Unification Theories: New Results and Examples

This paper is a continuation of a previous article that appeared in AXIOMS in 2018. A Euler’s formula for hyperbolic functions is considered a consequence of a unifying point of view. Then, the unification of Jordan, Lie, and associative algebras is revisited. We also explain that derivations and co-derivations can be unified. Finally, we consider a “modified” Yang–Baxter type equation, which unifies several problems in mathematics.


Introduction
Voted the most famous formula by undergraduate students, the Euler's identity states that e πi + 1 = 0. This is a particular case of the Euler's-De Moivre formula: cos x + i sin x = e ix ∀x ∈ R, (1) and, for hyperbolic functions, we have an analogous formula: where we consider the matrices In fact, R(x) = cosh(x)I + sinh(x)J = cosh x + J sinh x = e xJ also satisfies the equation called the colored Yang-Baxter equation. This fact follows easily from J 12 • J 23 = J 23 • J 12 and xJ 12 + (x + y) J 23 + yJ 12 = yJ 23 + (x + y) J 12 + xJ 23 , and it shows that the formulas (1) and (2) are related. While we do not know a remarkable identity related to (2), let us recall an interesting inequality from a previous paper: There is an open problem to find the matrix version of this inequality.
The above analysis is a consequence of a unifying point of view from previous papers ( [1,2]).
In the remainder of this paper, we first consider the unification of the Jordan, Lie, and associative algebras. In Section 3, we explain that derivations and co-derivations can be unified. We suggest applications in differential geometry. Finally, we consider a "modified" Yang-Baxter equation which unifies the problem of the three matrices, generalized eigenvalue problems, and the Yang-Baxter matrix equation. There are several versions of the Yang-Baxter equation (see, for example, [3,4]) presented throughout this paper.
We work over the field k, and the tensor products are defined over k.

Weak Ujla Structures, Dual Structures, Unification
Definition 2. Given a vector space V, with a linear map ∆ : is called a "weak co-UJLA structure" if this co-product satisfies the identity Definition 3. Given a vector space V, with a linear map φ : We now use the axiom of the "weak UJLA structure". Theorem 2. Let (V, ∆) be a weak co-UJLA structure with the co-unity ε : Proof. The proof is dual to the above proof. We refer to [6][7][8] for a similar approach. A direct proof should use the property of the co-unity: (ε ⊗ I) • ∆ = I = (I ⊗ ε) • ∆. After computing one just checks that the properties of the linear map Id + S + S 2 will help to obtain the desired result.
Theorem 3. Let (V, η) be a weak UJLA structure with the unity 1 ∈ V. Let φ : Proof. One can formulate a direct proof, similar to the proof of Theorem 1.
Alternatively, one could use the calculations from [7] and the axiom of the "weak UJLA structure".

Unification of (Co)Derivations and Applications
Definition 4. Given a vector space V, a linear map d : V → V, and a linear map φ : the triple (V, d, φ) is called a "generalized derivation" if the maps d and φ satisfy the identity Here, we have used our usual notation: Theorem 4. If A is an associative algebra and d : A → A is a derivation, and φ : Proof. According to [7], φ verifies conditions (10) and (11). Recall now that d(ab Theorem 5. If (C, ∆, ε) is a co-algebra, d : C → C is a co-derivation, and ψ = ∆ ⊗ ε + ε ⊗ ∆ − Id : (We use the sigma notation for co-algebras.) Proof. The proof is dual to the above proof.