Unification Theories: New Results and Examples

Simion Stoilow Institute of Mathematics of the Romanian Academy 21 Calea Grivitei Street, 010702 Bucharest, Romania

Axioms2019,8(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 IssueNon-associative Structures and Other Related Structures)

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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} \forall x \in R,

(1)

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

cosh x + J sinh x = e^{x J} \forall x \in C,

(2)

where we consider the matrices

J = (\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix})

(3)

I = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})

(4)

I^{'} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) .

(5)

In fact,

R (x) = cosh (x) I + sinh (x) J = cosh x + J sinh x = e^{x J}

also satisfies the equation

(R \otimes I^{'}) (x) \circ (I^{'} \otimes R) (x + y) \circ (R \otimes I^{'}) (y) = (I^{'} \otimes R) (y) \circ (R \otimes I^{'}) (x + y) \circ (I^{'} \otimes R) (x)

(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} - 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. InSection 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 fieldk, and the tensor products are defined overk.

2. Weak Ujla Structures, Dual Structures, Unification

Definition 1.

(Ref. [5]) Given a vector space V, with a linear map

η : V \otimes V \to V, η (a \otimes b) = a b,

the couple

(V, η)

is called a “weak UJLA structure” if the product

a b = η (a \otimes b)

satisfies the identity

(a b) c + (b c) a + (c a) b = a (b c) + b (c a) + c (a b) \forall a, b, c \in V .

(7)

Definition 2.

Given a vector space V, with a linear map

Δ : V \to V \otimes V,

the couple

(V, Δ)

is called a “weak co-UJLA structure” if this co-product satisfies the identity

(I d + S + S^{2}) \circ (Δ \otimes I) \circ Δ = (I d + S + S^{2}) \circ (I \otimes Δ) \circ Δ

(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

I d : 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

ϕ : V \otimes V \to V \otimes V,

the couple

(V, ϕ)

is called a “weak (co)UJLA structure” if the map ϕ satisfies the identity

(I d + S + S^{2}) \circ ϕ^{12} \circ ϕ^{23} \circ ϕ^{12} \circ (I d + S + S^{2}) = (I d + S + S^{2}) \circ ϕ^{23} \circ ϕ^{12} \circ ϕ^{23} \circ (I d + S + S^{2})

(9)

where

ϕ^{12} = ϕ \otimes I, ϕ^{23} = I \otimes ϕ

I d : 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, η)

be a weak UJLA structure with the unity

1 \in V

. Let

ϕ : V \otimes V \to V \otimes V, a \otimes b \mapsto a b \otimes 1

. Then,

(V, ϕ)

is a “weak (co)UJLA structure”.

Proof.

(I d + S + S^{2}) \circ ϕ^{23} \circ ϕ^{12} \circ ϕ^{23} \circ (I d + S + S^{2}) (a \otimes b \otimes c) = (I d + S + S^{2}) \circ ϕ^{23} \circ ϕ^{12} \circ ϕ^{23} (a \otimes b \otimes c + b \otimes c \otimes a + c \otimes a \otimes b) = (I d + S + S^{2}) \circ ϕ^{23} \circ ϕ^{12} (a \otimes b c \otimes 1 + b \otimes c a \otimes 1 + c \otimes a b \otimes 1) = (I d + S + S^{2}) \circ ϕ^{23} (a (b c) \otimes 1 \otimes 1 + b (c a) \otimes 1 \otimes 1 + c (a b) \otimes 1 \otimes 1) = (I d + S + S^{2}) (a (b c) \otimes 1 \otimes 1 + b (c a) \otimes 1 \otimes 1 + c (a b) \otimes 1 \otimes 1) = a (b c) \otimes 1 \otimes 1 + b (c a) \otimes 1 \otimes 1 + c (a b) \otimes 1 \otimes 1 + 1 \otimes 1 \otimes a (b c) + 1 \otimes 1 \otimes b (c a) + 1 \otimes 1 \otimes c (a b) + 1 \otimes a (b c) \otimes 1 + 1 \otimes b (c a) \otimes 1 + 1 \otimes c (a b) \otimes 1

Similarly,

(I d + S + S^{2}) \circ ϕ^{12} \circ ϕ^{23} \circ ϕ^{12} \circ (I d + S + S^{2}) (a \otimes b \otimes c) = (I d + S + S^{2}) \circ ϕ^{12} \circ ϕ^{23} \circ ϕ^{12} (a \otimes b \otimes c + b \otimes c \otimes a + c \otimes a \otimes b) = (a b) c \otimes 1 \otimes 1 + (b c) a \otimes 1 \otimes 1 + (c a) b \otimes 1 \otimes 1 + 1 \otimes 1 \otimes (a b) c + 1 \otimes 1 \otimes (b c) a + 1 \otimes 1 \otimes (c a) b + 1 \otimes (a b) c \otimes 1 + 1 \otimes (b c) a \otimes 1 + 1 \otimes (c a) b \otimes 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 \to k

. Let

ϕ = Δ \otimes ε : V \otimes V \to V \otimes 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 \otimes I) \circ Δ = I = (I \otimes ε) \circ Δ

. After computing

ϕ^{12} \circ ϕ^{23} \circ ϕ^{12} (a \otimes b \otimes c) = ε (b) ε (c) {(a_{1})}_{1} \otimes {(a_{1})}_{2} \otimes a_{2}

and

ϕ^{23} \circ ϕ^{12} \circ ϕ^{23} (a \otimes b \otimes c) = ε (b) ε (c) a_{1} \otimes {(a_{2})}_{1} \otimes {(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 \in V

. Let

ϕ : V \otimes V \to V \otimes V, a \otimes b \mapsto a b \otimes 1 + 1 \otimes a b - a \otimes 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 \to V

, and a linear map

ϕ : V \otimes V \to V \otimes V,

with the properties

ϕ^{12} \circ ϕ^{23} \circ ϕ^{12} = ϕ^{23} \circ ϕ^{12} \circ ϕ^{23}

(10)

ϕ \circ ϕ = I d,

(11)

the triple

(V, d, ϕ)

is called a “generalized derivation” if the maps d and ϕ satisfy the identity

ϕ \circ (d \otimes I + I \otimes d) = (d \otimes I + I \otimes d) \circ ϕ

Here, we have used our usual notation:

ϕ^{12} = ϕ \otimes I, ϕ^{23} = I \otimes ϕ

I d : 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

ϕ : A \otimes A \to A \otimes A, a \otimes b \mapsto a b \otimes 1 + 1 \otimes a b - a \otimes 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) \forall a, b \in A, d (1_{A}) = 0

(d \otimes I + I \otimes d) \circ ϕ (a \otimes b) = (d \otimes I + I \otimes d) (a b \otimes 1 + 1 \otimes a b - a \otimes b) = d (a b) \otimes 1 - d (a) \otimes b + 1 \otimes d (a b) - a \otimes d (b)

ϕ \circ (d \otimes I + I \otimes d) (a \otimes b) = ϕ (d (a) \otimes b + a \otimes d (b) = d (a) b \otimes 1 + 1 \otimes d (a) b - d (a) \otimes b + a d (b) \otimes 1 + 1 \otimes a d (b) - a \otimes d (b) .

□

Theorem 5.

(C, Δ, ε)

is a co-algebra,

d : C \to C

is a co-derivation, and

ψ = Δ \otimes ε + ε \otimes Δ - I d : C \otimes C \to C \otimes C, c \otimes d \mapsto ε (d) c_{1} \otimes c_{2} + ε (c) d_{1} \otimes d_{2} - c \otimes 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}) \otimes c_{2} + c_{1} \otimes d (c_{2}) \forall c \in C

ψ \circ (d \otimes I + I \otimes d) (c \otimes a) = ε (a) d {(c)}_{1} \otimes d {(c)}_{2} - d (c) \otimes a + ε (c) d {(a)}_{1} \otimes d {(a)}_{2} - c \otimes d (a)

(d \otimes I + I \otimes d) \circ ψ (c \otimes a) = ε (a) d (c_{1}) \otimes c_{2} + ε (c) d (a_{1}) \otimes a_{2} - d (c) \otimes a + ε (a) c_{1} \otimes d (c_{2}) + ε (c) a_{1} \otimes d (a_{2}) - c \otimes 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 \to A

, M an A-bimodule and

D : M \to M

with the properties

D (a m) = d (a) m + a D (m) D (m a) = D (m) a + m d (a) \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

δ : A \oplus M \to A \oplus M, (a \oplus m) \mapsto (d (a) \oplus D (m))

is a derivation of this algebra.

Proof.

We just need to check that

δ ((a \oplus m) (b \oplus n)) = δ ((a b \oplus a n + m b)) = d (a b) \oplus D (a n + m b)

equals

δ ((a \oplus m) (b \oplus n)) = δ ((a \oplus m)) (b \oplus n) + (a \oplus m) δ (b \oplus n) = (d (a) \oplus D (m)) (b \oplus n) + (a \oplus m) (d (b) \oplus D (n)) = (d (a) b \oplus d (a) n + D (m) b) + (a d (b) \oplus 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 \in M_{n} (C)

and

D \in M_{n} (C)

, a diagonal matrix, we propose the problem of finding

X \in M_{n} (C)

, such that

A X A + X A X = 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

andC, where

A + B = C

. A good reference is the paper [11]. Note that ifA 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 \in M_{2} (C)

, a matrix with trace -1 and

D = - (\begin{matrix} d e t (A) & 0 \\ 0 & d e t (A) \end{matrix}),

(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 = 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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Nichita, F.F. Unification Theories: New Results and Examples.Axioms2019,8, 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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Nichita, F. F. (2019). Unification Theories: New Results and Examples.Axioms,8(2), 60. https://doi.org/10.3390/axioms8020060

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Abstract

1. Introduction

2. Weak Ujla Structures, Dual Structures, Unification

3. Unification of (Co)Derivations and Applications

4. Modified Yang–Baxter Equation

Funding

Acknowledgments

Conflicts of Interest

References

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