Structure fractals and para-quaternionic geometry

It is well known that starting with real structure, the Cayley– Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions p = 2, 4 and 8, respectively, but the procedure fails for p = 16 in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are n = 2. Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteration process p → p + 2 → p + 4 → . . . , they have constructed 2-dimensional “bipetals” for p = 9 and 2-dimensional “bisepals” for p = 13. The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the “pistil” and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the “stamens”. The first named author, M. Nowak-Kępczyk, and S. Marchiafava (2006, 2009a, b) gave an effective, explicit determination of the periods and expressed them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. In contrast to earlier results, the fractal bundle flower structure, in particular petals, sepals, pistils, and stamens are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. Physical concepts of dual and conjugate objects as well as of antiparticles led us to extend the periodicity theorem to structure fractals in para-quaternionic formulation, applying some results in this direction by the second named author. The paper is concluded by outlining some applications. 2000 Mathematics Subject Classification. 81R25, 32L25, 53A50, 15A66.

The difference in formula (1) here and in [4] is due to the replacement of matrix units 1, iσ 1 , iσ 2 , iσ 3 of the usual quaternions by the units 1, i = iσ 2 , j = σ 1 , k = σ 3 of para-quaternions, so that our i, j, and k mean j, (1/i)i, and (1/i)k in [4], respectively.This is due to our definition of the real Clifford algebra H of para-quaternions as generated by 1 and imaginary units i, j, k satisfying For a para-quaternionic structure the left module structure is defined up to conjugation in H.
Within a closed square Q α q consider its diameter and two segments, symmetric and equidistant with respect to L ∞ : where Let further where g α q is the gradating function equal a qk αj on the closed square Q qj αk corresponding to the pair (j, k); we suppose that the original square is divided into 4 p+q−2 squares with sides parallel to the sides of Q α q .Given z ∈ L 0 ∞ , consider the sequences where (6) ĝα as well as (z 2 + ), . . .for 2r + 1 = α > 2p, (10) with the notation (6), and where (15) We need two lemmas.
(ii) If dim V = 4n in both cases: quaternionic and para-quaternionic, there exists a basis of V of the following type: for any admissible basis (J 1 , J 2 , J 3 ).

We have
Periodicity Theorem (para-quaternionic formulation).(i) If a λ αλ = 0 for λ = 2 p+q−2 , the sequences (3) are periodic of period 2, starting from some term.The periods are: the sequences (3) are constant-valued, starting from some term; it amounts at ) holds, the sequences (4) are periodic of period 2, starting from some term.The periods are: (iv) If (19) holds, the sequences (5) are constant-valued, starting from some term; it amounts at (v) The sequences (7) and ( 9) are periodic of period 2, starting from some term.The periods are: in the case of (7), and in the case of (9).(vi) The sequences (8) and (10) are constant-valued, starting from some term; it amounts at in the case of (8), and in the case of (10).
(viii) The sequences (11) and (12) are periodic of period 2, starting from some term.The periods are: The proof is analogous to that of the corresponding theorem given in [4][5][6].However, in order to understand two important differences: ij, i, ik instead of i, j, k and i α A q α instead of A q α , α = 1, 2, . . ., 2p + 2q − 3, we give a proof in the case of α odd and α > 2p.
2. Proof of the Periodicity Theorem for α odd and α > 2p.In order to calculate (33) , m = 0, ±1, . . ., ±(2 n − 1); n = 0, 1, . .., we have to look for (34) . .. To this end in the case of the Clifford-type fractal 5 = (Q 5 q ), p = 2 [1,2,11] we consider the table (cf.[4], Fig. 1), where rows numbered with positive integers, corresponding to odd numbers, represent (35) 3) on the q-th iteration step and columns represent the configuration related to s m = m/2 n ; hereafter, for z ∈ L ∞ , we are also using the notation In principle, the configuration (35) coincides for p = 2 with that of Sec. 6 in [4], but we have to take into account that the number of basic squares was 4 p+q−2 = 4 q , and now it amounts at 4 p+q−1 = 4 q+1 .The complete configuration related to (35) consists of four values of the generating function if z belongs to the interior of the basic square Q αj qk corresponding to 16 values if z is a vertex (of course, it may happen that some of these values coincide).In our case we have the following possibilities for the four values corresponding to the direction perpendicular to L ∞ determined by the points z − − (q − 1) and z + − (q − 1) starting from some term of ( 9): .
As far as the direction of L ∞ corresponding to z ∈ L ∞ is concerned, we have only one possibility: i −i or −i i .The complete configuration becomes: For each n the first period is indicated with help of the upper parts of two bigger squares: each upper part containing four small squares i and four small rectangles .Since the direct proof of the statement (v) of the Periodicity Theorem seems to be more complicated that the preceding ones, it is instructive to illustrate the determining procedure (34) for the complete structure (33) of the gradating function with respect to (z − − , z + − , z − + , z + + ) in the case of 5 and p = 2 by decomposing the lifting inverse to the projection (34) with respect to s, into three subliftings, covering the range of Fig. 1 in [4], as indicated on Fig. 2 in [4]: The data for the initial lifting s = 11 corresponding to q = 3 can easily be checked out from Fig. 3 in [4], the square corresponding to p = 2, α = 4, 5; q = 3, h = 2.
In the case of 7 = (Q 7 q ), p = 3 thanks to starting with q always from 1, the corresponding table differs from the previous one by minor changes only, caused by the fact that the number of basic squares has increased, for q = 1 from 16 to 64; cf.Fig. in [4], the squares corresponding to p = 2, α = 4, 5; q = 1, h = 0, p = 2, h = 6, 7; q = 1, h = 0.
Consequently, within the range of Fig. 1 in [4], the changes concern q ≤ 9 only; they are shown in Fig. 4 in [4].
Then we can proceed by induction with respect to p, considering α , α = 2p + 1.We observe that, by the definition of i, j, and k, the matrices expressing the periods can be given in the form (25) and ( 26), as desired.

The second and third Periodicity Theorem in para-quaternionic formulation.
The following generalizations of Periodicity Theorems (iii) and (iv) are obtained by analogy with those established in [4].
(ii) The periodicity of the sequences (4) always starts from their n-th term or earlier whenever m in the definition of z ∈ L 0 ∞ is odd.If m is even, of the form m = µ • 2 ν , where ν is odd, the periodicity of (4) starts from the n − ν-th term or earlier.In particular, such a situation appears in the cases of Pauli matrices σ 1 and σ 2 , where the periodicity, for m odd, starts exactly from the n-th term.For m even the periodicity starts exactly from the n − ν-th term.
If we replace (19) by a less restrictive condition (36), then the one-element periods (22) have a more general form (39) , where η = 1 or −1, and the para-quaternionic form, instead of (22), reads: (ii) The constancy of the sequences (5) always starts from their n-th term or earlier, whenever m in the definition of z ∈ L 0 ∞ is odd.If m is even, of the form m = µ • 2 ν , where ν is odd, the constancy of (5) starts from the n − ν-th term or earlier.In particular, such a situation appears in the cases of Pauli matrices σ 1 and σ 2 , where the constancy, for m odd, starts exactly from the n-th term.For m even the periodicity starts exactly from the n − ν-th term.
4. Perspectives of combining structure fractals with para-quaternionic geometry.The periodicity theorems for structure fractals in para--quaternionic formulation give wide research perspectives which we plan to explore in the future: 1. Physical interpretation of a generic subspace of a para-quaternionic Hermitian vector space [3,14].
3. Noncommutativity and phase-space theorem referring to the well--known Gelfand-Naǐmark theorem (1943): Let A be a commutative C *algebra and M denote the set of maximal ideals of A. Then, equipped with a natural topology, M is a locally compact topological space, and A = C 0 (M ), where C 0 denotes the C * -algebra of continuous functions of M vanishing at infinity [5,16].
4. Finsler geometry of holomorphical and projectivized bundles, in particular, extension of the quaternionic Randers models to a para-quaternionic geometry [8].
13.The solenoidal and nanosolenoidal parts of the generalized Yang-Mills equations as observed on the canonical principal fibre bundle.
14. Simplifying the external field in terms of the metric and connection.
15. General mathematical and physical conclusions of combining structure fractals with para-quaternionic geometry.