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On the Algebras Formed by the Cayley-Dickson Process
... These algebras extend the real numbers to more complex structures, incorporating properties such as addition, multiplication, and conjugation. Originally introduced in 1845 by the mathematician Arthur Cayley [1], and later analyzed by Leonard Eugene Dickson in 1919 [2], the Cayley-Dickson construction has found significant applications in various branches of mathematics, including algebra, analysis, and geometry [3][4][5][6][7][8][9][10][11][12][13][14] along with mathematical physical applications [15][16][17][18][19][20][21][22][23][24][25][26][27][28]. ...
... noting that Equations (19)- (22) comprise an implementation of the 4-tensor-action defined in the previous section. It is readily checked that the formula (14) for the product in A × A is the same as the one in the Cayley-Dickson construction [1,2]. It is further observed that the addition is no longer the usual component-wise addition as it would be expected in the Cayley-Dickson case. ...
... whereas the + and · operations on B are given, respectively, by the formulas p+p ′ 2 and pp ′ . The resulting algebra in X × B = C * × C * × C * can then be trivially obtained by applying this structure onto Equation (14). ...
A generalized construction procedure for algebraic number systems is hereby presented. This procedure offers an efficient representation and computation method for complex numbers, quaternions, and other algebraic structures. The construction method is then illustrated across a range of examples. In particular, the novel developments reported herein provide a generalized form of the Cayley–Dickson construction through involutive dimagmas, thereby allowing for the treatment of more general spaces other than vector spaces, which underlie the associated algebra structure.
... Each generation comes with its own copy of SU (3) C thereby also requiring three generations of gluons, for which there is currently no experimental evidence. Additionally, it turns out that Aut (S) = Aut (O) × S 3 , where Aut (O) = G 2 , and S 3 is the permutation group of three objects [37,38]. The S 3 automorphisms of S were however not given any clear physical interpretation, in part because these automorphisms stabilize the octonion subalgebras in S. ...
... Schafer [37] showed that for CD algebras A n with n ≥ 4 (A 0 = R), the derivation algebra der(A n ) consists of derivations of the form a +bu → a D+(bD)u, where a, b ∈ A n−1 , u is the new anticommuting imaginary unit introduced in the CD construction of A n from A n−1 , and D is a derivation of A n−1 . Brown [38] demonstrated that if θ ∈ Aut (A n−1 ), then ...
... Since the CD process generates an infinite series of algebras, one might question whether going beyond the division algebras and including S is a wise idea, or if it opens the door to considering ever larger algebras. The derivation algebra for all CD algebras A n , n ≥ 3 is equal to g 2 however [37]. Furthermore, at least for the cases n = 4, 5, 6, the automorphism group of each successive CD algebra only picks up additional factors of S 3 [38]. ...
An algebraic representation of three generations of fermions with SU(3) color symmetry based on the Cayley–Dickson algebra of sedenions S is constructed. Recent constructions based on division algebras convincingly describe a single generation of leptons and quarks with Standard Model gauge symmetries. Nonetheless, an algebraic origin for the existence of exactly three generations has proven difficult to substantiate. We motivate S as a natural algebraic candidate to describe three generations with SU(3) gauge symmetry. We initially represent one generation of leptons and quarks in terms of two minimal left ideals of Cℓ(6) , generated from a subset of all left actions of the complex sedenions on themselves. Subsequently we employ the S3 automorphism of order three, which is an automorphism of S but not of O, to generate two additional generations. Given the relative obscurity of sedenions, efforts have been made to present the material in a self-contained manner.
... Each generation comes with its own copy of SU (3) C thereby also requiring three generations of gluons, for which there is currently no experimental evidence. Additionally, Aut(S) = Aut(O) × S 3 , where Aut(O) = G 2 , and S 3 is the permutation group of three objects [37,38]. The S 3 automorphisms of S were however not given any clear physical interpretation, in part because these automorphisms stabilize the octonion subalgebras in S. ...
... Schafer [37] showed that for CD algebras A n with n ≥ 4 (A 0 = R), the derivation algebra der(A n ) consists of derivations of the form a + bu → aD + (bD)u, where a, b ∈ A n−1 , u is the new anticommuting imaginary unit u introduces in the CD construction of A n from A n−1 , and D is a derivation of A n−1 . Brown [38] demonstrated that if θ ∈ Aut(A n−1 ), then ...
... Since the CD process generates an infinite series of algebras, one might question whether going beyond the division algebras and including S is a wise idea, or if it opens the door to considering ever larger algebras. The derivation algebra for all CD algebras A n , n ≥ 3 is equal to g 2 however [37]. Furthermore, at least for the cases n = 4, 5, 6, the automorphism group of each successive CD algebra only picks up additional factors of S 3 [38]. ...
An algebraic representation of three generations of fermions with color symmetry based on the Cayley-Dickson algebra of sedenions is constructed. Recent constructions based on division algebras convincingly describe a single generation of leptons and quarks with Standard Model gauge symmetries. Nonetheless, an algebraic origin for the existence of exactly three generations has proven difficult to substantiate. We motivate as a natural algebraic candidate to describe three generations with gauge symmetry. We initially represent one generation of leptons and quarks in terms of two minimal left ideals of , generated from a subset of all left actions of the complex sedenions on themselves. Subsequently we employ the finite group , which are automorphisms of but not of to generate two additional generations. Given the relative obscurity of sedenions, efforts have been made to present the material in a self-contained manner.
... In this section we briefly discuss the important properties and notations used in the theory of Cayley-Dickson algebras that will be needed in the article. For more in-depth background on Cayley-Dickson algebras see, for example, the renowned paper [20], the classical monographs [16,23], and references therein. We start with the inductive definition of Cayley-Dickson algebras over an arbitrary field F. ...
... We remark that for char(F) 2 the algebra A 1 is isomorphic to F{γ 0 }, where γ 0 = (4µ + 1)/4 0, so one can start the Cayley-Dickson process from A 0 = F, with a →ā being the identity map. We recall a convenient notation for Cayley-Dickson algebras which extends the common notation used for quaternions and octonions over an arbitrary field (see [23, [20] for an alternative notation.) ...
... An element a ∈ A n is called alternative if a(ab) = a 2 b and (ba)a = ba 2 for all b ∈ A n . The next proposition implies that all elements of E n and L n are alternative (see also [20,Lemma 4] for a different proof of alternativity of E n ). Proposition 2.2. ...
We study the roots of polynomials over Cayley--Dickson algebras over an arbitrary field and of arbitrary dimension. For this purpose we generalize the concept of spherical roots from quaternion and octonion polynomials to this setting, and demonstrate their basic properties. We show that the spherical roots (but not all roots) of a polynomial f(x) are also roots of its companion polynomial (defined to be the norm of f(x)). For locally-complex Cayley--Dickson algebras, we show that the spherical roots of (defined formally) belong to the convex hull of the roots of , and we also prove that all roots of are contained in the snail of f(x), as defined by Ghiloni and Perotti for quaternions. The latter two results generalize the classical Gauss--Lucas theorem to the locally-complex Cayley--Dickson algebras, and we also generalize Jensen's classical theorem on real polynomials to this setting.
... On the other hand, the octonions, made public by Cayley, are known to a general mathematical audience mostly for their automorphism group, associated with the simple Lie algebra of type G 2 . Moving further, the automorphism groups of the higher Cayley-Dickson algebras F = A 0 ⊂ A 1 ⊂ · · · were computed by Schafer [21], also see [11]. Namely, for n > 3, Aut(A n ) is a direct product of Aut(A n−1 ) and an abelian group. ...
... Again considering only isomorphisms fixing the base field F , the same arguments prove a loop counterpart to Schafer's theorem on Cayley-Dickson algebras [21]: ...
We develop a theory of loops with involution. On this basis we define a Cayley-Dickson doubling on loops, and use it to investigate the lattice of varieties of loops with involution, focusing on properties that remain valid in the Cayley-Dickson double. Specializing to central-by-abelian loops with elementary abelian 2-group quotients, we find conditions under which one can characterize the automorphism groups of iterated Cayley-Dickson doubles. A key result is a corrected proof that for , the automorphism group of the Cayley-Dickson loop is .
... , e 2 n −1 } consists of alternative elements and that the Euclidean structure in R 2 n and the C − D algebra structure are related by 2 x, y = xy + yx ||x|| 2 = xx for all x and y in A n . (See [4]). §1. ...
... It is known [4] that the elements in the canonical basis are alternative. Clearly a scalar multiple of an alternative element is alternative but the sum of two alternative elements is not necessarily alternative. ...
An element a in A_n, the Cayley-Dickson algebra is alternative if (a,a,x)=0 for all x. In this paper we characterise such elements for n>3.To do so,we prove first the so called Yui's conjecture:For a and b pure elements in A_n. If (a,x,b)=0 for all x then a and b are linearly dependent.Also we study alternativity between every two elements and relate this with the norm of their product .
... One of the first well-known examples is the field of complex numbers C, which is a simple field extension of real numbers R. The direct generalization of this construction leads to the hypercomplex numbers (see, e.g., [7,8]) defined as finite D-dimensional algebras A over the reals with a special basis (with squares restricted to 0, ±1). Among numerous versions of hypercomplex number systems [9,10] (for a modern review, see, e.g., [11,12]), only the complex numbers A = C (D = 2), quaternions A = H (D = 4), and octonions A = O (D = 8) are classical division algebras (with no zero divisors or nilpotents) [13][14][15], and the latter two can be obtained via the Cayley-Dickson doubling procedure [16][17][18]. ...
... The standard method of obtaining the higher hypercomplex algebras is the Cayley-Dickson construction [18,49,50]. It is well known that all four binary division algebras D = R, C, H, O can be built in this way [8]. ...
We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element, we define a new norm which is polyadically multiplicative, and the corresponding map is a n-ary homomorphism. We define a polyadic analog of the Cayley-Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, which is proportional to the intermediate arities, and which are not isomorphic to those obtained by the previous constructions. Second, a new polyadic product of vectors in any vector space is defined, which is consistent with the polyadization procedure using vectorization. Endowed with this introduced product, the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process (we call it the "imaginary tower"), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call "half-quaternions" and "half-octonions". The latter are not the subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced "half-quaternion" norm, we obtain the ternary analog of the sum of two squares identity. We show that the ternary division algebra of imaginary "half-octonions" is unitless and totally associative.
... α ij and f k being uniquely determined by f i and f j . From [7], Lemma 4, the results show that in any algebra A t with the basis {1, f 1 , . . . , f n−1 } satisfying relations (4) and (5), we have: ...
... for all i ∈ {1, 2, . . . , n − 1} and for every x ∈ A. The field K is the center of the algebra A t ,for t ≥ 2 [7]. Algebras A t of dimension 2 t obtained by the Cayley-Dickson process, described above, are flexible and powerassociative, for all t ≥ 1, and, in general, are not division algebras, for all t ≥ 1. ...
In this paper, we provide some properties of k-potent elements in algebras obtained by the Cayley–Dickson process over Zp. Moreover, we find a structure of nonunitary ring over Fibonacci quaternions over Z3 and we present a method to encrypt plain texts, by using invertible elements in some of these algebras.
... , c n ∈ A. The substitution of λ ∈ A in f (x) is defined by f (λ) = c n (λ n ) + · · · + c 1 λ + c 0 . This expression is well-defined as Cayley-Dickson algebras are always power-associative, see [15]. We say that λ ∈ A is a root of ...
... Consider the real-valued symmetric bilinear form a, b associated with the quadratic form Norm(a) on an arbitrary Cayley-Dickson algebra A over R, with a, a = Norm(a) for all a ∈ A. The bilinear form satisfies the identities a, bc = ac, b = b a, c for all a, b, c ∈ A (see [15,Lemma 6]). Consequently, if αx = β, it follows that 2 = β, β = αx, β = − x, αβ = − x, 0 = 0, a contradiction. ...
Over a composition algebra A, a polynomial has a root if and only for some . We examine whether this is true for general Cayley-Dickson algebras. The conclusion is that it is when f(x) is linear or monic quadratic, but it is false in general. Similar questions about the connections between f and its companion are studied. Finally, we compute the left eigenvalues of octonion matrices.
... In general terms, a hypercomplex algebra is an algebra whose multiplication has additional algebraic or geometrical property [44]. Precisely, let us consider the following definition, which encompasses complex numbers, quaternions, Clifford algebras [45], Caylay-Dickson algebras [37], and the general framework provided by Kantor and Solodovnik [24]. ...
... is a compact set. Like the function considered previously, the components of f V given by (37) are quadratic forms on the components x 0 , x 1 , x 2 and x 3 of x. Therefore, f V is a continuous but non-linear vector-valued function. ...
The universal approximation theorem states that a neural network with one hidden layer can approximate continuous functions on compact sets with any desired precision. This theorem supports using neural networks for various applications, including regression and classification tasks. Furthermore, it is valid for real-valued neural networks and some hypercomplex-valued neural networks such as complex-, quaternion-, tessarine-, and Clifford-valued neural networks. However, hypercomplex-valued neu-ral networks are a type of vector-valued neural network defined on an algebra with additional algebraic or geometric properties. This paper extends the universal approximation theorem for a wide range of vector-valued neural networks, including hypercomplex-valued models as particular instances. Precisely, we introduce the concept of non-degenerate algebra and state the universal approximation theorem for neural networks defined on such algebras.
... Instead there are some reasons that suggest S constitutes a natural candidate to generalize the existing algebraic models from a single generation to three. The automorphism group of S is G 2 × S 3 , where S 3 is the symmetric group of three objects [24,25]. That is, Aut(S)= Aut(O) × S 3 , and thus the symmetries of S are the same as those of O but one finds a threefold multiplicity. ...
... Since the Cayley-Dickson process continues indefinitely, one might reasonably wonder why not continue beyond S to even larger (and stranger) algebras? One reason why S is a sensible place to stop is that the automorphism groups of larger Cayley-Dickson algebras do not change fundamentally, but only pick up additional factors of S 3 [24][25][26][27]. It is therefore unlikely that new physics will be generated by going beyond S. ...
We describe an algebraic model, currently being developed, based on the Cayley-Dickson algebra of sedenions, S, with the goal of describing three generations of Standard Model fermions. Although recent algebraic constructions based on the four (normed) division algebras convincingly describe a single generation of leptons and quarks, they offer little insight on how to generalize the results to three generations. We motivate S as a natural algebraic candidate to describe three generations, and argue that Cayley-Dickson algebras beyond the sedenions are unlikely to offer additional physical insights. In our approach, the three generations correspond to three intersecting octonion subalgebras of S. Depending on which maximal compact subgroup of G2 = Aut(O) is chosen, the generations are differentiated by either their SU(3)C color symmetry or their chiral SU(2)L weak gauge symmetry.
... The purpose of this paper is to catalog all possible variants of the Cayley-Dickson doubling products and to recommend a different way to number the basis vectors. The alternate numbering method has been used in the past, for example in Shafer's 1954 paper [14], but has fallen out of favor. By identifying the ordered pair of two sequences as the 'shuffling' of the two sequences the author demonstrates how this leads naturally to the numbering used by Shafer and others and how it illustrates the periodicity of the structure constants and corresponding 'twist' on the group product underlying the product of the basis vectors. ...
... The most commonly used of these eight doubling products is P ⊤ 3 [14] [5] [10] but P 3 has also been used [2] [3]. ...
The purpose of this paper is to identify all of the basic Cayley-Dickson doubling products. A Cayley-Dickson algebra A N+1 of dimension 2 N+1 consists of all ordered pairs of elements of a Cayley-Dickson algebra A N of dimension 2 N where the product (a, b)(c, d) of elements of A N+1 is defined in terms of a pair of second degree binomials (f (a, b, c, d), g(a, b, c, d)) satisfying certain properties. The polynomial pair(f, g) is called a 'doubling product.' While A 0 may denote any ring, here it is taken to be the set R of real numbers. The binomials f and g should be devised such that A 1 = C the complex numbers, A 2 = H the quaternions, and A 3 = O the octonions. Historically, various researchers have used different yet equivalent doubling products.
... for any a, b, c, d ∈ D (n) . This procedure is known as a doubling procedure of a smashed product, as introduced in [17,18]. The multiplication relations in (2.1) can be expressed in the following way: For any a, b ∈ D (n) , we have ag n = g nā , (2.7) g n a =āg n , (2.8) (g n a)b = g n (ba), (2.9) a(bg n ) = (ba)g n , (2.10) (g n a)(g n b) = −bā, (2.11) (ag n )(bg n ) = −ba. ...
... In the following section, we focus on split Cayley-Dickson algebras and provide explicit formulas for their twist functions. We begin by defining an involution and regular involution on an algebra over a field F using the doubling product definition from [17]. A Cayley-Dickson algebra is an algebra with a regular involution satisfying specific conditions. ...
The Cayley–Dickson algebra has long been a challenge due to the lack of an explicit multiplication table. Despite being constructible through inductive construction, its explicit structure has remained elusive until now. In this article, we propose a solution to this long-standing problem by revealing the Cayley–Dickson algebra as a twisted group algebra with an explicit twist function σ(A,B). We show that this function satisfies the equation eAeB=(-1)σ(A,B)eA⊕Band provide a formula for the relationship between the Cayley–Dickson algebra and split Cayley–Dickson algebra, thereby giving an explicit expression for the twist function of the split Cayley–Dickson algebra. Our approach not only resolves the lack of explicit structure for the Cayley–Dickson algebra and split Cayley–Dickson algebra but also sheds light on the algebraic structure underlying this fundamental mathematical object.
... We refer the reader to [13,19] for auxiliary definitions and general properties of the Cayley-Dickson algebras. Definition 2.5. ...
... This is equivalent to the equality [ e 0 , e 0 , a] = 0 for all a ∈ A n , that is, to the alternativity of the element e 0 . But e 0 = (0, e 0 ) is one of the standard basis elements, whence it is alternative by [19,Lemma 4]. ...
Zero divisors of Cayley–Dickson algebras over an arbitrary field 𝔽, char 𝔽 ≠= 2, are studied. It is shown that the zero divisors whose components alternate strongly pairwise and have nonzero norm form hexagonal structures in the zero-divisor graph of a Cayley–Dickson algebra. Properties of the doubly alternative zero divisors at least one of whose components has nonzero norm are established, and explicit forms of their annihilators, orthogonalizers, and centralizers are obtained. Properties of the zero divisors in Cayley–Dickson algebras with anisotropic norm are described, and it is shown that in this case, directed hexagons in the zero-divisor graph can be extended to undirected double hexagons in the orthogonality graph. A criterion of C-equivalence for elements of Cayley–Dickson algebras with anisotropic norm is obtained. Possible values of dimension for the annihilators of elements in Cayley–Dickson algebras are considered.
... Its discriminant is dis(a) = t(a) 2 − 4n(a). In this section, we use [19,24] to recall the classical nonassociative algebras, the so-called Cayley-Dickson algebras. Definition 3.6 ( [24]). ...
... In this section, we use [19,24] to recall the classical nonassociative algebras, the so-called Cayley-Dickson algebras. Definition 3.6 ( [24]). The algebra A{γ} produced by the Cayley-Dickson process, when applied to A with the parameter γ ∈ F, γ = 0, is defined as the set of ordered pairs of elements of A with operations α(a, b) = (αa, αb), Thus if we begin with a one-dimensional algebra and successively apply the Cayley-Dickson process to it, we will get a 2 n -dimensional algebra in the nth step. ...
Our paper is devoted to the investigations of doubly alternative zero divisors of the real Cayley–Dickson split-algebras. We describe their annihilators and orthogonalizers and also establish the relationship between centralizers and orthogonalizers for such elements. Then we obtain an analogue of the real Jordan normal form in the case of the split-octonions. Finally, we describe commutativity, orthogonality, and zero divisor graphs of the split-complex numbers, the split-quaternions, and the split-octonions in terms of their diameters and cliques.
... In general terms, a hypercomplex algebra is an algebra whose multiplication has additional algebraic or geometrical property [44]. Precisely, let us consider the following definition, which encompasses complex numbers, quaternions, Clifford algebras [45], Caylay-Dickson algebras [37], and the general framework provided by Kantor and Solodovnik [24]. ...
... is a compact set. Like the function considered previously, the components of f V given by (37) are quadratic forms on the components x 0 , x 1 , x 2 and x 3 of x. Therefore, f V is a continuous but non-linear vector-valued function. ...
The universal approximation theorem asserts that a single hidden layer neural network approximates continuous functions with any desired precision on compact sets. As an existential result, the universal approximation theorem supports the use of neural networks for various applications, including regression and classification tasks. The universal approximation theorem is not limited to real-valued neural networks but also holds for complex, quaternion, tessarines, and Clifford-valued neural networks. This paper extends the universal approximation theorem for a broad class of hypercomplex-valued neural networks. Precisely, we first introduce the concept of non-degenerate hypercomplex algebra. Complex numbers, quaternions, and tessarines are examples of non-degenerate hypercomplex algebras. Then, we state the universal approximation theorem for hypercomplex-valued neural networks defined on a non-degenerate algebra.
... Далi ми коротко опишемо процес Келi -Дiксона i властивостi алгебр, отриманих за допомогою цього процесу (див. [5,6] ...
... де \beta ij i f k визначаються однозначно через f i i f j (див. [6]). При t = 2 отримуємо алгебру узагальнених кватернiонiв, а при t = 3 -алгебру узагальнених октонiонiв. ...
УДК 512.55Дотримуючись iдей Бейлiса [J. W. Bales, A tree for computing the Cayley-Dickson twist, Missouri J. Math. Sci., 21 , No. 2, 83-93 (2009)], у цiй статтi запропоновано алгоритм обчислення елементiв базису алгебри, отриманої за допомогою процесу Келi – Дiксона. Як наслiдок доведено, що алгебра, отримана за допомогою процесу Келi – Дiксона, є скрученою груповою алгеброю для групи G = ℤ 2 n , n = 2 t , t ∈ ℕ над полем K з K ≠ 2 . Наведено властивостi i деякi застосування неасоцiативних алгебр кватернiонiв.
... By the Cayley-Dickson construction, D (n+1) is built up from D (n) by adding a new imaginary unit g n subject to the rules (2.1) (a + bg n )(c + dg n ) = (ac − db) + (da + bc)g n for any a, b, c, d ∈ D (n) . This is a doubling procedure of a smashed product; see [15,14]. We rewrite the multiplication relations in (2.1) as follows. ...
... Based on [14], we have a doubling product definition for both Cayley-Dickson algebras and split Cayley-Dickson algebras. Definition 6. ...
A long-standing unresolved issue in the Cayley-Dickson algebra of dimension is that there lacks of an explicit multiplication table although it can be constructed via inductive construction. In this article we solve this open problem. We show that the Cayley-Dickson algebra is a twisted group algebra with an \textit{explicit} twist function such that The same approach also yields similar results for split Cayley-Dickson algebras.
... The Cayley-Dickson process and the properties of the obtained algebras, will be briefly presented in the following. For other details see [6,7]. Let A be a finite dimensional unitary algebra over a field K, with a map called a scalar involution ...
... , n} and for every x ∈ A t . ( [7], Lemma 4) From the above, it results that each x ∈ A t can be written under the form ...
Due to the computational aspects which appear in the study of algebras obtained by the Cayley–Dickson process, it is difficult to obtain nice properties for these algebras. For this reason, finding some identities in such algebras plays an important role in obtaining new properties of these algebras and facilitates computations. In this regard, in the first part of this paper, we present some new identities and properties in algebras obtained by the Cayley–Dickson process. As another remark regarding the computational aspects in these algebras, in the last part of this paper, we solve some quadratic equations in the real division quaternion algebra when their coefficients are some special elements. These special coefficients allowed us to solve interesting quadratic equations, providing solutions directly, without using specialized softs.
... This leads to the construction, starting from the real numbers, of the complex numbers, quaternions, and octonions, through a doubling process which originated in the works of Cayley and Dickson. For the early history of these developments, see Dickson [13], and for the completion of the classical theory, see Albert [1] and Schafer [25]. The purpose of the present work is to determine the natural generalization of the Cayley-Dickson process to the setting of dialgebras. ...
We adapt the algorithm of Kolesnikov and Pozhidaev, which converts a polynomial identity for algebras into the corresponding identities for dialgebras, to the Cayley-Dickson doubling process. We obtain a generalization of this process to the setting of dialgebras, establish some of its basic properties, and construct dialgebra analogues of the quaternions and octonions.
... This construction of a new division algebra from 2 copies of another is a special case of the Cayley-Dickson process; for modern treatments, see[2][3][4][5][6]. ...
The eigenvalue problem for 3x3 octonionic Hermitian matrices contains some surprises, which we have reported elsewhere. In particular, the eigenvalues need not be real, there are 6 rather than 3 real eigenvalues, and the corresponding eigenvectors are not orthogonal in the usual sense. The nonassociativity of the octonions makes computations tricky, and all of these results were first obtained via brute force (but exact) Mathematica computations. Some of them, such as the computation of real eigenvalues, have subsequently been implemented more elegantly; others have not. We describe here the use of Mathematica in analyzing this problem, and in particular its use in proving a generalized orthogonality property for which no other proof is known.
... The Cayley-Dickson doubling process, [21], can give us new examples of ternary Jordan algebras. Let us recall such process. ...
Based on the relation between the notions of Lie triple systems and Jordan algebras, we introduce the n-ary Jordan algebras,an n-ary generalization of Jordan algebras obtained via the generalization of the following property , where is an n-ary algebra. Next, we study a ternary example of these algebras. Finally, based on the construction of a family of ternary algebras defined by means of the Cayley-Dickson algebras, we present an example of a ternary -derivation algebra (n-ary -derivation algebras are the non-commutative version of n-ary Jordan algebras).
... For auxiliary definitions and general properties of the Cayley-Dickson algebras, we refer the reader to [9,14]. ...
The commutativity graph of the real sedenion algebra is considered. It is shown that the elements whose imaginary parts are not zero divisors correspond to isolated vertices of this graph. All other elements form a connected component whose diameter equals 3.
... Among numerous versions of hypercomplex number systems HAWKES [1902], TABER [1904] (for modern review, see, e.g. YAGLOM [1968], BURLAKOV AND BURLAKOV [2020]), only the complex numbers A " C (D " 2), quaternions A " H (D " 4) and octonions A " O (D " 8) are classical division algebras (with no zero divisors or nilpotents) SCHAFER [1966], SALTMAN [1992], GUBARENI [2021], and the two latter can be obtained via the Cayley-Dickson doubling procedure DICKSON [1919], ALBERT [1942], SCHAFER [1954]. ...
https://arxiv.org/abs/2312.01366 (v3: added Section 7 containing a new (polyadic) product of vectors, updated Bibliography from 37 to 60 entries; v4: minor corrections, added Acknowledgements).
We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element we define a new norm which is polyadically multiplicative, and the corresponding map is a n-ary homomorphism. We define a polyadic analog of the Cayley-Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, that is proportional to the intermediate arities, and which are not isomorphic to those obtained by the previous constructions. Second, a new polyadic product of vectors in any vector space is defined, which is consistent with the polyadization procedure by using vectorization. Endowed with this introduced product the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process (we call it the "imaginary tower"), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call "half-quaternions" and "half-octonions". The latter are not subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced "half-quaternion" norm we obtain the ternary analog of the sum of two squares identity. We show that the ternary division algebra of imaginary "half-octonions" is unitless and totally associative.
... Among numerous versions of hypercomplex number systems HAWKES [1902], TABER [1904] (for modern review, see, e.g. YAGLOM [1968], BURLAKOV AND BURLAKOV [2020]), only complex numbers A " C (D " 2), quaternions A " H (D " 4) and octonions A " O (D " 8) are classical division algebras (with no zero divisors and nilpotents) SCHAFER [1966], SALTMAN [1992], GUBARENI [2021], and the two latter can be obtain by the Cayley-Dickson doubling procedure DICKSON [1919], ALBERT [1942], SCHAFER [1954]. ...
https://arxiv.org/abs/2312.01366 (Version 3, added: Section 7 containing a new (polyadic) product of vectors). We introduce a new class of division algebras, hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O, without considering new elements. First, we use the proposed earlier matrix polyadization procedure which increases the algebra dimension. The obtained algebras obey the binary addition and nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element we define a new multiplicative norm. We define a polyadic analog of the Cayley-Dickson construction which corresponds to the consequent embedding of monomial matrices. Then we obtain another series of n-ary algebras corresponding to the binary division algebras which have more dimension, that is proportional to intermediate arities. Second, a new polyadic product of vectors in any vector space is defined, which is consistent with the polyadization procedure by using vectorization. Endowed with the introduced product the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process (we call it "imaginary tower"), which leads to nonunital nonderived ternary division algebras of half dimension, we call them "half-quaternions" and "half-octonions". The latter are not subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced "half-quaternion" norm we obtain the ternary analog of the sum of two squares identity. We show that the ternary division algebra of imaginary "half-octonions" is unitless and totally associative.
... The automorphism group of S is known to be Aut [42,43]. Explicitly, the automorphisms of S are given by ...
Three generations of fermions with SU (3) C symmetry are represented algebraically in terms of the algebra of sedenions, 𝕊, generated from the octonions, 𝕆, via the Cayley-Dickson process. Despite significant recent progress in generating the Standard Model gauge groups and particle multiplets from the four normed division algebras, an algebraic motivation for the existence of exactly three generations has been difficult to substantiate. In the sedenion model, one generation of leptons and quarks with SU (3) C symmetry is represented in terms of two minimal left ideals of ℂ ℓ (6), generated from a subset of all left actions of the complex sedenions on themselves. Subsequently, the finite group S 3 , which are automorphisms of 𝕊 but not of 𝕆, is used to generate two additional generations. The present paper highlight the key aspects and ideas underlying this construction.
... A similar loss of properties occurs in the general case as well, which makes A n extremely hard to work with as n grows (it becomes very hard to work with from n ≥ 4). Nevertheless, all Cayley-Dickson algebras are flexible, i.e., satisfy (ab)a = a(ba) for any a and b, and power-associative [6]. ...
We introduce the notions of alternating roots of polynomials and alternating polynomials over a Cayley-Dickson algebra, and prove a connection between the alternating roots of a given polynomial and the roots of the corresponding alternating polynomial over the Cayley-Dickson doubling of the algebra. We also include a detailed Octave code for the computation of alternating roots over Hamilton's quaternions.
... By [57] an algebra obtained from a flexible algebra by the Cayley-Dickson process is flexible. An example of a flexible algebra that is not Lie-admissible (so not associator-cyclic) is the imaginary octonions im O equipped with the commutator bracket ...
The sectional nonassociativity of a metrized (not necessarily associative or unital) algebra is defined analogously to the sectional curvature of a pseudo-Riemannian metric, with the associator in place of the Levi-Civita covariant derivative. For commutative real algebras nonnegative sectional nonassociativity is usually called the Norton inequality, while a sharp upper bound on the sectional nonassociativity of the Jordan algebra of Hermitian matrices over a real Hurwitz algebra is closely related to the Böttcher–Wenzel-Chern-do Carmo-Kobayashi inequality. These and other basic examples are explained, and there are described some consequences of bounds on sectional nonassociativity for commutative algebras. A technical point of interest is that the results work over the octonions as well as the associative Hurwitz algebras.
... Definition 1 (Cayley-Dickson form [32] and Equivalent complex matrix of a quaternion matrix [33]). For any quater- ...
Color image completion is a challenging problem in computer vision, but recent research has shown that quaternion representations of color images perform well in many areas. These representations consider the entire color image and effectively utilize coupling information between the three color channels. Consequently, low-rank quaternion matrix completion (LRQMC) algorithms have gained significant attention. We propose a method based on quaternion Qatar Riyal decomposition (QQR) and quaternion -norm called QLNM-QQR. This new approach reduces computational complexity by avoiding the need to calculate the QSVD of large quaternion matrices. We also present two improvements to the QLNM-QQR method: an enhanced version called IRQLNM-QQR that uses iteratively reweighted quaternion -norm minimization and a method called QLNM-QQR-SR that integrates sparse regularization. Our experiments on natural color images and color medical images show that IRQLNM-QQR outperforms QLNM-QQR and that the proposed QLNM-QQR-SR method is superior to several state-of-the-art methods.
... Algebras that are obtained by this process are called Cayley-Dickson algebras. In particular, such algebras are power-associative (see [5]). Moreover, every element λ in a Cayley-Dickson algebra A with involution σ over F satisfies λ 2 − Tr(λ) · λ + Norm(λ) = 0 where Tr(λ) = λ + σ (λ) ∈ F and Norm(λ) = λ · σ (λ) ∈ F. ...
We prove that given an octonion algebra A over a field F, a subring E⊆F and an octonion E-algebra R inside A, the set S of polynomials f(x)∈A[x] satisfying f(R)⊆R is an octonion (S∩F[x])-algebra, under the assumption that either 12∈R or char(F)≠0, and R contains the standard generators of A and their inverses. The project was inspired by a question raised by Werner on whether integer-valued octonion polynomials over the reals form a nonassociative ring. We also prove that for any prime p, the polynomial is integer-valued in the ring of polynomials A[x] over any real nonsplit Cayley–Dickson algebra A.
... Moreover, there are reasons to suspect that exhibits the algebraic structure necessary to describe three full generations. As mentioned above, the automorphism group of is G 2 × S 3 , where S 3 is the symmetric group of three objects [25,26]. Thus, Aut( ) = Aut( ) × S 3 , and so the symmetries of are the same as those of but with an additional threefold multiplicity. ...
A certain class of models based on the four division algebras attempts to find a suitable algebraic structure capable of providing a unified description of both the gauge symmetries as well as the specific particle content of the Standard Model. So far, partial success has been found, although typically only one generation of fermions is convincingly described. We propose to consider the Cayley-Dickson algebra of the sedenions (S), generated from the octonions (O), as a natural candidate for a three-generation model. In this approach, the three generations are associated with three intersecting subalgebras of , differentiated by either their SU(3) color or chiral SU(2) weak gauge symmetry. Finally we argue that larger Cayley-Dickson algebras beyond are unlikely to yield further interesting structure relevant to particle physics.
... Ÿ Example 3.10. By [57] an algebra obtained from a flexible algebra by the Cayley-Dickson process is flexible. An example of a flexible algebra that is not Lie-admissible (so not associator-cyclic) is the imaginary octonions im O equipped with the commutator bracket rx, ys " xy´yx. ...
The sectional nonassociativity of a metrized (not necessarily associative or unital) algebra is defined analogously to the sectional curvature of a pseudo-Riemannian metric, with the associator in place of the Levi-Civita covariant derivative. For commutative real algebras nonnegative sectional nonassociativity is usually called the Norton inequality, while a sharp upper bound on the sectional nonassociativity of the Jordan algebra of Hermitian matrices over a real Hurwitz algebra is closely related to the B\"ottcher-Wenzel-Chern-do Carmo-Kobayashi inequality. These and other basic examples are explained, and there are described some consequences of bounds on sectional nonassociativity for commutative algebras. A technical point of interest is that the results work over the octonions as well as the associative Hurwitz algebras.
... Example 2 (Cayley-Dickson Algebras). Dickson developed in 1919 a recursive process that generates algebras of doubling dimension [64]. For example, complex numbers are obtained from real numbers using this recursive process. ...
This paper aims to establish a framework for extreme learning machines (ELMs) on general hypercomplex algebras. Hypercomplex neural networks are machine learning models that feature higher-dimension numbers as parameters, inputs, and outputs. Firstly, we review broad hypercomplex algebras and show a framework to operate in these algebras through real-valued linear algebra operations in a robust manner. We proceed to explore a handful of well-known four-dimensional examples. Then, we propose the hypercomplex-valued ELMs and derive their learning using a hypercomplex-valued least-squares problem. Finally, we compare real and hypercomplex-valued ELM models’ performance in an experiment on time-series prediction and another on color image auto-encoding. The computational experiments highlight the excellent performance of hypercomplex-valued ELMs to treat multi-dimensional data, including models based on unusual hypercomplex algebras.
... For example, the multiplication of the octonions is alternative but not associative. In 1954, Richard Schafer [20] examined the algebras generated by the Cayley-Dickson process over a field and showed that they satisfy the flexible identity. ...
We initiate the study of a quantitative measure for the failure of a binary operation to be commutative and associative. We call this measure the associative-commutative spectrum as it extends the so-called associative spectrum (also known as the subassociativity type), which measures the nonassociativity of a binary operation. In fact, the associative-commutative spectrum (resp. associative spectrum) is the cardinality of the symmetric (resp. nonsymmetric) operad obtained naturally from a groupoid (a set with a binary operation). In this paper we provide some general results on the associative-commutative spectrum, precisely determine this measure for certain binary operations, and propose some problems for future study.
... Recall that any Hurwitz algebra A is isomorphic to a Cayley-Dickson algebra A n of dimension 2 n , where 0 ≤ n ≤ 3. For more information on Cayley-Dickson algebras, see [14,15,31] and references therein. ...
We suggest a new method which allows us to compute the lengths of (possibly non-unital) standard composition algebras over an arbitrary field F with char F≠2.
Some results regarding eigenvalues, companion matrices and fixed points in some algebras obtained by the Cayley-Dickson process are presented in this paper. The paper contains several examples which emphasize the obtained results.
The low-rank quaternion matrix approximation has been successfully applied in many applications involving signal processing and color image processing. However, the cost of quaternion models for generating low-rank quaternion matrix approximation is sometimes considerable due to the computation of the quaternion singular value decomposition (QSVD), which limits their application to real large-scale data. To address this deficiency, an efficient quaternion matrix CUR (QMCUR) method for low-rank approximation is suggested, which provides significant acceleration in color image processing. We first explore the QMCUR approximation method, which uses actual columns and rows of the given quaternion matrix, instead of the costly QSVD. Additionally, two different sampling strategies are used to sample the above-selected columns and rows. Then, the perturbation analysis is performed on the QMCUR approximation of noisy versions of low-rank quaternion matrices. And we also employ the proposed QMCUR method to color image recovery problem. Extensive experiments on both synthetic and real data further reveal the superiority of the proposed algorithm compared with other algorithms for getting low-rank approximation, in terms of both efficiency and accuracy.
The universal approximation theorem states that a neural network with one hidden layer can approximate continuous functions on compact sets with any desired precision. This theorem supports using neural networks for various applications, including regression and classification tasks. Furthermore, it is valid for real-valued neural networks and some hypercomplex-valued neural networks such as complex-, quaternion-, tessarine-, and Clifford-valued neural networks. However, hypercomplex-valued neural networks are a type of vector-valued neural network defined on an algebra with additional algebraic or geometric properties. This paper extends the universal approximation theorem for a wide range of vector-valued neural networks, including hypercomplex-valued models as particular instances. Precisely, we introduce the concept of non-degenerate algebra and state the universal approximation theorem for neural networks defined on such algebras.
Following some ideas proposed in [J. W. Bales, Missouri J. Math. Sci., 21, No. 2, 83–93 (2009)], we present an algorithm for computing basis elements in an algebra obtained by the Cayley–Dickson process. As a consequence of this result, we prove that an algebra obtained with the help of the Cayley–Dickson process is a twisted group algebra for the group G = ℤ2n, n = 2t, t ∈ ℕ, over a field K with charK ≠ 2. We also present some properties and applications of the quaternion nonassociative algebras.
The roots of polynomials over Cayley–Dickson algebras over an arbitrary field and of arbitrary dimension are studied. It is shown that the spherical roots of a polynomial f(x) are also roots of its companion polynomial Cf(x). We generalize the classical theorems for complex and real polynomials by Gauss–Lucas and Jensen to locally-complex Cayley–Dickson algebras: it is proved that the spherical roots of f′(x) belong to the convex hull of the roots of Cf(x), and we also show that all roots of f′(x) are contained in the snail of f(x), as defined by Ghiloni and Perotti.
We define and derive basic properties of the notion of Rota-Baxter operator on anti-flexible algebra. Pre-anti-flexible algebra structure and associated left(right)-symmetric algebra are constructed from a given Rota-Baxter operator on an anti-flexible algebra. The notion of -operator on anti-flexible algebra are recalled and use to build left(right)-symmetric algebra as well as related properties. Furthermore, we introduce Nijenhuis anti-flexible algebra and derive associated properties. Nijenhuis operator on anti-flexible algebra is used to build pre-anti-flexible algebra structure and related left(right)-symmetric algebra.