Lie Algebras & Lie Groups

Best way to study abstract math is to use concrete examples, visualizable 2- or 3- dimensional mathematical ‘objects’.

Groups:
(Do not need to search too far…) Best example is the Integer Group (Z) with + operation, denoted as {Z, +}. (Note: Easy to verify it satisfies the group 4 properties: CAN I“.

It has infinite elements (infinite group)

It is a Discrete Group, because it jumps from one integer to another (1,2,3…, in digital sense).

The group of rotation of a round table, which consists of all points on a circle, is an infinite group. We can change the angle of rotation continously between 0 and 360 degrees, rotating around a geometrical shape – a circle. Such shapes are called Manifolds (流形).

All points of a manifold forms a Lie group.

Example: The group of rotations of a sphere around a central…