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’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.

Notes:

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** http://www.amazon.co.uk/dp/0465050743/ref=cm_sw_r_udp_awd_53swtb16779PY