Dual algebra

We have just defined a quadratic algebra .

Now we will define the dual algebra, which is denoted .

Consider the linear space which is dual to :

As we introduced the basis in , we can introduce the dual basis in . “Dual basis” means:
Remember that was defined using the set of constant matrices , which were used to define the quadratic constraints (3). Consider the linear space of all the matrices orthogonal to every . “Orthogonal” means that satisfies the equation:


Suppose that , is a set of linearly independent solutions of this equation. Let us consider:


where is generated by the tensors of the form:

Conclusion: We started from a quadratic algebra and constructed another quadratic algebra , which is called quadratic dual to .