Decomposition: A tensor product deformed by a cube A defined on a vector space built over a commutative field K of dimension D and acting on a pair (a, b) of E(D, K) x E(D, K) is decomposed when there exists a pair ([P], z) of M(D, K) x E(D, K) such that it is possible to write in the dual space E*(D, K):
| ⊗A(a, b) > = [P]. | b > + | z > ∈ E*(D, K)
In what follows, K will often be the field of real numbers R or the field of complex numbers C.
Trivial decomposition: A decomposition is called "trivial" whenever its vector part is zero.
Non-trivial decomposition: A decomposition is called "non-trivial" whenever its vector part is non-zero.
Inspired by the terminology commonly used when discussing division of numbers, the following semantics apply to the decompositions of deformed tensor products.
Projectile: This refers to the first argument of the pair (a, b) upon which the deformed tensor product acts; in this case, the vector a.
Target: This refers to the second argument of the pair (a, b) upon which the deformed tensor product acts; in this case, the vector b.
Main part of a decomposition: This refers to the matrix part of the decomposition; in this case, the matrix [P].
Residual part of a decomposition: This refers to the vector part of the decomposition; in this case, the vector z.
It is not difficult to show that there is always at least one, the simplest:
| ⊗A(a, b) > = AΦ(a). | b > + | 0 > ∈ E*(D, K)
With:
AΦ(a) = [Aχαβ. aα] ∈ M(D, K)
Through a relatively simple line of reasoning, it is possible to demonstrate the counter-intuitive non-uniqueness of trivial decompositions.
Assuming that there exists a priori at least one other trivial decomposition denoted [P] for a given deformed tensor product, it is then possible to write:
| ⊗A(a, b) > = [P]. | b > + | 0 > = {[P] - AΦ(a)}. | b > + AΦ(a). | b > + | 0 >
But anyway:
| ⊗A(a, b) > = AΦ(a). | b > + | 0 >
The other trivial decomposition whose existence is presumed must necessarily be subject to the condition:
{[P] - AΦ(a)}. | b > = | 0 >
The possibility of a null target (b = 0) must be repositioned within the study of the decompositions of a null deformed tensor product. Since it does not allow for the specification of the [T] matrix formalism but does not prevent its existence, there is an infinite number of trivial decompositions.
In all other cases (the target is not null), there is a certain number of trivial decompositions subject to the sole constraint:
Λ(a) = |[P] - AΦ(a)| = 0K
It amounts to the same thing as saying that « the discriminant of the system (S) of the D linear combinations written as a function of the D components of the target b :
|⊗A(a, b) > - [P]. | b > = | z >
…must be null".
Or, equivalently, that this system is degenerate. Conversely, if the system (S) exists and is degenerate, then its discriminant is zero.