The cubes

Introduction to the concept

Definition: a cube

Per definition, in that theory, a cube is a mathematical object of which the components (i) are elements arbitrarily chosen in a set K and (ii) are situated at each knot of a three-dimensional Euclidean crystalline cubic structure.

More precisely, they are involve in procedures deforming the classical tensor products.

For example, the set of all cubes with knots in C (the set of all complex numbers) acting on elements in E(D, C) is denoted ⊞(D, C). Any cube in that set contains at most D . D . D = D³ knots.

Definition: a symmetric cube

A given cube is "symmetric" when its nodes have the proprety:

∀ α, β, χ:

A^χ_αβ - A^χ_βα = 0

The subset of all symmetric cubes is denoted ⊞⁺(D, K)

Definition: an antisymmetric cube

A given cube is "antisymmetric" when its nodes have the proprety:

∀ α, β, χ:

A^χ_αβ + A^χ_βα = 0

The subset of all symmetric cubes is denoted ⊞^-(D, K).

Definition: a reduced cube

A given cube is "reduced" when its nodes have the proprety:

∀ α, β, χ:

A^χ_αβ - A^β_αχ = 0

The subset of all symmetric cubes is denoted ⊞^↑(D, K).

Definition: an anti-reduced cube

A given cube is "reduced" when its nodes have the proprety:

∀ α, β, χ:

A^χ_αβ + A^β_αχ = 0

The subset of all symmetric cubes is denoted ⊞^↓(D, K).

Basic properties

Remark: compatibilities between different types of cubes

1) Symmetric and reduced cubes are compatible:

A^χ_αβ = A^χ_βα = A^α_β_χ = A^α_χ_β = A^β_χα = A^β_αχ = A^χ_αβ

2) Antisymmetric and anti-reduced cubes are compatible:

A^χ_αβ = - A^χ_βα = A^α_β_χ = - A^α_χ_β = A^β_χα = - A^β_αχ = A^χ_αβ

3) Symmetric and anti-reduced cubes are null.

4) Antisymmetric and reduced cubes are null too.

Remark: number of knots in a cube

It should be kept in mind that any element of ⊞(D, K) is equivalent of a superposition of D elements in M(D, K); therefore:

1) A symmetric cube is an element in ⊞⁺(D, K) which is the the superposition of D symmetric elements in M(D, K); each of them contains:

D terms along the diagonal

D - 1 terms in the first line

D - 2 terms in the second line

etc...

1 term in the (D - 1)^th line

Each symmetric matrix in M(D, K) has:

D + (1/2).{D² - D) = D.(D +1)/2

terms

As a result, each symmetric cube contains:

D².(D +1)/2 knots

2) An antisymmetric cube is an element in ⊞^-(D, K) which is the the superposition of D antisymmetric elements in M(D, K); each of them contains D.(D - 1) non necessarily vanishing terms

But since half of these terms are equal to the other half up to a sign, it is more accurate to say that each skew-symmetric matrix contains at most D.(D - 1)/2 terms whose absolute value is potentially different from the others and not necessarily zero.

As a result, each anti-symmetric cube contains at most D².(D -1)/2 effectively different knots.

Example: A byside consequence of this statement is that any element in ⊞^-(3, R) contains at most 9 knots. They can be reorganized inside an element in M(3, R).