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Unification Theories: Examples and Applications

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Article

# Unification Theories: New Results and Examples

by
Florin F. Nichita
Simion Stoilow Institute of Mathematics of the Romanian Academy 21 Calea Grivitei Street, 010702 Bucharest, Romania
Axioms 2019, 8(2), 60; https://doi.org/10.3390/axioms8020060
Submission received: 3 May 2019 / Revised: 16 May 2019 / Accepted: 17 May 2019 / Published: 18 May 2019
(This article belongs to the Special Issue Non-associative Structures and Other Related Structures)

## Abstract

:
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.
MSC:
17C05; 17C50; 16T15; 16T25; 17B01; 17B40; 15A18; 11J81

## 1. Introduction

Voted the most famous formula by undergraduate students, the Euler’s identity states that $e π i + 1 = 0$. This is a particular case of the Euler’s–De Moivre formula:
$cos x + i sin x = e i x ∀ x ∈ R ,$
and, for hyperbolic functions, we have an analogous formula:
$cosh x + J sinh x = e x J ∀ x ∈ C ,$
where we consider the matrices
$J = 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0$
$I = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1$
$I ′ = 1 0 0 1 .$
In fact, $R ( x ) = cosh ( x ) I + sinh ( x ) J = cosh x + J sinh x = e x J$ also satisfies the equation
$( R ⊗ I ′ ) ( x ) ∘ ( I ′ ⊗ R ) ( x + y ) ∘ ( R ⊗ I ′ ) ( y ) = ( I ′ ⊗ R ) ( y ) ∘ ( R ⊗ I ′ ) ( x + y ) ∘ ( I ′ ⊗ R ) ( x )$
called the colored Yang–Baxter equation. This fact follows easily from $J 12 ∘ J 23 = J 23 ∘ J 12$ and $x J 12 + ( x + y ) J 23 + y J 12 = y J 23 + ( x + y ) J 12 + x J 23$, 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: $| e i − π | > e$. 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.

## 2. Weak Ujla Structures, Dual Structures, Unification

Definition 1.
(Ref. [5]) Given a vector space V, with a linear map $η : V ⊗ V → V , η ( a ⊗ b ) = a b ,$ the couple $( V , η )$ is called a “weak UJLA structure” if the product $a b = η ( a ⊗ b )$ satisfies the identity
$( a b ) c + ( b c ) a + ( c a ) b = a ( b c ) + b ( c a ) + c ( a b ) ∀ a , b , c ∈ V .$
Definition 2.
Given a vector space V, with a linear map $Δ : V → V ⊗ V ,$ the couple $( V , Δ )$ is called a “weak co-UJLA structure” if this co-product satisfies the identity
$( I d + S + S 2 ) ∘ ( Δ ⊗ I ) ∘ Δ = ( I d + S + S 2 ) ∘ ( I ⊗ Δ ) ∘ Δ$
where $S : V ⊗ V ⊗ V → V ⊗ V ⊗ V , a ⊗ b ⊗ c ↦ b ⊗ c ⊗ a$, $I : V → V , a ↦ a$ and $I d : V ⊗ V ⊗ V → V ⊗ V ⊗ V , a ⊗ b ⊗ c ↦ a ⊗ b ⊗ c$.
Definition 3.
Given a vector space V, with a linear map $ϕ : V ⊗ V → V ⊗ V ,$ the couple $( V , ϕ )$ is called a “weak (co)UJLA structure” if the map ϕ satisfies the identity
$( I d + S + S 2 ) ∘ ϕ 12 ∘ ϕ 23 ∘ ϕ 12 ∘ ( I d + S + S 2 ) = ( I d + S + S 2 ) ∘ ϕ 23 ∘ ϕ 12 ∘ ϕ 23 ∘ ( I d + S + S 2 )$
where $ϕ 12 = ϕ ⊗ I , ϕ 23 = I ⊗ ϕ$, $I d : V ⊗ V ⊗ V → V ⊗ V ⊗ V , a ⊗ b ⊗ c ↦ a ⊗ b ⊗ c$ and $I : V → V , a ↦ a$.
Theorem 1.
Let $( V , η )$ be a weak UJLA structure with the unity $1 ∈ V$. Let $ϕ : V ⊗ V → V ⊗ V , a ⊗ b ↦ a b ⊗ 1$. Then, $( V , ϕ )$ is a “weak (co)UJLA structure”.
Proof.
$( I d + S + S 2 ) ∘ ϕ 23 ∘ ϕ 12 ∘ ϕ 23 ∘ ( I d + S + S 2 ) ( a ⊗ b ⊗ c ) = ( I d + S + S 2 ) ∘ ϕ 23 ∘ ϕ 12 ∘ ϕ 23 ( a ⊗ b ⊗ c + b ⊗ c ⊗ a + c ⊗ a ⊗ b ) = ( I d + S + S 2 ) ∘ ϕ 23 ∘ ϕ 12 ( a ⊗ b c ⊗ 1 + b ⊗ c a ⊗ 1 + c ⊗ a b ⊗ 1 ) = ( I d + S + S 2 ) ∘ ϕ 23 ( a ( b c ) ⊗ 1 ⊗ 1 + b ( c a ) ⊗ 1 ⊗ 1 + c ( a b ) ⊗ 1 ⊗ 1 ) = ( I d + S + S 2 ) ( a ( b c ) ⊗ 1 ⊗ 1 + b ( c a ) ⊗ 1 ⊗ 1 + c ( a b ) ⊗ 1 ⊗ 1 ) = a ( b c ) ⊗ 1 ⊗ 1 + b ( c a ) ⊗ 1 ⊗ 1 + c ( a b ) ⊗ 1 ⊗ 1 + 1 ⊗ 1 ⊗ a ( b c ) + 1 ⊗ 1 ⊗ b ( c a ) + 1 ⊗ 1 ⊗ c ( a b ) + 1 ⊗ a ( b c ) ⊗ 1 + 1 ⊗ b ( c a ) ⊗ 1 + 1 ⊗ c ( a b ) ⊗ 1$.
Similarly,
$( I d + S + S 2 ) ∘ ϕ 12 ∘ ϕ 23 ∘ ϕ 12 ∘ ( I d + S + S 2 ) ( a ⊗ b ⊗ c ) = ( I d + S + S 2 ) ∘ ϕ 12 ∘ ϕ 23 ∘ ϕ 12 ( a ⊗ b ⊗ c + b ⊗ c ⊗ a + c ⊗ a ⊗ b ) = ( a b ) c ⊗ 1 ⊗ 1 + ( b c ) a ⊗ 1 ⊗ 1 + ( c a ) b ⊗ 1 ⊗ 1 + 1 ⊗ 1 ⊗ ( a b ) c + 1 ⊗ 1 ⊗ ( b c ) a + 1 ⊗ 1 ⊗ ( c a ) b + 1 ⊗ ( a b ) c ⊗ 1 + 1 ⊗ ( b c ) a ⊗ 1 + 1 ⊗ ( c a ) b ⊗ 1$.
We now use the axiom of the “weak UJLA structure”. □
Theorem 2.
Let $( V , Δ )$ be a weak co-UJLA structure with the co-unity $ε : V → k$. Let $ϕ = Δ ⊗ ε : V ⊗ V → V ⊗ V$. Then, $( V , ϕ )$ is a “weak (co)UJLA structure”.
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: $( ε ⊗ I ) ∘ Δ = I = ( I ⊗ ε ) ∘ Δ$. After computing
$ϕ 12 ∘ ϕ 23 ∘ ϕ 12 ( a ⊗ b ⊗ c ) = ε ( b ) ε ( c ) ( a 1 ) 1 ⊗ ( a 1 ) 2 ⊗ a 2$ and $ϕ 23 ∘ ϕ 12 ∘ ϕ 23 ( a ⊗ b ⊗ c ) = ε ( b ) ε ( c ) a 1 ⊗ ( a 2 ) 1 ⊗ ( a 2 ) 2$,
one just checks that the properties of the linear map $I d + S + S 2$ will help to obtain the desired result. □
Theorem 3.
Let $( V , η )$ be a weak UJLA structure with the unity $1 ∈ V$. Let $ϕ : V ⊗ V → V ⊗ V , a ⊗ b ↦ a b ⊗ 1 + 1 ⊗ a b − a ⊗ b$. Then, $( V , ϕ )$ is a “weak (co)UJLA structure”.
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”. □

## 3. Unification of (Co)Derivations and Applications

Definition 4.
Given a vector space V, a linear map $d : V → V$, and a linear map $ϕ : V ⊗ V → V ⊗ V ,$ with the properties
$ϕ 12 ∘ ϕ 23 ∘ ϕ 12 = ϕ 23 ∘ ϕ 12 ∘ ϕ 23$
$ϕ ∘ ϕ = I d ,$
the triple $( V , d , ϕ )$ is called a “generalized derivation” if the maps d and ϕ satisfy the identity
$ϕ ∘ ( d ⊗ I + I ⊗ d ) = ( d ⊗ I + I ⊗ d ) ∘ ϕ$.
Here, we have used our usual notation: $ϕ 12 = ϕ ⊗ I , ϕ 23 = I ⊗ ϕ$, $I d : V ⊗ V → V ⊗ V , a ⊗ b ↦ a ⊗ b$ and $I : V → V , a ↦ a$.
Theorem 4.
If A is an associative algebra and $d : A → A$ is a derivation, and $ϕ : A ⊗ A → A ⊗ A , a ⊗ b ↦ a b ⊗ 1 + 1 ⊗ a b − a ⊗ b$, then $( A , d , ϕ )$ is a “generalized derivation”.
Proof.
According to [7], $ϕ$ verifies conditions (10) and (11). Recall now that $d ( a b ) = d ( a ) b + a d ( b ) ∀ a , b ∈ A , d ( 1 A ) = 0$.
$( d ⊗ I + I ⊗ d ) ∘ ϕ ( a ⊗ b ) = ( d ⊗ I + I ⊗ d ) ( a b ⊗ 1 + 1 ⊗ a b − a ⊗ b ) = d ( a b ) ⊗ 1 − d ( a ) ⊗ b + 1 ⊗ d ( a b ) − a ⊗ d ( b )$.
$ϕ ∘ ( d ⊗ I + I ⊗ d ) ( a ⊗ b ) = ϕ ( d ( a ) ⊗ b + a ⊗ d ( b ) = d ( a ) b ⊗ 1 + 1 ⊗ d ( a ) b − d ( a ) ⊗ b + a d ( b ) ⊗ 1 + 1 ⊗ a d ( b ) − a ⊗ d ( b ) .$ □
Theorem 5.
If $( C , Δ , ε )$ is a co-algebra, $d : C → C$ is a co-derivation, and $ψ = Δ ⊗ ε + ε ⊗ Δ − I d : C ⊗ C → C ⊗ C , c ⊗ d ↦ ε ( d ) c 1 ⊗ c 2 + ε ( c ) d 1 ⊗ d 2 − c ⊗ d$, then $( C , d , ψ )$ is a “generalized derivation”. (We use the sigma notation for co-algebras.)
Proof.
The proof is dual to the above proof.
According to [7], $ψ$ verifies conditions (10) and (11). From the definition of the co-derivation, we have $ε ( d ( c ) ) = 0$ and $Δ ( d ( c ) ) = d ( c 1 ) ⊗ c 2 + c 1 ⊗ d ( c 2 ) ∀ c ∈ C$.
$ψ ∘ ( d ⊗ I + I ⊗ d ) ( c ⊗ a ) = ε ( a ) d ( c ) 1 ⊗ d ( c ) 2 − d ( c ) ⊗ a + ε ( c ) d ( a ) 1 ⊗ d ( a ) 2 − c ⊗ d ( a )$,
$( d ⊗ I + I ⊗ d ) ∘ ψ ( c ⊗ a ) = ε ( a ) d ( c 1 ) ⊗ c 2 + ε ( c ) d ( a 1 ) ⊗ a 2 − d ( c ) ⊗ a + ε ( a ) c 1 ⊗ d ( c 2 ) + ε ( c ) a 1 ⊗ d ( a 2 ) − c ⊗ d ( a )$.
The statement follows on from the main property of the co-derivative. □
Definition 5.
Given an associative algebra A with a derivation $d : A → A$, M an A-bimodule and $D : M → M$ with the properties
$D ( a m ) = d ( a ) m + a D ( m ) D ( m a ) = D ( m ) a + m d ( a ) ∀ a ∈ A , ∀ m ∈ M ,$
the quadruple $( A , d , M , D )$ is called a “module derivation”.
Remark 1.
A“module derivation” is a module over an algebra with a derivation. It can be related to the co-variant derivative from differential geometry. Definition 5 also requires us to check that the formulas for D are well-defined.
Note that there are some similar constructions and results in [9] (see Theorems 1.27 and 1.40).
Theorem 6.
In the above case, $A ⊕ M$ becomes an algebra, and $δ : A ⊕ M → A ⊕ M , ( a ⊕ m ) ↦ ( d ( a ) ⊕ D ( m ) )$ is a derivation of this algebra.
Proof.
We just need to check that $δ ( ( a ⊕ m ) ( b ⊕ n ) ) = δ ( ( a b ⊕ a n + m b ) ) = d ( a b ) ⊕ D ( a n + m b )$
equals $δ ( ( a ⊕ m ) ( b ⊕ n ) ) = δ ( ( a ⊕ m ) ) ( b ⊕ n ) + ( a ⊕ m ) δ ( b ⊕ n ) = ( d ( a ) ⊕ D ( m ) ) ( b ⊕ n ) + ( a ⊕ m ) ( d ( b ) ⊕ D ( n ) ) = ( d ( a ) b ⊕ d ( a ) n + D ( m ) b ) + ( a d ( b ) ⊕ a D ( n ) + m d ( b ) )$. □
Remark 2.
A dual statement with a co-derivation and a co-module over that co-algebra can be given.
Remark 3.
The above theorem leads to the unification of module derivation and co-module derivation.

## 4. Modified Yang–Baxter Equation

For $A ∈ M n ( C )$ and $D ∈ M n ( C )$, a diagonal matrix, we propose the problem of finding $X ∈ M n ( C )$, such that
$A X A + X A X = D .$
This is an intermediate step to other “modified” versions of the Yang–Baxter equation (see, for example, [10]).
Remark 4.
Equation (12) is related to the problem of the three matrices. This problem is about the properties of the eigenvalues of the matrices $A , B$ and C, where $A + B = C$. A good reference is the paper [11]. Note that if A is “small” then $D − A X A$ could be regarded as a deformation of D.
Remark 5.
Equation (12) can be interpreted as a “generalized eigenvalue problem” (see, for example, [12]).
Remark 6.
Equation (12) is a type of Yang–Baxter matrix equation (see, for example, [13,14]) if $D = O n$ and $X = − Y$.
Remark 7.
For $A ∈ M 2 ( C )$, a matrix with trace -1 and
$D = − d e t ( A ) 0 0 d e t ( A ) ,$
.
Equation (12) has the solution X = I’.
Remark 8.
There are several methods to solve (12). For example, for $A 3 = I n$, one could search for solutions of the following type: $X = α I n + β A + γ A 2$. Now, (12) implies that $( 2 α β + γ 2 + α ) A 2 + ( α 2 + 2 β γ + γ ) A + ( 2 α γ + β 2 + β ) I n − D = 0$.
It can be shown that we can produce a large class of solutions in this way, if D is of a certain type.

## Funding

This research received no external funding.

## Acknowledgments

I would like to thank Dan Timotin for the discussions and the reference on the problem of the three matrices. I also thank the editors and the referees.

## Conflicts of Interest

The author declares no conflict of interest.

## References

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Nichita, F.F. Unification Theories: New Results and Examples. Axioms 2019, 8, 60. https://doi.org/10.3390/axioms8020060

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Nichita FF. Unification Theories: New Results and Examples. Axioms. 2019; 8(2):60. https://doi.org/10.3390/axioms8020060

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Nichita, Florin F. 2019. "Unification Theories: New Results and Examples" Axioms 8, no. 2: 60. https://doi.org/10.3390/axioms8020060

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