Lie Groups–Notes

Algebraic equations vs. Differential equations

Galois Theory – dealt with symmetries in algebraic equations

Lie – Tried to do the same (look for underlying symmetries).

Lie Groups

Are a Group

Are a Manifold

Groups vs. Rings vs. Fields

1. A group is an abstraction of addition and subtraction—except that the group operation might not be commutative. But the important part is that there is an operation, which is something like addition, and the operation can be reversed, so there is also something like subtraction.
2. To this, a ring adds multiplication, but not necessarily division.
3. To this, a field adds division.

Fields – Real, Rationals and Complex Numbers. NOT Integers – since multiplying inverse integer by itself – does not give an integer.

Rings –  The main difference between groups and rings is that rings have two binary operations (usually called addition and multiplication) instead of just one binary operation.

If you forget about multiplication, then a ring becomes a group with respect to addition (the identity is 0 and inverses are negatives). This group is always commutative!

If you forget about addition, then a ring does not become a group with respect to multiplication. The binary operation of multiplication is associative and it does have an identity 1, but some elements like 0 do not have inverses. (This structure is called a monoid.)

A commutative ring is a field when all nonzero elements have multiplicative inverses. In this case, if you forget about addition and remove 0, the remaining elements do form a group under multiplication. This group is again commutative.

Groups – Most primitive

Mu – > Binary Operator – G x G –> G (An ordered pair of elements maps to another element in the group)

{1, e^i2Pi/5,e^i4Pi/5,e^i6Pi/5,e^i8Pi/5} – Five pentic roots of unity (solutions of x^5 – 1 = 0)

Apply binary operation of Multiplication – multiply any two elements of this group – to get another element in this group.

https://www.youtube.com/watch?v=5br-GWd_DpA