Unification Theories: New Results and Examples

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.

1. Introduction

Voted the most famous formula by undergraduate students, the Euler’s identity states that $e^{\pi i}+1=0$. This is a particular case of the Euler’s–De Moivre formula:

$$cosx+isinx=e^{ix}\forall x\in \mathbb{R},$$

(1)

and, for hyperbolic functions, we have an analogous formula:

$$coshx+Jsinhx=e^{xJ}\forall x\in \mathbb{C},$$

(2)

where we consider the matrices

$$J=\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 1& 0& 0& 0\end{array}\right)$$

(3)

$$I=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$

(4)

$$I^{\prime }=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right).$$

(5)

In fact, $R\left(x\right)=cosh\left(x\right)I+sinh\left(x\right)J=coshx+Jsinhx=e^{xJ}$ also satisfies the equation

$$(R\otimes I^{\prime })\left(x\right)\circ (I^{\prime }\otimes R)(x+y)\circ (R\otimes I^{\prime })\left(y\right)=(I^{\prime }\otimes R)\left(y\right)\circ (R\otimes I^{\prime })(x+y)\circ (I^{\prime }\otimes R)\left(x\right)$$

(6)

called the colored Yang–Baxter equation. This fact follows easily from $J^{12}\circ J^{23}=J^{23}\circ J^{12}$ 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: $|e^i-\pi |>e$. 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 $\eta :V\otimes V\to V,\eta (a\otimes b)=ab,$ the couple $(V,\eta )$ is called a “weak UJLA structure” if the product $ab=\eta (a\otimes b)$ satisfies the identity

$$\left(ab\right)c+\left(bc\right)a+\left(ca\right)b=a\left(bc\right)+b\left(ca\right)+c\left(ab\right)\forall a,b,c\in V.$$

(7)

Definition 2.

Given a vector space V, with a linear map $\Delta :V\to V\otimes V,$ the couple $(V,\Delta )$ is called a “weak co-UJLA structure” if this co-product satisfies the identity

$$(Id+S+S^2)\circ (\Delta \otimes I)\circ \Delta =(Id+S+S^2)\circ (I\otimes \Delta )\circ \Delta$$

(8)

where $S:V\otimes V\otimes V\to V\otimes V\otimes V,a\otimes b\otimes c\mapsto b\otimes c\otimes a$, $I:V\to V,a\mapsto a$ and $Id:V\otimes V\otimes V\to V\otimes V\otimes V,a\otimes b\otimes c\mapsto a\otimes b\otimes c$.

Definition 3.

Given a vector space V, with a linear map $\varphi :V\otimes V\to V\otimes V,$ the couple $(V,\varphi )$ is called a “weak (co)UJLA structure” if the map ϕ satisfies the identity

$$(Id+S+S^2)\circ {\varphi }^{12}\circ {\varphi }^{23}\circ {\varphi }^{12}\circ (Id+S+S^2)=(Id+S+S^2)\circ {\varphi }^{23}\circ {\varphi }^{12}\circ {\varphi }^{23}\circ (Id+S+S^2)$$

(9)

where ${\varphi }^{12}=\varphi \otimes I,{\varphi }^{23}=I\otimes \varphi$, $Id:V\otimes V\otimes V\to V\otimes V\otimes V,a\otimes b\otimes c\mapsto a\otimes b\otimes c$ and $I:V\to V,a\mapsto a$.

Theorem 1.

Let $(V,\eta )$ be a weak UJLA structure with the unity $1\in V$. Let $\varphi :V\otimes V\to V\otimes V,a\otimes b\mapsto ab\otimes 1$. Then, $(V,\varphi )$ is a “weak (co)UJLA structure”.

Proof. 

$$\begin{gathered} (Id+S+S^2)\circ {\varphi }^{23}\circ {\varphi }^{12}\circ {\varphi }^{23}\circ (Id+S+S^2)(a\otimes b\otimes c)=(Id+S+S^2)\circ {\varphi }^{23}\circ {\varphi }^{12}\circ {\varphi }^{23} \\ (a\otimes b\otimes c+b\otimes c\otimes a+c\otimes a\otimes b)=(Id+S+S^2)\circ {\varphi }^{23}\circ {\varphi }^{12}(a\otimes bc\otimes 1+b\otimes ca\otimes 1+c\otimes ab\otimes 1) \\ =(Id+S+S^2)\circ {\varphi }^{23}(a\left(bc\right)\otimes 1\otimes 1+b\left(ca\right)\otimes 1\otimes 1+c\left(ab\right)\otimes 1\otimes 1) \\ =(Id+S+S^2)(a\left(bc\right)\otimes 1\otimes 1+b\left(ca\right)\otimes 1\otimes 1+c\left(ab\right)\otimes 1\otimes 1)=a\left(bc\right)\otimes 1\otimes 1+b\left(ca\right) \\ \otimes 1\otimes 1+c\left(ab\right)\otimes 1\otimes 1+1\otimes 1\otimes a\left(bc\right)+1\otimes 1\otimes b\left(ca\right)+1\otimes 1\otimes c\left(ab\right)+ \\ 1\otimes a\left(bc\right)\otimes 1+1\otimes b\left(ca\right)\otimes 1+1\otimes c\left(ab\right)\otimes 1 \end{gathered}$$.

Similarly,

$$\begin{gathered} (Id+S+S^2)\circ {\varphi }^{12}\circ {\varphi }^{23}\circ {\varphi }^{12}\circ (Id+S+S^2)(a\otimes b\otimes c)=(Id+S+S^2)\circ {\varphi }^{12}\circ {\varphi }^{23}\circ {\varphi }^{12}(a\otimes b\otimes \\ c+b\otimes c\otimes a+c\otimes a\otimes b)=\left(ab\right)c\otimes 1\otimes 1+\left(bc\right)a\otimes 1\otimes 1+\left(ca\right)b\otimes 1\otimes 1+1\otimes 1\otimes \left(ab\right)c+1\otimes 1\otimes \\ \left(bc\right)a+1\otimes 1\otimes \left(ca\right)b+1\otimes \left(ab\right)c\otimes 1+1\otimes \left(bc\right)a\otimes 1+1\otimes \left(ca\right)b\otimes 1 \end{gathered}$$.

We now use the axiom of the “weak UJLA structure”. □

Theorem 2.

Let $(V,\Delta )$ be a weak co-UJLA structure with the co-unity $\epsilon :V\to k$. Let $\varphi =\Delta \otimes \epsilon :V\otimes V\to V\otimes V$. Then, $(V,\varphi )$ 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: $(\epsilon \otimes I)\circ \Delta =I=(I\otimes \epsilon )\circ \Delta$. After computing

${\varphi }^{12}\circ {\varphi }^{23}\circ {\varphi }^{12}(a\otimes b\otimes c)=\epsilon \left(b\right)\epsilon \left(c\right){\left(a_1\right)}_1\otimes {\left(a_1\right)}_2\otimes a_2$ and ${\varphi }^{23}\circ {\varphi }^{12}\circ {\varphi }^{23}(a\otimes b\otimes c)=\epsilon \left(b\right)\epsilon \left(c\right)a_1\otimes {\left(a_2\right)}_1\otimes {\left(a_2\right)}_2$,

one just checks that the properties of the linear map $Id+S+S^2$ will help to obtain the desired result. □

Theorem 3.

Let $(V,\eta )$ be a weak UJLA structure with the unity $1\in V$. Let $\varphi :V\otimes V\to V\otimes V,a\otimes b\mapsto ab\otimes 1+1\otimes ab-a\otimes b$. Then, $(V,\varphi )$ 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 $d:V\to V$, and a linear map $\varphi :V\otimes V\to V\otimes V,$ with the properties

$${\varphi }^{12}\circ {\varphi }^{23}\circ {\varphi }^{12}={\varphi }^{23}\circ {\varphi }^{12}\circ {\varphi }^{23}$$

(10)

$$\varphi \circ \varphi =Id,$$

(11)

the triple $(V,d,\varphi )$ is called a “generalized derivation” if the maps d and ϕ satisfy the identity

$\varphi \circ (d\otimes I+I\otimes d)=(d\otimes I+I\otimes d)\circ \varphi$.

Here, we have used our usual notation: ${\varphi }^{12}=\varphi \otimes I,{\varphi }^{23}=I\otimes \varphi$, $Id:V\otimes V\to V\otimes V,a\otimes b\mapsto a\otimes b$ and $I:V\to V,a\mapsto a$.

Theorem 4.

If A is an associative algebra and $d:A\to A$ is a derivation, and $\varphi :A\otimes A\to A\otimes A,a\otimes b\mapsto ab\otimes 1+1\otimes ab-a\otimes b$, then $(A,d,\varphi )$ is a “generalized derivation”.

Proof. 

According to [7], $\varphi$ verifies conditions (10) and (11). Recall now that $d\left(ab\right)=d\left(a\right)b+ad\left(b\right)\forall a,b\in A,d\left(1_A\right)=0$.

$(d\otimes I+I\otimes d)\circ \varphi (a\otimes b)=(d\otimes I+I\otimes d)(ab\otimes 1+1\otimes ab-a\otimes b)=d\left(ab\right)\otimes 1-d\left(a\right)\otimes b+1\otimes d\left(ab\right)-a\otimes d\left(b\right)$.

$\varphi \circ (d\otimes I+I\otimes d)(a\otimes b)=\varphi \left(d\right(a)\otimes b+a\otimes d(b)=d(a)b\otimes 1+1\otimes d(a)b-d(a)\otimes b+ad(b)\otimes 1+1\otimes ad(b)-a\otimes d(b).$ □

Theorem 5.

If $(C,\Delta ,\epsilon )$ is a co-algebra, $d:C\to C$ is a co-derivation, and $\psi =\Delta \otimes \epsilon +\epsilon \otimes \Delta -Id:C\otimes C\to C\otimes C,c\otimes d\mapsto \epsilon \left(d\right)c_1\otimes c_2+\epsilon \left(c\right)d_1\otimes d_2-c\otimes d$, then $(C,d,\psi )$ is a “generalized derivation”. (We use the sigma notation for co-algebras.)

Proof. 

The proof is dual to the above proof.

According to [7], $\psi$ verifies conditions (10) and (11). From the definition of the co-derivation, we have $\epsilon \left(d\right(c\left)\right)=0$ and $\Delta \left(d\left(c\right)\right)=d\left(c_1\right)\otimes c_2+c_1\otimes d\left(c_2\right)\forall c\in C$.

$\psi \circ (d\otimes I+I\otimes d)(c\otimes a)=\epsilon \left(a\right)d{\left(c\right)}_1\otimes d{\left(c\right)}_2-d\left(c\right)\otimes a+\epsilon \left(c\right)d{\left(a\right)}_1\otimes d{\left(a\right)}_2-c\otimes d\left(a\right)$,

$$\begin{gathered} (d\otimes I+I\otimes d)\circ \psi (c\otimes a)=\epsilon \left(a\right)d\left(c_1\right)\otimes c_2+\epsilon \left(c\right)d\left(a_1\right)\otimes a_2-d\left(c\right)\otimes a+\epsilon \left(a\right)c_1\otimes d\left(c_2\right)+\epsilon \left(c\right)a_1\otimes \\ d\left(a_2\right)-c\otimes d\left(a\right) \end{gathered}$$.

The statement follows on from the main property of the co-derivative. □

Definition 5.

Given an associative algebra A with a derivation $d:A\to A$, M an A-bimodule and $D:M\to M$ with the properties

$$D\left(am\right)=d\left(a\right)m+aD\left(m\right)D\left(ma\right)=D\left(m\right)a+md\left(a\right)\forall a\in A,\forall m\in M,$$

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, $A\oplus M$ becomes an algebra, and $\delta :A\oplus M\to A\oplus M,(a\oplus m)\mapsto \left(d\right(a)\oplus D(m\left)\right)$ is a derivation of this algebra.

Proof. 

We just need to check that $\delta \left(\right(a\oplus m\left)\right(b\oplus n\left)\right)=\delta \left(\right(ab\oplus an+mb\left)\right)=d\left(ab\right)\oplus D(an+mb)$

equals $$\begin{gathered} \delta \left(\right(a\oplus m\left)\right(b\oplus n\left)\right)=\delta \left(\right(a\oplus m\left)\right)(b\oplus n)+(a\oplus m)\delta (b\oplus n)=\left(d\right(a)\oplus D(m\left)\right)(b\oplus n)+(a\oplus \\ m)\left(d\right(b)\oplus D(n\left)\right)=\left(d\right(a)b\oplus d(a)n+D(m\left)b\right)+\left(ad\right(b)\oplus aD(n)+md(b\left)\right) \end{gathered}$$. □

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 $A\in M_n\left(\mathbb{C}\right)$ and $D\in M_n\left(\mathbb{C}\right)$, a diagonal matrix, we propose the problem of finding $X\in M_n\left(\mathbb{C}\right)$, such that

$$AXA+XAX=D.$$

(12)

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 $D-AXA$ 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 $A\in M_2\left(\mathbb{C}\right)$, a matrix with trace -1 and

$$D=-\left(\begin{array}{cc}det\left(A\right)& 0\\ 0& det\left(A\right)\end{array}\right),$$

(13)

.

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: $X=\alpha I_n+\beta A+\gamma A^2$. Now, (12) implies that $(2\alpha \beta +{\gamma }^2+\alpha )A^2+({\alpha }^2+2\beta \gamma +\gamma )A+(2\alpha \gamma +{\beta }^2+\beta )I_n-D=0$.

It can be shown that we can produce a large class of solutions in this way, if D is of a certain type.