Phy622 Mathematical Methods 3

01-06

Special Rings

[Ring#Special Rings]

Division Rings

• Multiplicative identity exists
• But all x may not have inverses
  • if x^- does not exist, x is singular
  • Multiplicative inverse of 0 always does not exist (${00}^{-}=0=1$)

Field

A field is a ring with 0 as the only singular point.

Example -

1. Reals
2. Rationals

Zero divisors

For $r,s$ in a [Ring], if $rs=0$, then $r,s$ are called Zero divisors.

$Z_n$

$Z_n,{+}_n,{\ast }_n$

• 0, 1 exist.
• Group over addition
• Monoid over multiplication
• But elements apart from 0 may have a lack of inverses
  • Except if n is prime
  • $Z_5$ is a field but $Z_6$ is not a field

Exercises

• Read pg 45-47 of P. Pal.
• Ex 3.4, 3.9

Vector Spaces

$\mathbb{V}$ is defined as a set $V$ over Field $F$ and a $\oplus$ vector addition

$\mathbb{V}=V,F,+,\ast ,\oplus$

Elements of $F$ are called scalars, of $V$ are called vectors

$|x⟩,|y⟩$

$|x⟩\oplus |y⟩\in V$

$a|x⟩\in V$

Hence,

$\alpha |x⟩+\beta |y⟩\in V$

More formally,

1. Vector addition is commutative
2. $\oplus$ is associative
3. There is an unique identity for $\oplus$, called $|0⟩$
4. Scalar multiplication defined, $\alpha |x⟩\in V$
5. Multiplicative identity of F is also identity of Scalar multiplication
6. Additive inverse exists, denoted by $-|x⟩$. Further, $-|x⟩=-1|x⟩$
7. Scalar Vector multiplication is distributive over vector addition. 8.Scalar Vector multiplication is distributive over scalar addition. $(\alpha +\beta )|x⟩=\alpha |x⟩+\alpha |x⟩$

Algebra

An algebra $\mathcal{A}$ is a [Ring] whose elements form a vector space over field F. The multiplication of this Ring is called a vector multiplication $\ast$.

$\ast :V\times V\to V$

Example

1. $N\times N$ matrices.
2. Lie Algebra: Lie Ring [Ring#Special Rings] whose elements form a vector space.
   1. $|x⟩\ast |y⟩=-|y⟩\ast |x⟩$
   2. Jacobi Identity
   3. Eg. Set of antisymmetric matrices, with $\ast =[.,.]$ - Algebra of orthogonal group $O(N)$