On Jordan algebras and some unification results

This paper is based on a talk given at the 14-th International Workshop on Differential Geometry and Its Applications, hosted by the Petroleum Gas University from Ploiesti, between July 9-th and July 11-th, 2019. After presenting some historical facts, we will consider some geometry problems related to unification approaches. Jordan algebras and Lie algebras are the main non-associative structures. Attempts to unify non-associative algebras and associative algebras led to UJLA structures. Another algebraic structure which unifies non-associative algebras and associative algebras is the Yang-Baxter equation. We will review topics relared to the Yang-Baxter equation and Yang-Baxter systems, with the goal to unify constructions from Differential Geometry.


Introduction
This paper is a survey paper, but it also presents new results. It emerged after our lectures given at the 9-th Congress of Romanian Mathematicians (CRM 9, Galati, Romania, June-July 2019) and 14-th International Workshop on Differential Geometry and Its Applications (DGA 14, UPG Ploiesti, July 2019).
We begin with some historical remarks. Vrȃnceanu proposed the study of spaces with constant affine connection associated to finite-dimensional real Jordan algebras in 1966. This study was afterwards developed by Iordanescu, Popovici and Turtoi (see the Comments on page 31 from [14]). Dan Barbilian proved that the rings which can be the underlying rings for projective geometries are (with a few exceptions) rings with a unit element in which any one-sided inverse is a two-sided inverse (see [14]). Another big Romanian geometer was G. Tzitzeica, and some of his contributions to mathematics will be presented in our next section.
A substantial part of this paper is dedicated to unifying structures. The apparition of the Yang-Baxter equation ( [36]) in theoretical physics and statistical mechanics (see [44,2,3]) has led to many applications in these fields and in quantum groups, quantum computing, knot theory, braided categories, analysis of integrable systems, quantum mechanics, etc (see [34]). The interest in this equation is growing, as new properties of it are found, and its solutions are not classified yet. This equation can be understood as an unifying equation (see, for example, [35,20,21,15]). Another unification of 1 non-associative structures, was obtained using the UJLA structures ( [15,16]), which could be seen as structures which comprise the information encapsulated in associative algebras, Lie algebras and Jordan algebras.
We now refer to important results obtained under the guidance of St. Papadima. Some presentations at DGA 14 were dedicated to his memory (see [4,38,28,41]). It was observed during DGA 14, that some results from [38] can be extended for the virtual braid groups from [4]. Also, the techniques of [41] can be considered in the framework of UJLA structures. The Yang-Baxter equation plays an important role in knot theory. Dȃscȃlescu and Nichita have shown in [8] how to associate a Yang-Baxter operator to any algebra structure over a vector space, using the associativity of the multiplication. Turaev has described in [43] a general scheme to derive an invariant of oriented links from a Yang-Baxter operator, provided this one can be "enhanced". The invariant which was obtained from those Yang-Baxter operators is the Alexander polynomial of knots. Thus, in a way, the Alexander polynomial is the knot invariant corresponding to the axioms of (unitary associative) algebras.
The current paper is organised as follows. The next section deals with Tzitzeica-Johnson's theorem. In Sections 3 and 4 we present the Yang-Baxter equations and the Yang-Baxter systems (including the set-theoretical Yang-Baxter equation). These will be followed by a section on Jordan algebras, UJLA structures and unification constructions in Differential Geometry. A conclusion section will end this article.

On Tzitzeica-Johnson's theorem and pictorial mathematics
In his talk at DGA 12, Florin Caragiu explained that there exists a special mathematical discourse, called "proofs without words", which uses pictures or diagrams in order to boost the intuition of the reader (see [7]). Pictorial (diagrammatic) style of mathematical language is much appreciated by both educators and researchers in mathematics (see, also, [26]). Very easy to be grasped, some pictorial style problems need an entirely "artillery" in order to be cracked.
We now consider the Tzitzeica-Johnson's problem (see, for example, [26]). We start with three circles of the same radius r. The intersection points of pairs of circles are denoted by A, B, C and the common point of intersection of the three circles is denoted by O (see the Figure 1      The proof of the above theorem is based on the properties of the power of a circle and on the Desargues' theorem. It is the limit case of the following picture. The above problems can be included in a more general scheme (see the Figure 5). The dual of the Desargues' Theorem can be interpretated in the light of the above scheme by using arrangements of (three) colored lines.
The hypothesis: There exist a triple (l, m, n) through A, A , B, B , C, C .
From the phylosophical point of view, one could include in this scheme the inclussionexclusion principle: Let us explain this analogy. Once we have information about the cardinality of X, Y, Z, X ∩ Y, X ∩ Z, Y ∩ Z and X ∩ Y ∩ Z, then there exists a T , related to X, Y and Z, whose cardinality can be also computed. Of course, in this case T = X ∪Y ∪Z.
Notice that Tzitzeica-Johnson's Theorem has many generalizations ( [26]) and interpretations related to the above mathematical scheme (from Figure 5).
Thus, Tzitzeica-Johnson's Theorem can be interpretated in terms of disks: (i) "If three disks of the same radius have a common point of intersection, then they contain inside of their union a forth disk with the same radius." (ii) "If three disks X , Y and Z of the same radius have a common point, then there exists a forth disk of the same radius which includes ( It is an open problem to prove a similar statement for a domain bounded by an arbitary closed convex curve. In other words we conjecture that if we consider a domain X bounded by an arbitary closed convex curve, and two copies of X , denoted by Y and Z such that X ∩ Y ∩ Z = ∅, then there exists another copy of X , denoted by T such that ( We now consider other types of geometry problems. In his best-seller book [10], P. Greco (see [37] -there is a puzzle there!) refers to a problem which can be generalized as follows.
OPEN PROBLEMS. For an arbitrary convex closed curve, we consider the length of largest barycentric diameter (D). Here, the largest barycentric diameter is the longest segment through the center of mass with the endpoints on the given curve. In a similar manner, one can define the smallest barycentric diameter (d).
i) If L is the length of the given curve and the domain inside the given curve is a convex set, then we conjecture that: (ii) If A is the area inside the given curve, the equation and its implications are not completely understood. For example, if the given curve is an ellipse, solving this equation in terms of the semi-axes of the ellipse is an unsolved problem.
The second part of the above conjecture is related to [1].
Another idea related to [10] and to the Euler's relation, e πi + 1 = 0 , is the following relation containing π: |e i − π| > e. It is very good approximation, and it has a pictorial interpretation (see the Figure 6).

Yang-Baxter equations
In the next sections, we will work over a generic field k. The tensor products will be defined over k.
This section was inspired by two papers, [25,42], and it begins with an introduction to the Yang-Baxter equation.
Let V be a vector space over k.
We denote by τ : Now, we will give the main notations for introducing the Yang-Baxter equation.
In a similar manner, we denote by R 13 a linear map acting on the first and third component of which satisfies the braid condition (the Yang-Baxter equation): We also require that the map R is invertible.

Remark 3.2.
Some examples of Yang-Baxter operators are the following:
Remark 3.5. The following construction of Yang-Baxter operators is described in [8].
If A is a k-algebra, then for all non-zero r, s ∈ k, the linear map Turaev has described in [43] a general scheme to derive an invariant of oriented links from a Yang-Baxter operator, provided this one can be "enhanced". The Jones polynomial [18] and its two-variable extensions, namely the Homflypt polynomial [9,39] and the Kauffman polynomial [19], can be obtained in that way by "enhancing" some Yang-Baxter operators obtained in [17]. Those solutions of the Yang-Baxter equation are associated to simple Lie algebras and their fundamental representations. The Alexander polynomial can be derived from a Yang-Baxter operator as well, using a slight modification of Turaev's construction [27].
The only invariant which can be obtained from the Yang-Baxter operators (4) is the Alexander polynomial of knots. Thus, in a way, the Alexander polynomial is the knot invariant corresponding to the axioms of (unitary associative) algebras (cf. [24]).
Note that specializations of the Homflypt polynomial had to be expected from those Yang-Baxter operators since they have degree 2 minimal polynomials.

Remark 3.6.
There is a similar terminology for the set-theoretical Yang-Baxter equation. In this case V is replaced by a set X and the tensor product by the Cartesian product. We will explain this definition in the next examples below.
Let X be a set containing three logical sentences p, q, r (i.e., p, q, r ∈ X). We can choose X as rich as we wish for the moment. Later, we will try to find the smallest X which fits for our theory.
It is now the moment to discuss about X. Open problems. What can be said about X in general? What is the smallest X for which R is well-defined? One could consider a set X containing more than three logical sentences. What is the interpretation of the set-theoretical Yang-Baxter solutions in this case ?
We can go a step further and consider an algebra of type (2,2), (A, * , •), and call the operations * and • YB conjugated if R(a, b) = (a  *  b, a • b) satisfies the set-theoretical Yang-Baxter equation.
We propose the study of algebras with two operations which are YB conjugated. Groups, distributive lattices, and self-distributive laws can be considered as objects with operations which are YB conjugated (compare with [5]). (7) and (8) then R is a common solution for (2) and (3). (2)

and (3) then it is a solution for (9). If a k-map R : V ⊗ V → V ⊗ V verifies (2) or (3) then R is a solution for (10). If a k-map R : V ⊗ V → V ⊗ V verifies (2) or (3) then it is a solution for (11).
Proof. We only prove the first claim: The other claims follow in a similar manner.
We will write the above results in the following manner:

Remark 3.10. The set of equations of operators of type
, has a natural distributive latice structure (see [25]).

Yang-Baxter systems
We present the Yang-Baxter systems theory following the paper [6]. Yang-Baxter systems were introduced in [11] as a spectral-parameter independent generalisation of quantum Yang-Baxter equations related to non-ultralocal integrable systems.
Yang-Baxter systems are conveniently defined in terms of Yang-Baxter commutators. Consider three vector spaces V, V , V and three linear maps R : Then a Yang-Baxter commutator is a map [R, S, T ] : In terms of a Yang-Baxter commutator, the quantum Yang-Baxter equation (3) is expressed simply as [R, R, R] = 0.

Definition 4.1. Let V and V be vector spaces. A system of linear maps
is called a WXZ-system or a Yang-Baxter system, provided the following equations are satisified: There are several algebraic origins and applications of WXZ-systems. WXZ-systems with invertible W , X and Z can be used to construct dually-paired bialgebras of the FRT type, thus leading to quantum doubles.
More precisely, consider a WXZ-system with finite-dimensional V = V , so that each of W , X, Z is an N 2 × N 2 -matrix. Suppose that W, X, Z are invertible. Since W and Z satisfy Yang-Baxter equations (15)- (16), one can consider two matrix bialgebras A and B with N × N matrices of generators U and T respectively, and relations The existence of an invertible operator X that satisfies equations (17)- (18), means that A and B are dually paired with a non-degenerate pairing < U 1 , T 2 >= X 12 . Furthermore, the tensor product A ⊗ B has an algebra (quantum double) structure with crossed relations X 12 U 1 T 2 = T 2 U 1 X 12 .
Given a WXZ-system as in Definition 4.1 one can construct a Yang-Baxter operator on V ⊕ V , provided the map X is invertible. This is a special case of a gluing procedure described in [ Then the linear map is a Yang-Baxter operator.
Entwining structures were introduced in order to recapture the symmetry structure of non-commutative (coalgebra) principal bundles or coalgebra-Galois extensions.

Definition 4.2.
An algebra A is said to be entwined with a coalgebra C if there exists a linear map ψ : C ⊗ A → A ⊗ C satisfying the following four conditions: The map ψ is known as an entwining map, and the triple (A, C) ψ is called an entwining structure.

Remark 4.3.
To denote the action of an entwining map ψ on elements it is convenient to use the following α-notation, for all a, b ∈ A and c ∈ C,

Theorem 4.4. ([6]) Let
A be an algebra and let C be a coalgebra. For any s, r, t, p ∈ k define linear maps Then W, X, Z is a Yang-Baxter system if and only if A is entwined with C by the map ψ := X • τ C,A .
We have introduced some structures which unify Jordan algebras, Lie algebras and (non-unital) associative algebras. These structures were called UJLA (from "unification", "Jordan", "Lie" and "associative") structures, and one could "decode" the results obtained for UJLA structures in results for Jordan algebras, Lie algebras or (non-unital) associative algebras.
Changhing the perspective, one can consider the UJLA structures as generalizations of Jordan algebras. [ , ] The study of filiform Lie algebras ( [40]) can be extended to filiform UJLA structures.
Definition 5.1. We have defined the unifying structure (V, η), also called a "UJLA structure", in the following way. Let V be a vector space, and η : V ⊗ V → V, η(a ⊗ b) = ab, be a linear map, which satisfies the following axioms ∀a, b, c ∈ V : a(a 2 b).

OPEN PROBLEM.
A UJLA structure is power associative. We know that a UJLA structure is power associative for dimensions less or equal to 5.
If C is a coalgebra, d : C → C a coderivative, and ψ : If τ is the twist map, the condition represents the unification of the comutativity and the co-comutativity conditions. In other words, if the algebra A is comutative, then φ verifies the above condition. If the coalgebra C is cocomutative, then ψ verifies the same condition.  D(m)) is a derivative in this algebra.
Translated into the "language" of Differential Geometry, the above theorem says that the Lie derivative is a derivative (i. e., d(ab) = d(a)b + ad(b) ) on the product of the algebra of functions defined on the manifold M with the set of vector fields on M (see [22]).

Remark 5.13.
A dual construction would refer to a coalgebra structure, ∆ : A → A ⊗ A, f → f ⊗ 1 + 1 ⊗ f , and a comodule structure on forms, ρ : A × Ω becomes a coalgebra with the following comultiplication: We can see now that the Lie derivative is a coderivative with the above comultiplication, ∆. The key ingredient in the above remark is a theorem given below. Dually, W (f ⊗ g) = ε(f ) ⊗ ∆(g), X(f ⊗ v) = ε(f ) ⊗ ρ(v) and Z(v ⊗ w) = 0 forms a Yang-Baxter systems.
We have explained that gluing procedures for Yang-Baxter systems lead to Yang-Baxter operators. The algebraic part of this unification is presented below. More details and interpretations will follows in later studies.

Theorem 5.16. Let V and V be vector spaces, and
such that the following equations are satisified: (29) [W, W, W ] = 0 , [W, X, X] = 0 ,  Proof. The proof is based on observations on the decomposition of (V ⊕ V ) ⊗3 into direct summands of tensor products.

Conclusions
We proposed an open problem which generalities Tzitzeica-Johnson's theorem: the circles from that theorem are replaced by an arbitrary closed convex curve, and the conclusion is slightly weakened.
The story of the number π was then continued with a new episode. A review (and new results) on Yang-Baxter systems/equations were needed for unifying two constructions. Theorem 5.16 unifyies Theorem 5.12 (i) and Theorem 5.14.
The Vrȃnceanu -Vergne basis was mentioned in a talk at DGA 13, which later became a paper ( [40]).