Lie Algebra

Related to Rodrigues’ Rotation Formula. Also we covered this in Mechanization

A major motivation for using Lie algebra is to do optimization.

So basically we want to represent certain operations like matrices or exponentials in different ways (like Rotation Matrix Time Derivative)

Group

A group is an algebraic structure of one set plus one operator. We denote the set as $A$ and the operation as $\cdot$, then the group can be denoted as $G=(A,\cdot )$. We say $G$ is a group if the operation satisfies the following conditions (“well-defined operations”):

1. Closure:
   $\forall a_1,a_2\in A,a_1\cdot a_2\in A$

2. Combination (Associativity):
   $\forall a_1,a_2,a_3\in A,(a_1\cdot a_2)\cdot a_3=a_1\cdot (a_2\cdot a_3)$

3. Unit element:
   $\exists a_0\in A,\text{s.t. }\forall a\in A,a_0\cdot a=a\cdot a_0=a$

4. Inverse element:
   $\forall a\in A,\exists a^{-1}\in A,\text{s.t. }a\cdot a^{-1}=a_0$