His paper is succinct and full of gem, a great add on on his video.

First quote Howe about beauty of Lie Theory “The essential phenomenon of Lie theory is that one may associate in a natural way to a Lie group G its Lie algebra g. The Lie algebra g is first of all a vector space and secondly is endowed with a bilinear non-associative product called the Lie bracket […]. Amazingly, the group G is almost completely determined by g and its Lie bracket. Thus for many purposes one can replace G with g. Since G is a complicated nonlinear object and g is just a vector space, it is usually vastly simpler to work with g. […] This is one source of the power of Lie theory.” Although he explicitly stated he won’t introduce Lie bracket at all. Therefore, the connection between the Lie group and its Lie algebra will not be made here as profound as it should. It’s still a great paper.

It makes computation of Jacobian much easier using by taking differentiation on the group and vector directly, to elaborate:

Lie algebra originated from abstract algebra and is used in the study of algebraic structures related to differentiable manifolds. On the other hand, Jacobians are a concept from calculus used in the study of vector fields and transformations.

How these two concepts connect is mostly seen in the field of Robotics. One of the primary applications of Lie algebra in robotics involves the representation of the space of 3D rotations and translations i.e the Special Euclidean group SE(3), which play a vital role in describing the kinematics of robots.

Robot manipulators move in the 3-dimensional space, and the motion of these manipulators can be nicely described using concepts from Lie groups and Lie algebra. Particularly, Lie groups enable the expression of both translation and rotation in a uniform way, which is very useful for robot kinematics.

The Jacobian in robotics gives us an understanding of the robot manipulator’s velocity. That is, given a differential change in the joint space (the inputs), the Jacobian allows us to determine the differential change in the task space (the outputs).

Now, let’s bring them together. In the context of robotics and manipulator kinematics, there’s a thing called the “Twist”, which is an element of the Lie algebra se(3) associated with SE(3). The twist contains both linear and angular velocity components of the manipulator, and is often represented as a spatial velocity.

The Jacobian that we compute in the kinematic analysis of robots can be regarded as a map from the joint velocities (represented in the joint space) to this spatial velocity (represented as a twist in the Lie algebra se(3)). Thus, in this context, the Jacobian serves as a bridge between the joint space and the Lie algebra associated with the task space.

This connection between Lie algebra and the Jacobian is a remarkable example of how abstract mathematical concepts find concrete and practical applications in fields like robotics.