {\wedge}$$ Right multiply with $\mathbf{R}(t)$ on both sides, since $\mathbf{R}(t)$ is an orthogonal matrix, we have: $$\dot{\mathbf{R}} (t) = \pmb{\phi} (t)^{\wedge} \mathbf{R}(t) = \left[ {\begin{array}{*{20}{c}} 0&{ - {\phi _3}}&{ {\phi _2}}\\ { {\phi _3}}&0&{ - {\phi _1}}\\ { - {\phi _2}}&{ {\phi _1}}&0 \end{array}} \right] \mathbf{R} (t)$$ Time derivative of a rotation matrix is just by multiply a $\phi^ {\wedge}(t)$ matrix on the left. Consider $\mathbf{R}(0) = \mathbf{I}$ , we use the first-order Taylor expansion around $t=t_0$ to write $\mathbf{R}(t)$ as: $$\begin{aligned} \mathbf{R} \left( t \right) & \approx \mathbf{R} \left( t_0 \right) + \dot {\mathbf{R}} \left( { {t_0}} \right)\left ( {t - {t_0}} \right)\\ &= \mathbf{I} + \pmb{\phi} {\left( { {t_0}} \right)^ \wedge } \left( t \right). \end{aligned}$$ We see that $\pmb{\phi}$ reflects the derivative of $\mathbf{R}$, so it is called the tangent space near the origin of $\mathrm{SO}(3)$

Solve for the above differential equation for $\mathbf{R}$, and with the initial value $\mathbf{R}(0) = \mathbf{I}$, we have: $$\mathbf{R}(t) = \exp \left( \pmb{\phi}_0^\wedge t \right).$$

4.1.3 The definition of Lie Algebra

Each Lie group has a Lie algebra corresponding to it.

Definition of Lie Algebra:

A Lie algebra consists of a set $\mathbb{V}$, a scalar field $\mathbb{F}$, and a binary operation $[,]$. If they satisfy the following properties, then $(\mathbb{V}, \mathbb{F}, [,])$ is a Lie algebra, denoted as $\mathfrak{g}$.

• Closure (封闭性):

  $\forall \mathbf{X}, \mathbf{Y} \in \mathbb{V}; [\mathbf{X}, \mathbf{Y}] \in \mathbb{V}$

• Bilinear Composition (双线性) :

  $\forall \mathbf{X},\mathbf{Y},\mathbf{Z} \in \mathbb{V}; a,b \in \mathbb{F }$

  $[a\mathbf{X}+b\mathbf{Y}, \mathbf{Z}] = a[\mathbf{X}, \mathbf{Z}] + b [ \mathbf{Y}, \mathbf{Z} ], \quad [\mathbf{Z}, a \mathbf{X}+b\mathbf{Y}] = a [\mathbf{Z}, \mathbf{X} ]+ b [\mathbf{Z},\mathbf{Y}]$

• Reflexive (自反省): (Reflexive means that an element operates with itself results in zero) $$\forall \mathbf{X} \in \mathbb{V}; [\mathbf{X},\mathbf{X}] = \mathbf{0}$$

• Jacobi Identity (雅克比等价性)

$$\forall \mathbf{X},\mathbf{Y},\mathbf{Z} \in \mathbb{V}; [\mathbf{X}, [ \mathbf{Y},\mathbf{Z}] ] + [\mathbf{Z}, [\mathbf{X},\mathbf{Y}] ] + [\mathbf{Y}, [\mathbf{Z}, \mathbf{X}]] =\mathbf{0}$$

The binary operations $[,]$$ are called Lie brackets

4.1.4 Lie Algebra $\mathfrak{so}(3)$

The Lie algebra corresponding to $\mathrm{SO}(3)$ is a vector defined on $\mathbb{R}^3$, which we will denote as $\pmb{\phi}$. $$\pmb{\varPhi} = \pmb{\phi}^{\wedge} = \left[ {\begin{array}{*{20}{c}} 0&{ - {\phi _3}}&{ {\phi _2}}\\ { {\phi _3}}&0&{ - {\phi _1}}\\ { - {\phi _2}}&{ {\phi _1}}&0 \end{array}} \right] \in \mathbb{R}^{3 \times 3}.$$ Lie bracket: $$[\pmb{\phi}_1, \pmb{\phi}_2] = \left( \mathbf{ \varPhi }_1 \mathbf{ \varPhi }_2 - \mathbf{ \varPhi }_2 \mathbf{ \varPhi }_1 \right)^\vee.$$ Since the vector $\pmb{\phi}$ is one-to-one with the skew-symmetric matrix, we say the elements of $\mathfrak{so}(3)$ are three-dimensional vectors or three-dimensional skew-symmetric matrices, without any ambiguity: $$\mathfrak{so}(3) = \left\{ \pmb{\phi} \in \mathbb{R}^3 \ \text{or}\ \pmb{\varPhi} = \pmb{\phi^\wedge} \in \mathbb{ R}^{3 \times 3} \right\}.$$ $\mathfrak{so}(3)$ are just a set of 3D vectors that can express the derivative of the rotation matrix. Its relationship to $\mathrm{SO}(3)$ is given by the exponential map: $$\mathbf{R} = \exp ( \pmb{\phi}^\wedge ).$$

4.1.5 Lie Algebra $\mathfrak{se}(3)$

Similar to $\mathfrak{so}(3)$, $\mathfrak{se}(3)$ is located in the $\mathbb{R}^6$ space: $$\mathfrak{se}(3) = \left\{ { \pmb{\xi} = \left[ \begin{array}{l} \pmb{\rho} \\ \pmb{\phi} \end{array} \right] \in { \mathbb{R}^6} , \pmb{\rho} \in { \mathbb{R}^3}, \pmb{\phi} \in \mathfrak{so} \left( 3 \right),{ \pmb{\xi} ^ \wedge } = \left[ {\begin{array}{*{20}{c}} { { \pmb{\phi} ^ \wedge }}& \pmb{\rho} \\ { {\mathbf{0}^T}}&0 \end{array}} \right] \in { \mathbb{R}^{4 \times 4}}} \right\}$$ The first three dimensions are 'translation part' which is denoted as $\pmb{\rho}$; the second part is a rotation part $\pmb{\phi}$, which is essentially a $\mathfrak{so}(3)$ element

$\mathfrak{se}(3)$ as a 'vector consisting of a translation plus a $\mathfrak{so}(3)$ element'

Lie bracket: $$[ \pmb{\xi}_1, \pmb{\xi}_2 ] = \left( \pmb{\xi}_1^\wedge \pmb{\xi}_2^\wedge -\pmb{\xi}_2^ \wedge \pmb{\xi}_1^\wedge \right) ^\vee.$$