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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. [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.

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