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Unification Theories: Examples and Applications

Axioms 2019, 8(2), 60; https://doi.org/10.3390/axioms8020060

Article
Unification Theories: New Results and Examples
Simion Stoilow Institute of Mathematics of the Romanian Academy 21 Calea Grivitei Street, 010702 Bucharest, Romania
Received: 3 May 2019 / Accepted: 17 May 2019 / Published: 18 May 2019

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.
Keywords:
Euler’s formula; hyperbolic functions; Yang–Baxter equation; Jordan algebras; Lie algebras; associative algebras; UJLA structures; (co)derivation
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. ) 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  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 , $ϕ$ 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 , $ψ$ 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  (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, ).
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 . 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, ).
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.

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