# Lie algebra

## Cross products

### The cross product

$$v\times u$$

#### Cross product is a bilinear map

This is a bilinear map from two vectors in $$\mathbb{R}^3$$ to another vector in the same space.

$$V\times V \rightarrow V$$

#### Calculating the cross product

This is calcualted by:

$$u\times v=||u|| ||v|| \sin(\theta )n$$

The resulting vector is perpendicular to both input vectors.

## Lie algebra

### Lie algebra

Lie groups have symmetries. We can consider only the infintesimal symmetries.

For example the unit circle has many symmetries, but we can consider only those which rotate infintesimally.

#### Example

Take a continous group, such as $$U(1)$$. Its Lie algebra is all matrices such that their exponential is in the Lie group.

$$\mathfrak{u}(1)=\{X\in \mathbb {C}^{1\times 1}|e^{tX}\in U(1) \forall t\in \mathbb{R}\}$$

This is satisfied by the matrices where $$M=-M^*$$. Note that this means the diagonals are all $$0$$.

#### Scale of specific Lie algebra matrices doesnâ€™t matter

Because of $$t$$.

#### Commutation of Lie group algebra

Consider two members of the Lie algebra: $$A$$ and $$B$$. The commutator is:

$$A$$.

The corresponding Lie group member is:

$$e^{t(A+B)}=e^{tA}e^{tB}$$

While the Lie group multiplication may not commute, the corresponding addition of the Lie algebra does.

### The Lie bracket

We can define the Lie bracket from the ring commutator.

We use the Lie bracket, rather than multiplication, as the operator over a field homomorphism.

$$[A, B]$$

This generates another element in the algebra.

This satisifies:

• Bilinearity: $$[xA+yB,C]=x[A,C]+y[B,C]$$

• Alternativity: $$[A,A]=0$$

• Jacobi identity: $$[A, [B,C]]+[C,[A,B]]+[B,[C,A]]=0$$

• Anticommutivity: $$[A, B]=-[B,A]$$

One option for the Lie bracket is the ring commutor. So that:

$$[A,B]=AB-BA$$

### Commutation of Lie groups

We can measure commutation of Lie groups using:

$$ABA^{-1}B^{-1}$$

If the group commutes then:

$$ABA^{-1}B^{-1}=BA^{-1}B^{-1}=I$$

#### Commutation of Lie algebra: COMPLETE THIS

This corresponds to $$[A,B]=AB-BA$$ in the underlying lie algebra, if we expand.

$$A=e^{ta}$$

$$B=e^{tb}$$

$$ABA^{-1}B^{-1}=e$$

## Lie algebra of specific Lie groups

### Lie algebra of $$O(n)$$

#### Lie algebra of $$O(n)$$

The Lie algebra of $$O(n)$$ is defined as:

$$\mathfrak{o}(n)=\{X\in \mathbb {R}^{n\times n}|e^{tX}\in O(n) \forall t\in \mathbb{R}\}$$

This is satisfied by the skew-symmetric matrices where $$M=-M^T$$. Note that this means the diagonals are all $$0$$.

### Lie algebra of $$U(n)$$

#### Lie algebra of $$U(n)$$

The Lie algebra of $$U(n)$$ is defined as:

$$\mathfrak{u}(n)=\{X\in \mathbb {C}^{n\times n}|e^{tX}\in U(n) \forall t\in \mathbb{R}\}$$

This is satisfied by the skew-Hermitian matrices where $$M=-M^*$$. Note that this means the diagonals are all $$0$$ or pure imaginary.

### Lie algebra of $$SO(n)$$

#### Lie algebra of $$SO(n)$$

The Lie algebra of $$SO(n)$$ is defined as:

$$\mathfrak{so}(n)=\{X\in \mathbb {R}^{n\times n}|e^{tX}\in SO(n) \forall t\in \mathbb{R}\}$$

This is satisfied by the skew-symmetric matrices where $$M=-M^T$$. Note that this means the diagonals are all $$0$$.

### Lie algebra of $$SU(n)$$

#### Lie algebra of $$SU(n)$$

The Lie algebra of $$SU(n)$$ is defined as:

$$\mathfrak{su}(n)=\{X\in \mathbb {C}^{n\times n}|e^{tX}\in SU(n) \forall t\in \mathbb{R}\}$$

This is satisfied by the skew-Hermitian matrices where $$M=-M^*$$ and the trace is $$0$$. Note that this means the diagonals are all $$0$$ or pure imaginary.