On the Ternary Approach to Cliﬀord Structures and Ising Lattices

. We continue to modify and simplify the Ising-Onsager-Zhang procedure for analyzing simple orthorhombic Ising lattices by consider-ing some fractal structures in connection with Jordan and Cliﬀord algebras and by following Jordan-von Neumann-Wigner (JNW) approach. We concentrate on duality of complete and perfect JNW-systems, in particular ternary systems, analyze algebras of complete JNW-systems, and prove that in the case of a composition algebra we have a self-dual perfect JNW-system related to quaternion or octonion algebras. In this context, we are interested in the product table of the sedenion algebra. Mathematics Subject Classiﬁcation (2010). Primary 82C44; Secondary 82D25, 81R05, 15A66.


Introduction and the organization of the paper
After recalling Jordan-von-Neumann-Wigner (JNW) procedure and Ising-Onsager-Zhang (IOZ) approach, we discuss a JNW-system and related fractal structures. Then, we turn our attention to the duality structure of complete and perfect ternary systems. We consider algebras of JNW-systems, in particular algebras of Clifford type. We conclude with a study of a composition algebra and self-dual perfect JNW-systems. Continuing our previous papers [10,11,13] we prove in this case it is one of quaternion and octonion algebras, and this needs a study of the product table of sedenion algebra. The idea of applying the JNW-systems, considered in Section 3 and thereafter, to binary and ternary alloys may be to some extent illustrated for instance by the first step of fractal construction related to an AB 3 binary alloy of an fcc lattice and (111) surface orientation [2, Fig. 1]. The choice in this first step of the fractal construction is tightly related to the theory of cooperative phenomena of Kikuchi [9]. From our point of view it is important that like in the case of modern building model constructions, Kikuchi takes into account collections of adjacent edges, mostly three or four, of the crystal lattice. This suggests our subsequent steps: duality of fractals, duality fractals, JNW-systems, duality structure of complete JNW-systems, composition algebras of self-dual perfect JNW-systems, etc. These ideas can also be expressed by the following scheme:

Advances in Applied Cliff ord Algebras
fractals including their duality ←→ binary and ternary alloys Vol. 22 (2012) On the Ternary Approach to Cliff ord Structures 759 The content of the paper may be visualized as follows:

Jordan-von Neumann-Wigner procedure
Jordan, von Neumann, and Wigner proved [7] (cf. also [4][5][6]) that among irreducible algebras there are algebras H q (F) of Hermitian q × q-matrices with entries in a field F, namely As it was observed in [11, p. 737], and then repeated in [21, p. 7], the natural appearance of multiplication instead of the usual matrix multiplication replaces in elegant way the desire of Ising [3], Onsager [16], Zhang [20], and Staples [18] of finding commutative subalgebras of the given algebra and studying their properties related to the superposition of bilinear forms.

Ising-Onsager-Zhang approach
The models of Ising [3] and Onsager [16] are extended by Zhang [20] so that the resulting model applies to a simple orthorhombic crystal lattice in a threedimensional space. This extension involves, in particular, opening knots by a rotation in a higher dimensional space, and studying important commutators in the corresponding algebra. In our approach, we extend Zhang's idea of decomposing knots related to lattice vertices to deformations of formal knot sequences arising from Reidemeister moves [19]. The model consisting of an orthombic lattice constructed by m rows and n sites per row in one of the planes involves [11] 2 n·dimensional matrices s α r,s , α = 1, 2, 3, 1 ≤ s ≤ , 1 ≤ r ≤ n, related to the so-called split-quaternions and, correspondingly, to the familiar Pauli matrices namely: It is worthwhile to notice that both Onsager and Zhang analyze continued fractions and nested series, e.g., which give rise to two kinds of fractals: of flower type and of branch type, and to the problem of their duality [8,10]. The objects of type (3.1) appear as a result of calculations in [16] and [20], but they inspire in us a relationship with the two types of fractals mentioned.

Jordan-von Neumann-Wigner system
We begin by introducing two kinds of systems for a set {e 1 , e 2 , . . . , e n }. (4.1) The first system is called a triad. It is obtained by choosing three distinct elements from the set (4.1), for example, (e 1 , e 2 , e 3 ). The second system is our JNW-system: we choose two elements, for example, (e i , e j ). Then, we select the third element, for example, e k / ∈ {e i , e j }, and by doing so we define a JNW-element: the triple (e i , e j , e k ). The most important elements arise from Vol. 22 (2012) On the Ternary Approach to Cliff ord Structures 7 61 the binary product of an algebra. All JNW-elements selected from (4.1) make up a JNW-system.

Fractal structure of Jordan-von Neumann-Wigner procedure
We recall the following result from [11].
The relation * 1 , connected with the vertex e 1 , is visualized in Fig. 3. Repeating this process and making a network, understood as a connected system of non intersecting triangles, such that each pair of neighboring element has a common vertex or common site, we can obtain a complete set of all perfect JNW-systems, in question.
It is clear that the structure in question is perfectly suitable for investigating ternary alloys without vacancies, binary, as well as ternary alloys with vacancies (cf. the triangle (1,8,10) in Fig. 3(c)).

Duality structure of complete JNW-systems and the characterization of self-dual perfect JNW-systems
In this section we shall determine a duality structure for complete ternary systems and introduce a concept of a self-dual perfect JWN-system which characterizes the composition algebra directly. At first, we describe the duality structure for a complete JNW-system.

The duality structure of complete ternary systems
We discuss the duality structure of complete ternary systems. By use of the basic construction of complete ternary systems, let us define Λ : Λ(1) = 1, Λ(t) = 2t, 2 ≤ t ≤ 2 k−1 .
Hence, we have a mapping called a duality mapping of genetic type, cf. [12]. We refer to each of the following three sets as a branch of the system. A ternary system is called self dual when each branch of the system is preserved under the duality mapping.

The dual structure of perfect ternary systems
We are going to discuss the duality structure of perfect ternary systems: Hence, we have the mapping which is natural to call duality of JNW-type. Moreover, we have the following theorem.

Algebras of JNW-systems
We are going to introduce algebras of a JNW-system. Here we restrict ourselves to a special class of the following unital real (not necessarily associative) algebras.

Definition 8.1.
A finitely generated algebra A = R[e 1 , e 2 , . . . , e n ] is said to be of monic product type if, for any two generators e i , e j ∈ A, we have e i e j = η ij e k for some generator e k and η ij ∈ R.
We concentrate on algebras of a perfect JNW-system. We begin with the following definition. In the case when A is an associative, it becomes a Clifford algebra when it additionally satisfies the following condition: e 2 j = η j e i for some i and η j ∈ R. The following theorem has been proven in [11]: (B1) Let A be a finitely generated algebra A = R[e 1 , e 2 , . . . , e n ] of Clifford type. Then the JNW-system of A is perfect. (B2) Conversely, choosing a perfect system, we can find an algebra of Clifford type which realizes the system.
We illustrate Theorem B with a few examples.  Table 5.

Composition algebra and self-dual perfect JNW-systems
Finally, we characterize composition algebras in terms of the JNW-systems. At first, we recall the definition of a composition algebra.  Table 1. Sedenion algebra Definition 9.2. An algebra A with a norm N : A → R + is said to be ternary of composition type if its two ternaries (e i , e j , e k ) and (e r , e s , e t ) have a common vertex, for example e k = e t . When studying the commutation relations we may use the notation (e i , e j : e k ), etc..
We know from the Hurtwitz's Theorem that every (unital) composition algebra (over the reals) which, at the same time, is a normed division algebra, is isomorphic to one of the following four algebras: the real numbers R, the complex numbers C, the quaternions H, or the Cayley numbers (the octonions) O.
Here we shall treat this characterization in terms of the perfect JNWsystems. We consider the characterization of the problem under the assumption that the algebra is of monic product type. We have the following theorem.  [17] we may suppose that A is an alternative algebra. Precisely, in [17, p. 162], a relation is shown for an algebra A with a unit element e and an involutive anti-automorphism a → a * such that a + a * and aa * are spanned by e for all a in A. With the notation n(a) = aa * , if n is a nonsingular mapping into the field of A and A is alternative, then (A, n) is a composition algebra [1].
Proof of (T3). We notice that we have the following identities from the assumption that A is an alternative algebra: for every X, Y ∈ A. Then, letting X = α i e i and Y = β j e j , we find the following equalities from, respectively, (9.1), (9. Thus, we can see that the condition may not be satisfied for non-self dual perfect systems. In order to prove the assertion, we realize the system by use of the realization theorem, reduce the problem to sedenions, and have to prove it. The proof follows from the relations (9.7) and the table of Example 5.