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:
| (6) |
Suppose that
,
is a set of linearly independent solutions
of this equation. Let us consider:
| (7)
|
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 .
