Lie Algebras & Lie Groups

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

(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’s 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 axis, (eg. The Earth), is a Lie group SO(3) – a special orthogonal (meaning preserve all distances) group.

SO(2): Rotations of a circle in 2-dim space.

SO(n): Rotations of a manifold in n-dim space.

Note: SO(2), SO(3), SO(n) are both infinite groups and manifolds, so they are Lie groups.


Example of infinite dimensional Lie group is a Loop group.


Approximation at a given point
1. For SO(2) – manifold circle – it can be approximated at a given point by the tangent.
2. For SO(3) – manifold Sphere – at a given point by a tangent plane.
(Note: The cross-product operation makes the 3-dimensional space into a Lie algebra.)
3. For Lie group in general – SO(n) – its special point is the identity element of this group. The approximation tangent space at this special point is the Lie algebra.

Why study Lie Algebra instead of Lie Group ?
Unlike a Lie group which is usually curved (like a circle), Lie algebra is a flat space (like a line, plane, etc). This makes Lie Algebras easier to study than Lie groups.

1. The Lie algebra of an n-dimensional Lie group is an n-dimensional flat space, also known as a Vector Space.
2. Kac-Moody Algebras are the Lie algebras – which we should think of as the simplified versions – of the loop groups.
3. Quantum groups are certain deformations of Lie groups.

Reference :
Love and Math by Edward Frenkel


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