## Geometric Algebra and Lie-Algebra (Commutator Symbol)

### Commutator Symbol

The commutator symbol is defined as [a,b] = ab-ba.

If we apply the commutator symbol on vectors **a**, **b** with the geometric product as operation,
we get the following expression for the outer product:

2**a**∧**b** = [**a**, **b**] = **a****b**-**b****a**;

2**a**∧**b** = [**a**, **b**] = -[**b**, **a**] = -(**b****a**-**a****b**) = -2**b**∧**a**;

2**a**∧**a** = [**a**,**a**] = **aa** - **aa** = 0;

λ[**a**, **b**] = [λ**a**, **b**] = λ(**a****b** - **b****a**) = [**a**, λ**b**];

[**a**+**b**,**c**] = [**a**,**c**] + [**b**,**c**] = (**a**+**b**)**c** - **c**(**a**+**b**) =
**ac** + **bc** - **ca** - **cb** = (**ac** - **ca**) + (**bc** - **cb**);

[**a**, [**b**, **c**]] = **a**(**bc** - **cb**) - (**bc** - **cb**)**a** = **abc** - **acb** - **bca** + **cba**;

[[**a**, **b**], **c**] = (**ab** - **ba**)**c** - **c**(**ab** - **ba**) = **abc** - **bac** - **cab** + **cba**;

### Jacobi Identity:

[**a**, [**b**, **c**]] + [**b**, [**c**, **a**]] + [**c**, [**a**, **b**]] = 0;

[**a**, **bc** - **cb**] + [**b**, **ca** - **ac**] + [**c**, **ab** - **ba**] = 0;

(**abc** - **acb**) - (**bca** - **cba**) + (**bca** - **bac**) - (**cab** - **acb**) + (**cab** - **cba**) - (**abc** - **bac**) = 0;

**abc** - **abc** + **acb** - **acb** + **bca** - **bca** + **cba** - **cba** + **bac** - **bac** + **cab** - **cab** = 0;

### Anticommutator Symbol

2**a**⋅**b** = {**a**, **b**} = **ab** + **ba**;

### The Geometric Product with Commutator and Anticommutator Symbol

2**ab** = 2**a**⋅**b** + 2**a**∧**b** = {**a**, **b**} + [**a**, **b**] = **ab** + **ba** + **ab** - **ba**;

### Motivation for a Lie-Algebra

With a matrix **A** we can define a element *a* = (**I**+ε**A**) and
a second element *b* = (**I**+ε**B**) from a matrix **B**, both close to the identity element **I** for small ε.

The product will be:

*a**b* = (**I**+ε**A**)(**I**+ε**B**) = **I** + ε(**A**+**B**) +ε^{2}**AB**;

If we ingnore the **AB** term because ε^{2} is very small we get:

*a**b* = **I** + ε(**A**+**B**);

With *a*^{-1} = (1+ε**A**)^{-1} = **I** - ε**A** + ε^{2}**A**^{2} - ε^{3}**A**^{3}... ;

For non commutating elements *a*,*b* the expression *aba*^{-1}*b*^{-1} is not equal **I**. We can express this term as follows:

*aba*^{-1}*b*^{-1} = (**I**+ε**A**)(1+ε**B**)(**I** - ε**A** + ε^{2}**A**^{2})(**I** - ε**B** + ε^{2}**B**^{2});

Suppressing terms with ε^{n} and n>2 will give:

*aba*^{-1}*b*^{-1} = **I** + ε^{2}(**AB** - **BA**).

Definiton of a Lie Group can be found on the Abstract Algebra page:

Lie Group