On Some Unification Theorems: Yang–Baxter Systems; Johnson–Tzitzeica Theorem

Abstract

This paper investigates the properties of the Yang–Baxter equation, which was initially formulated in the field of theoretical physics and statistical mechanics. The equation’s framework is extended through Yang–Baxter systems, aiming to unify algebraic and coalgebraic structures. The unification of the algebra structures and the coalgebra structures leads to an extension for the duality between finite dimensional algebras and finite dimensional coalgebras to the category of finite dimensional Yang–Baxter structures. In the same manner, we attempt to unify the Tzitzeica–Johnson theorem and its dual version, obtaining a new theorem about circle configurations.

1. Introduction

The Yang–Baxter equation emerged from a theoretical physics article written by the Nobel laureate C.N. Yang [1] and from the statistical mechanics work of R.J. Baxter. [2,3]. Besides several Special Issues on the Yang–Baxter equation published in the journal Axioms (see, for example [4]), we would like to mention some other recent papers on this equation: [5,6,7,8,9,10,11].

In Knot Theory, invariants of links can be obtained from “enhanced” Yang–Baxter operators [12,13]. For example, the Jones polynomial [14], the Homflypt polynomial [15,16] and the Kauffman polynomial [17] are produced from “enhanced” Yang–Baxter operators.

Attempts to unify the algebra structures and the coalgebra structures have eventually led to new solutions of the Yang–Baxter equation [18]. The only invariant which can be obtained from these Yang–Baxter operators is the Alexander polynomial of knots [19].

The unification of the algebra structures and the coalgebra structures leads to an extension for the duality between finite dimensional algebras and finite dimensional coalgebras. We will develop this point of view below.

We now refer to another notably duality extension, which starts with the Pontryagin duality. In this case, we have a duality between the category of compact Hausdorff Abelian groups and the category of discrete Abelian groups. There exists a duality extension given be the Pontryagin–van Kampen duality theorem. This theorem extends the duality between the category of compact Hausdorff Abelian groups and the category of discrete Abelian groups to the self-duality of all locally compact Hausdorff Abelian topological groups. So, the compact Hausdorff Abelian groups and discrete Abelian groups are also locally compact Hausdorff Abelian topological groups (we recommend the book [20]). This duality extension is illustrated in the following diagram:

$$\begin{array}{ccc}\mathbf{LCA}& \underset{(-,T)}{\overset{(-,T)}{\overleftarrow{{\displaystyle \to }}}}& \mathbf{LCA}\\ {\displaystyle \uparrow }& & {\displaystyle \uparrow }\\ \mathbf{Cpct}& \underset{(-,T)}{\overset{(-,T)}{\overleftarrow{{\displaystyle \to }}}}& \mathbf{Disc}.\end{array}$$

With the Pontryagin–van Kampen duality theorem in mind, we consider the unification of the algebra structures and the coalgebra structures as an extension for the duality between finite dimensional algebras and coalgebras to the category of finite dimensional Yang–Baxter structures [21]. The resulting duality theorem can be illustrated by the following diagram:

$$\begin{array}{ccc}\mathbf{f}.\mathbf{d}.\mathbf{YB}\mathbf{str}.& \underset{D=\left(\right)*}{\overset{D=\left(\right)*}{\overleftarrow{{\displaystyle \to }}}}& \mathbf{f}.\mathbf{d}.\mathbf{YB}\mathbf{str}.\\ F{\displaystyle \uparrow }& & {\displaystyle \uparrow }G\\ \mathbf{f}.\mathbf{d}.k-\mathbf{alg}.& \underset{\left(\right)*}{\overset{\left(\right)*}{\overleftarrow{{\displaystyle \to }}}}& \mathbf{f}.\mathbf{d}.k-\mathbf{coalg}.\end{array}$$

Another example of duality extension was given for the duality between right finitely generated projective ring extensions and right finitely generated projective corings. This duality was extended to the category of right finitely generated projective generalized Yang–Baxter structures [22]. So, we constructed a new category with a self-dual functor acting on it, which extended the initial duality. That construction could be seen as the non-commutative generalization of the above duality extension. It is easier to explain this duality extension by the following diagram.

$$\begin{array}{ccc}\mathbf{r}.\mathbf{f}.\mathbf{g}.\mathbf{p}\mathbf{YB}{\mathbf{str}}_B& \underset{\left(\right)*}{\overset{\left(\right)*}{\overleftarrow{{\displaystyle \to }}}}& \mathbf{r}.\mathbf{f}.\mathbf{g}.\mathbf{p}\mathbf{YB}{\mathbf{str}}_{B^{\mathrm{op}}}\\ F{\displaystyle \uparrow }& & {\displaystyle \uparrow }G\\ \mathbf{r}.\mathbf{f}.\mathbf{g}.\mathbf{p}\mathbf{R}\mathbf{g}{\mathbf{e}}_B& \underset{\left(\right)*}{\overset{\left(\right)*}{\overleftarrow{{\displaystyle \to }}}}& \mathbf{r}.\mathbf{f}.\mathbf{g}.\mathbf{p}\mathbf{C}\mathbf{r}{\mathbf{g}}_{B^{\mathrm{op}}}.\end{array}$$

In this paper, we will continue our approach on the theory of the unification of the algebra structures and the coalgebra structures. In order to capture the information encapsulated in modules over algebras and comodules over coalgebras, we will need some kind of extensions of the Yang–Baxter equation, called Yang–Baxter systems. There are several types of Yang–Baxter systems, and many times they could be used to construct a Yang–Baxter operator. So, one can return with this new Yang–Baxter operator to Knot Theory and try to find new invariants.

Furthermore, the Tzitzeica–Johnson theorem is generalized in the current paper. Our theorem could be understood as a unification of the Tzitzeica–Johnson theorem and the dual Tzitzeica–Johnson theorem.

We could think of this duality procedure as a continuous transformation. This idea is illustrated by the following diagram:

$$\begin{array}{ccc}\mathbf{Our}\mathbf{new}\mathbf{theorem}& \overleftarrow{{\displaystyle \to }}& \mathbf{Our}\mathbf{new}\mathbf{theorem}\\ {\displaystyle \uparrow }& & {\displaystyle \uparrow }\\ \mathbf{Tzitzeica}-\mathbf{Johnson}\mathbf{theorem}& \overleftarrow{{\displaystyle \to }}& \mathbf{Dual}\mathbf{Tzitzeica}-\mathbf{Johnson}\mathbf{thm}.\end{array}$$

The current paper is organized as follows. In the next section, we review the terminology related to the Yang–Baxter equation and Yang–Baxter systems (WXZ systems). We also discuss the applications of WXZ systems. Section 3 deals with the unification of modules over algebras and comodules over coalgebras. In this case, we will use another type of Yang–Baxter system. Section 4 presents the unification of the the Tzitzeica–Johnson theorem and the dual Tzitzeica–Johnson theorem. There exists a duality procedure: points from a given picture are replaced with circles of a certain radius, while the circles of that given radius are replaced by points. In this way, we obtain the dual of the Tzitzeica–Johnson theorem. However, the operation of taking the dual can be extended for circles of different radii, as we will see in our new theorem.

The sections on the Yang–Baxter equations and Yang–Baxter systems are more technical, while Section 4 is more illustrative. These topics are related to (differential) geometry, as we will explain in our last section. We also try to recreate the atmosphere of the discussions with Professor Stefan Papadima, but deeper connections might be included in future works. Furthermore, we appeal to imagery in order to burst the imagination of the reader.

Several conclusions and explanations are given at the end of the paper.

2. Yang–Baxter Equations and Yang–Baxter Systems

As usual, we work over a field k, and our tensor products are defined over k. Let $I=I_V:V\to V$ be the identity map of the k-space V. For $R:V\otimes V\to V\otimes V$ a k-linear map, let $R^{12}=R\otimes I,R^{23}=I\otimes R:V\otimes V\otimes V\to V\otimes V\otimes V.$

Let $R^{13}$ be the linear map acting on the first and third components of $V\otimes V\otimes V$.

Definition 1. 

A Yang–Baxter operator is an invertible k-linear map $R:V\otimes V\to V\otimes V$, which satisfies the braid condition (the Yang–Baxter equation):

$$R^{12}\circ R^{23}\circ R^{12}=R^{23}\circ R^{12}\circ R^{23}.$$

(1)

Remark 1. 

The quantum Yang–Baxter equation is the following:

$$R^{12}\circ R^{13}\circ R^{23}=R^{23}\circ R^{13}\circ R^{12}.$$

(2)

Remark 2. 

It is well known that (1) and (2) are equivalent.

Remark 3. 

If A is a k-algebra, then for all non-zero $r,s\in k$, the linear map

$$R_{r,s}^A:A\otimes A\to A\otimes A,a\otimes b\mapsto sab\otimes 1+r1\otimes ab-sa\otimes b$$

(3)

is a Yang–Baxter operator [23].

Definition 2. 

Let $V,V^{\prime },V^{″}$ be k-spaces. Let $R:V\otimes V^{\prime }\to V\otimes V^{\prime }$, $S:V\otimes V^{″}\to V\otimes V^{″}$ and $T:V^{\prime }\otimes V^{″}\to V^{\prime }\otimes V^{″}$ be k-maps. A Yang–Baxter commutator is a map $[R,S,T]:V\otimes V^{\prime }\otimes V^{″}\to V\otimes V^{\prime }\otimes V^{″}$, defined by

$$[R,S,T]=R^{12}\circ S^{13}\circ T^{23}-T^{23}\circ S^{13}\circ R^{12}.$$

(4)

There are several types of Yang–Baxter systems, and in this section we consider the following example and its implications.

Definition 3. 

For V and $V^{\prime }$ k–spaces, we consider the following maps:

$$W:V\otimes V\to V\otimes V,Z:V^{\prime }\otimes V^{\prime }\to V^{\prime }\otimes V^{\prime },X:V\otimes V^{\prime }\to V\otimes V^{\prime }$$

A WXZ-system verifies the following equations:

$$[W,W,W]=0,$$

(5)

$$[Z,Z,Z]=0,$$

(6)

$$[W,X,X]=0,$$

(7)

$$[X,X,Z]=0.$$

(8)

Of course, the relations (5) and (6) are just quantum Yang–Baxter equations. There are many applications of WXZ-systems (see, for example, [24,25]). Furthermore, they can be used to construct dually paired bialgebras of the FRT type [26] or algebra factorizations of generalized tensor algebras [27].

Given a WXZ-system as in Definition 3, one can construct a Yang–Baxter operator on $V\oplus V^{\prime }$, provided the map X is invertible. This is a special case of a gluing procedure described in [28], Theorem 2.7 (cf. [28], Example 2.11).

Entwining structures were introduced in order to recapture the symmetry structure of non-commutative (coalgebra) principal bundles or coalgebra-Galois extensions.

Theorem 1 

([29]). Let A be an algebra and let C be a coalgebra. For any $s,r,t,p\in k$, define linear maps

$$W:A\otimes A\to A\otimes A,a\otimes b\mapsto sba\otimes 1+r1\otimes ba-sb\otimes a,$$

$$Z:C\otimes C\to C\otimes C,c\otimes d\mapsto t\epsilon \left(c\right)\sum d_{\left(1\right)}\otimes d_{\left(2\right)}+p\epsilon \left(d\right)\sum c_{\left(1\right)}\otimes c_{\left(2\right)}-pd\otimes c.$$

Let $X:A\otimes C\to A\otimes C$ be a linear map such that $X\circ (\iota \otimes {\mathit{Id}}_C)=\iota \otimes {\mathit{Id}}_C$ and $({\mathit{Id}}_A\otimes \epsilon )\circ X={\mathit{Id}}_A\otimes \epsilon$. Then $W,X,Z$ is a Yang–Baxter system if and only if A is entwined with C by the map $\psi :=X\circ {\tau }_{C,A}$.

3. Unifications Using Yang–Baxter Systems

In this section, we propose another kind of Yang–Baxter system, with the purpose to unify the following theorems:

(i) For an A–bimodule, M, $A\times M$ becomes an algebra.

(ii) For a C–bicomodule, N, $C\times N$ becomes a coalgebra.

Theorem 2. 

Let V and $V^{\prime }$ be vector spaces, and

$$W:V\otimes V\to V\otimes V,X:V\otimes V^{\prime }\to V\otimes V^{\prime },Y:V^{\prime }\otimes V\to V^{\prime }\otimes V$$

such that the following equations are satisfied:

$$[W,W,W]=0,$$

(9)

$$[W,X,X]=0,$$

(10)

$$[Y,Y,W]=0,$$

(11)

$$[X,W,Y]=0.$$

(12)

Then, the linear map $R:(V\oplus V^{\prime })\otimes (V\oplus V^{\prime })\to (V\oplus V^{\prime })\otimes (V\oplus V^{\prime })$, given by ${R|}_{V\otimes V}=W$, ${R|}_{V\otimes V^{\prime }}=X$, ${R|}_{V^{\prime }\otimes V}=Y$, and ${R|}_{V^{\prime }\otimes V^{\prime }}=0$, has the property $[R,R,R]=0$.

Proof. 

The proof is based on observations on the decomposition of ${(V\oplus V^{\prime })}^{\otimes 3}$ into direct summands of tensor products. More precisely, we write ${(V\oplus V^{\prime })}^{\otimes 3}=(V\otimes V\otimes V)\oplus (V\otimes V\otimes V^{\prime })\oplus (V\otimes V^{\prime }\otimes V)\oplus (V\otimes V^{\prime }\otimes V^{\prime })\oplus (V^{\prime }\otimes V\otimes V)\oplus$ $(V^{\prime }\otimes V\otimes V^{\prime })\oplus (V^{\prime }\otimes V^{\prime }\otimes V)\oplus (V^{\prime }\otimes V^{\prime }\otimes V^{\prime })$.

The only operator which acts on $V\otimes V\otimes V$ is W. Therefore, on $V\otimes V\otimes V$, the fact that the operator R satisfies the quantum Yang–Baxter equation will imply $[W,W,W]=0$.

Likewise, on $V\otimes V\otimes V^{\prime }$, we obtain $[W,X,X]=0$.

On $V\otimes V^{\prime }\otimes V$, we have $[X,W,Y]=0$; on $V^{\prime }\otimes V\otimes V$, and we obtain $[Y,Y,W]=0$.

All the other terms will eventually vanish. □

Remark 4. 

If A is an associtive algebra, $W:A\otimes A\to A\otimes A,a\otimes b\mapsto ab\otimes 1+1\otimes ab-b\otimes a$, M an A-bimodule, $X:A\otimes M\to A\otimes M,a\otimes m\mapsto 1\otimes am$, and $Y:M\otimes A\to M\otimes A,n\otimes b\mapsto nb\otimes 1$, then we have the conditions of the above theorem fulfilled.

Indeed, according to Remark 3, $W(a\otimes b)=ab\otimes 1+1\otimes ab-b\otimes a$ verifies $[W,W,W]=0$. Notice that a twist operator was applied to $R_{r,s}^A$, and that $r=s=1$. The other conditions could be checked directly.

Remark 5. 

Let C be a coalgebra, $W:C\otimes C\to C\otimes C,c\otimes d\mapsto c_1\otimes c_2\epsilon \left(d\right)+\epsilon \left(c\right)d_1\otimes d_2-d\otimes c$, N a C-bicomodule, $X:C\otimes N\to C\otimes N,c\otimes n\mapsto \epsilon \left(c\right)n_{-1}\otimes n_0$, and $Y:N\otimes C\to N\otimes C,n\otimes d\mapsto n_0\otimes n_1\epsilon \left(d\right)$, then we have the conditions of the above theorem fulfilled.

Indeed, this remark is dual to the previous one. The other way is to check the conditions directly.

4. On the Tzitzeica–Johnson Theorem

Let us start with the Tzitzeica–Johnson problem (see [30,31,32,33,34]).

We consider three circles of radius r that intersect at a single point O. The intersection points of pairs of circles are denoted by A, B and C. Then, the circle through the points A, B and C also has radius r (see Figure 1 and Figure 2 below).

We now propose a new theorem.

We start with a circle of radius r. Next, we construct three circles of radius R, tangent to the circle of radius r. We then consider three circles of radius r tangent to pairs of circles of radius R (see Figure 3).

At this moment, we can present our Theorem: under the above assumptions, there exists a circle of radius R, tangent to the three new r-circles (see Figure 4).

The proof of our theorem is based on reducing the above figure to the Tzitzeica–Johnson theorem for circles of radius $r+R$.

More precisely, in Figure 2 we consider three circles of radius $r+R$, which intersect at a single point O (the center of the circle of radius r). We now apply the Tzitzeica–Johnson theorem, and we obtain a fourth circle of radius $r+R$. It is now easy to revert to the picture from Figure 4 (there exist four circles of radius R and four circles of radius r).

One can construct dual pictures (see Figure 5 and Figure 6 below).

Furthermore, the conclusion is the following.

The limit cases (either $r\to 0$ or $R\to 0$) lead to the Tzitzeica–Johnson theorem and the dual Tzitzeica–Johnson theorem (see Figure 7 and Figure 8).

5. Conclusions and Further Comments

The main objectives of this work are to unify algebraic structures and geometric theorems. These approaches lead to generalizations of classical theorems.

The first objective is related to the search for solutions of the Yang–Baxter equation. In this case, the initial motivation was to unify modules and comodules. It would be interesting to look for similar results in the Lie algebra theory: the unification of Lie modules and Lie comodules.

The second objective concerns the famous Tzitzeica–Johnson theorem, considered a wonder of geometry. The unification of the the Tzitzeica–Johnson theorem and the dual Tzitzeica–Johnson theorem could be understood as a duality extension problem. More precisely, there exists a duality procedure, in which points from a given figure are replaced with circles, while circles (of specified radius) are replaced by points. In this way, we obtained the dual of the Tzitzeica–Johnson theorem. However, the operation of taking the dual can be extended for circles of different radii, leading to a continuous process.

The sections on the Yang–Baxter equations and Yang–Baxter systems are more technical, while the section on the Tzitzeica–Johnson theorem is more illustrative.

Some of the results of this paper were presented at the 14th International Workshop on Differential Geometry and Its Applications, hosted by the Petroleum Gas University from Ploiesti, between 9 July and 11 July 2019. Even the unification of the modules and comodules was inspired by differential geometry. That initial problem was to relate the vector fields on a manifold and the forms on the same manifold.

The fruitful interactions with the participants motivated us to elaborate this paper, and could lead to future collaborations. For example, Professor Nicu Anghel (Texas) was interested in Tzitzeica–Johnson theorems.