Daniel K. Biss, J. Daniel Christensen, Daniel Dugger, Daniel C. Isaksen
We establish many previously unknown properties of zero-divisors in Cayley-Dickson algebras. The basic approach is to use a certain splitting that simplifies computations surprisingly.
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Cayley-Dickson algebras are an infinite sequence of non-associative algebras starting with the reals, complexes, quaternions, and octonions. We study the zero-divisors in the higher Cayley-Dickson algebras. In particular, we show that the annihilator of any element in the 2^n-dimensional Cayley-Dickson algebra has dimension at most 2^n-4n+4. Moroever, every multiple of four between 0 and this upper bound actually occurs as the dimension of some annihilator (a theorem of Moreno says that only multiples of four can occur). We completely describe all the zero-divisors whose annihilator has dimens…
In this paper we study the algebra monomorphisms from A_m =R^(2^m) into A_n=R^(2^n) for 0<m<n where the A_k 's are the Cayley- Dickson algebras over the real numbers. We show that for m>2 there are many different types of monomorphisms and we describe them in terms of the zero divisors in A_n.
In this paper we give new methods to construct zero divisors in A_n =R^(2^n) the Cayley_Dickson algebras over the real numbers, for n larger than 4, and we also relate the set of zero divisors in A_{n+1} with the Stiefel Manifold V_{2^n -1,2} for n>3. We also introduce the notion of "Spectrum" (of a no zero double pure element) wich synthesize the information regarding the structure of the linear operators left and right multiplication by the element. We use this as a main technical tool to construct the zero divisors.
The effect of some properties of twisted groups on the associated algebras, particularly Cayley-Dickson and Clifford algebras. It is conjectured that the Hilbert space of square-summable sequences is a Cayley-Dickson algebra.
Regarding the Cayley-Dickson algebras as twisted group algebras, this paper reveals some basic periodic properties of these twists.
This paper is a short survey about some properties of algebras ob- tained by the Cayley-Dickson process and some of their applications.
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 $$e_Ae_B=(-1)^{σ(A,B)}e_{A\oplus B}$$ and provide a formula for the relationship between the Cayley-Dickson algebra and split Cayley-Dickson algebra, t…
In this paper we will prove that the Hall identity is true in all algebras obtained by the Cayley-Dickson process and, in some conditions, the converse is also true.
In this paper we improve the level and sublevel of algebras obtained by the Cayley-Dickson process when their level and sublevel are greater than dimension of the algebras.
A catalog of all 32 Cayley-Dickson-like doubling products on ordered pairs (a,b),(c,d) for which (1,0) is the left and right identity and for which xx* and x*x equal the square of the norm of x given the conjugate (a,b)*=(a*,-b). Only eight of these are true Cayley-Dickson doubling products, since 24 of them do not satisfy the quaternion properties. Each of the eight Cayley-Dickson products has a distinctive representation in the Fano Plane.
We show how eigentheory clarifies many algebraic properties of Cayley-Dickson algebras. These notes are intended as background material for those who are studying this eigentheory more closely.
We work to find a basis of graded identities for the octonion algebra. We do so for the $\mathbb{Z}_2^2$ and $\mathbb{Z}_2^3$ gradings, both of them derived of the Cayley-Dickson process, the later grading being possible only when the characteristic of the scalars is not two.
The Cayley-Dickson loop Q_n is the multiplicative closure of basic elements of the algebra constructed by n applications of the Cayley-Dickson doubling process (the first few examples of such algebras are real numbers, complex numbers, quaternions, octonions, sedenions). We discuss properties of the Cayley-Dickson loops, show that these loops are Hamiltonian, and describe the structure of their automorphism groups.
In this paper we will prove that any number n\in N-{0} could be realised as level of an algebra obtained by the Cayley-Dickson process over a suitable field.
In the first two sections of this paper we provide a brief account of the Cayley-Dickson algebras and prove that the radius on these algebras is given by the Euclidean norm. With this observation we resort to three related topics: a variant of the Gelfand formula, stability of subnorms, and the functional power equation.
In this paper, we define a special class of elements in the algebras obtained by the Cayley Dickson process, called l elements. We find conditions such that these elements to be invertible. These conditions can be very useful for finding new identities, identities which can help us in the study of the properties of these algebras.
Finding identities in nonassociative algebras plays an important role in the study of properties of these algebras. In this paper, we present some identities in alternative algebras and in algebras obtained by the Cayley-Dickson process. Moreover, the spectrum of matrices with coefficients in such algebras are investigated.
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.
Starting from some ideas given by Bales in [Ba; 09], in this paper we present an algorithm for computing the elements of the basis in an algebra obtained by the Cayley-Dickson process. As a consequence of this result, we prove that an algebra obtained by the Cayley-Dickson process is a twisted group algebra for the group G = Zn2 ; n = 2t, t 2 N, over a field K, with charK = 0. In the last section, we give some properties and applications of the quaternion nonassociative algebras.
The article is devoted to Beta and Gamma functions of Cayley-Dickson numbers. It is shown that there are specific features in comparison with the complex case. These functions serve as examples of meromorphic functions of Cayley-Dickson variables and illustrate applications of line integrals. Useful identities for Beta and Gamma functions are proved.
Although the Cayley-Dickson algebras are twisted group algebras, little attention has been paid to the nature of the Cayley-Dickson twist. One reason is that the twist appears to be highly chaotic and there are other interesting things about the algebras to focus attention upon. However, if one uses a doubling product for the algebras different from yet equivalent to the ones commonly used and if one uses a numbering of the basis vectors different from the standard basis a quite beautiful and highly periodic twist emerges. This leads easily to a simple closed form formula for the product of an…
Involutions in the Cayley-Dickson construction
本文确定了Cayley-Dickson构造中所有扩展原$*$-代数对合的对合，并分类了由此得到的$*$-代数同构类，应用包括证明无零因子的Cayley双代数恰好有一个额外对合，以及分类实数域上至多4维的$*$-代数。
Some remarks regarding special elements in algebras obtained by the Cayley-Dickson process over Zp
该研究探讨了Zp上Cayley-Dickson代数中k-幂等元的性质，构建了Z3上Fibonacci四元数的非幺环结构，并提出基于该类代数可逆元的明文加密方法。
On doubly alternative zero divisors in Cayley-Dickson algebras
本文研究特征非2的域上Cayley-Dickson代数的零因子，明确双交替零因子的性质与显式形式，刻画各向异性范数情形下零因子图的结构，得到C等价准则及零化子维数的可能取值。
In this paper we investigated holomorphic functions (belonging to the kernel of the Dirac operator) in Cayley-Dickson algebras. For this purpose, we study the structure of Cayley-Dickson algebras. We also provide an algorithm for the construction of such functions.
Flipped non-associative polynomial rings and the Cayley-Dickson construction
本文引入翻转非结合多项式环，证明 Cayley-Dickson 代数可自然实现为其商，并推广 McCrimmon 的相关经典结果。
We present a new polar representation of quaternions inspired by the Cayley-Dickson representation. In this new polar representation, a quaternion is represented by a pair of complex numbers as in the Cayley-Dickson form, but here these two complex numbers are a complex 'modulus' and a complex 'argument'. As in the Cayley-Dickson form, the two complex numbers are in the same complex plane (using the same complex root of -1), but the complex phase is multiplied by a different complex root of -1 in the exponential function. We show how to calculate the amplitude and phase from an arbitrary quate…
We show that if a non-associative unital ring is graded by a hypercentral group, then the ring is simple if and only if it is graded simple and the center of the ring is a field. Thereby, we extend a result by Jespers to a non-associative setting. By applying this result to non-associative crossed products, we obtain non-associative analogues of results by Bell, Jordan and Voskoglou. We also apply our result to Cayley-Dickson doublings, thereby obtaining a new proof of a classical result by McCrimmon.
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 $C_f(x)$ (defined to be the norm of $f(x)$). For locally-complex Cayley--Dickson algebras, we show that the spherical roots of $f'(x)$ (defined formally) belong to the convex hull of the roots of $C_f(x)$,…
A Mnemonic Matrix Rule for (Split) Octonionic Multiplication and its Extension to the Cayley--Dickson Tower
本文提出了一种基于记忆矩阵的八元数乘法规则，并将其扩展到凯莱-迪科森塔中的所有代数，为非结合代数提供了高效的计算方法。
The paper begins with short proofs of classical theorems by Frobenius and (resp.) Zorn on associative and (resp.) alternative real division algebras. These theorems characterize the first three (resp. four) Cayley-Dickson algebras. Then we introduce and study the class of real unital nonassociative algebras in which the subalgebra generated by any nonscalar element is isomorphic to C. We call them locally complex algebras. In particular, we describe all such algebras that have dimension at most 4. Our main motivation, however, for introducing locally complex algebras is that this concept makes…
In this paper, we generalize the concepts of level and sublevels of a composition algebra to algebras obtained by the Cayley-Dickson process. In 1967, R. B. Brown constructed, for every $t\in \Bbb{N},$ a division algebra $A_{t}$ of dimension $2^{t}$ over the power-series field $K\{X_{1},X_{2},...,X_{t}\}.$ This gives us the possibility to construct a division algebra of dimension 2$^{t}$ and prescribed level 2$^{k}$ $ k, t\in \Bbb{N}^{*}.$
General Polynomials and Eigenvalues Over Cayley--Dickson Algebras
研究凯莱 - 迪克森代数上首一二次一般多项式零集，给出求根方法，证明实数上局部复凯莱 - 迪克森代数中此类多项式必有根，以及凯莱八元数上2×2矩阵右特征值存在并可计算部分右特征值。
Orthogonality graphs of real Cayley-Dickson algebras. Part II: The subgraph on pairs of basis elements
该研究针对实Cayley-Dickson代数的基元素对构成的正交图，证明至少16维时两代数同构等价于其图同构，并给出从图提取代数Cayley-Dickson参数的算法。
We describe associative center $N(R)$ and the center $Z(R)$ of the ring $R$ obtained by applying the generalized Cayley-Dickson construction and we find conditions under which the ring $R$ is $N$-essential or centrally essential. The obtained results are applied to generalized quaternion rings and octonion rings; we use them to construct an example of a non-associative centrally essential ring. Victor Markov is supported by the Russian Foundation for Basic Research, project 17-01-00895-A. Askar Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013.
Loops with involution and the Cayley-Dickson doubling process
本文建立带对合的环的理论，定义Cayley-Dickson加倍并研究相关簇格性质，进而刻画迭代加倍的自同构群，修正证明了n>3时Q_n的自同构群为GL_3(F_2)×{±1}^{n-3}。
This article is devoted to the noncommutative version of the Laplace transformation. New types of the direct and inverse transformations of the Laplace type over general Cayley-Dickson algebras, in particular, also the skew field of quaternions and the octonion algebra are investigated. Examples are given. Theorems are proved about properties of such transformations and also theorems about images and originals in conjunction with operations of multiplication, differentiation, integration, convolution, shift and homothety. The noncommutative analog of the Mellin transformation is studied as wel…
Alternating Roots of Polynomials over Cayley-Dickson Algebras
本文引入了 Cayley-Dickson 代数上多项式的交错根与交错多项式概念，证明了交错根与对应交错多项式根的联系，并提供了计算 Hamilton 四元数上交错根的详细 Octave 代码。
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 .
Rota-Baxter operators of weight zero on Cayley-Dickson algebra
该研究完成了特征不为2的域上分裂八元数代数权零Rota-Baxter算子的自同构与反自同构共轭分类，终结了复合代数上此类算子的分类问题，给出了通用域与二次闭域下的两类描述。
A set of valuable universal similarity factorization equalities is established over complex Clifford algebras $\Cn.$ Through them matrix representations of complex Clifford algebras $\Cn$ can directly be derived, and their properties can easily be determined.
Orthogonality graphs of real Cayley-Dickson algebras. Part I: Doubly alternative zero divisors and their hexagons
该研究探讨实Cayley-Dickson代数的正交图，构造定向六边形，确定双六边形顶点乘法表，给出生成交错子代数的条件，分类特定零因子并明确其正交条件。
The Cayley-Dickson algebras R (real numbers), C (complex numbers), H (quaternions), O (octonions), S (sedenions), and T (trigintaduonions) have attracted the attention of several mathematicians and physicists because of their important applications in both pure mathematics and theoretical physics. This paper deals with the determination of the subalgebra structure of the algebra T by analyzing the loop T_L of order 64 generated by its 32 basis elements. The analysis shows that T_L is a non-associative finite invertible loop (NAFIL) with 373 non-trivial subloops of orders 32, 16, 8, 4, and 2 al…
{\small \ We generalize the concepts of \thinspace level \thinspace and \thinspace sublevel of a composition algebra to division algebras obtained by the Cayley-Dickson process. In 1967, R. B. Brown constructed, for every}$t\in \mathbb{N},${\small \ a nonassociative division algebra}$A_{t}${\small \ of dimension}$2^{t}${\small \ over the power-series field}$K\{X_{1},X_{2},...,X_{t}\}.\,${\small \ This gives us the possibility to construct a division algebra of \thinspace dimension}$% {\small \,2}^{t}\,${\small and prescribed \thinspace level and sublevel}$% {\small \,\,2}^{k}${\small ,\thinspa…
In this paper, \thinspace \thinspace we generalize the concepts of \thinspace level \thinspace and \thinspace sublevel of a composition algebra to algebras obtained by the Cayley-Dickson process. In 1967, R. B. Brown constructed, for every}$t\in \Bbb{N},${\small % \ a division algebra}$A_{t}${\small \ of dimension}$2^{t}${\small \ over the power-series field}$K\{X_{1},X_{2},...,X_{t}\}.\,${\small \ This gives us the possibility to construct a division algebra of \thinspace dimension \thinspace 2}$^{t}\,${\small and prescribed \thinspace level and sublevel \thinspace \thinspace 2}$^{k}${\small …
In the recent years a lot of effort has been made to extend the theory of hyperholomorphic functions from the setting of associative Clifford algebras to non-associative Cayley-Dickson algebras, starting with the octonions. An important question is whether there appear really essentially different features in the treatment with Cayley-Dickson algebras that cannot be handled in the Clifford analysis setting. Here we give one concrete example. Cayley-Dickson algebras namely admit the construction of direct analogues of CM-lattices, in particular lattices that are closed under multiplication. Can…
Central Products of Cayley-Dickson Loops
研究 Cayley-Dickson 环中心积中换位子的平凡性，构造了随机换位子平凡概率趋于1的非交换双结合环序列，并证明了n≥3时中心积同构蕴含逐项同构。
Rota-Baxter operators of nonzero weight on the split Cayley-Dickson algebra
研究分裂奥克托恩的罗塔-巴克斯运算符，发现任何域上存在唯一非分裂运算符，并描述了分裂奥克托恩在二次闭合域上的分解，完成对任意权重组合代数罗塔-巴克斯运算符的分类。
We introduce what we call "alternative twisted tensor products" for not necessarily associative algebras, as a common generalization of several different constructions: the Cayley-Dickson process, the Clifford process and the twisted tensor product of two associative algebras, one of them being commutative. We show that some very basic facts concerning the Cayley-Dickson process (the equivalence between the two different formulations of it and the lifting of the involution) are particular cases of general results about alternative twisted tensor products of algebras. As a class of examples of …
Structure of the Cayley-Dickson algebras
本文研究了Cayley-Dickson代数的结构，揭示了非结合性与幂结合子代数的关系，并通过分层构造定义了八元数和sedenion的子代数。
The Cayley-Dickson doubling products
本文研究Cayley-Dickson代数的倍增乘积问题，通过明确二次二项式形式的乘积定义及约束条件，旨在找出所有满足从实数出发依次生成复数、四元数、八元数要求的倍增乘积，填补此前研究仅涉及部分乘积的空白。
New families of eight-dimensional real division algebras with large derivation algebra are presented: We generalize the classical Cayley-Dickson doubling process starting with a unital algebra with involution over a field F by allowing the scalar in the doubling to be an invertible element in the algebra. The resulting unital algebras are neither power-associative nor quadratic. Starting with a quaternion division algebra D, we obtain division algebras A for all invertible scalars chosen in D outside of F. This is independent on where the scalar is placed inside the product and three pairwise …
Roots and right factors of polynomials and left eigenvalues of matrices over Cayley-Dickson algebras
本文研究一般 Cayley-Dickson 代数中多项式有根与存在右因子的等价性，分析其适用边界，并计算八元数矩阵的左特征值。
In this paper, we give some properties of the levels and sublevels of algebras obtained by the Cayley-Dickson process. We will emphasize how isotropy of some quadratic forms can influence the levels and sublevels of algebras obtained by the Cayley-Dickson process.
Octonions as Clifford-like algebras
本文基于八元数与Clifford代数的相似性，引入Kingdon代数，通过泛性质刻画八元数及分裂八元数，并赋予其超代数结构。
Determinant Factorization of Left Multiplication in the 32 Dimensional Cayley Dickson Algebra
该研究针对32维Cayley-Dickson代数$A_5$，构造14次齐次多项式$D_{14}$并证明左乘法行列式的分解式，得到$A_5$左零因子的判据，还分析了其与十六元数$A_4$对应结构的差异。
The objects of the great Nonlinear Revolutions - Catastrophes and Chaos in the 1960s-70s (henceforth, CT); and, small-world and scale-free Network Theory (NT), emerging quite recently - will be spliced together by a New Kind of Number Theory, focused on digitizations (i.e., binary strings). NT nodes then become feature-rich representations (nodules in a "rhizosphere") of CT contents. The "Box-Kite" formalism of zero-divisors (ZD's) - first showing in the 16-D Sedenions, then in all higher 2^N-ions derived from Imaginaries by Cayley-Dickson Process (CDP) - can model such "enriched" nodules. Its…
Quasilinear multiplication in the real Cayley--Dickson tower
本文提出实Cayley-Dickson代数乘法的拟线性算法，将乘法化为虚子空间交错积并用两次递归求值，实现$O(N\log N)$复杂度，在$N\ge32$时优于现有方法，并已实现为fastCD库。
In this paper, we study the zero set of the Hopf construction map F_n : A_n_ x A_n --> A_n x A_0 given by F_n (x, y) = (2xy, | y|^2 - |x|^2) for n >3 where A_n is the Cayley-Dickson algebra of dimension 2^n over the real numbers.
We present here some results of applying the Cayley-Dickson process to certain alternative algebras (notably built upon Galois fields and congruence rings), in a manner which might yield new building blocks for cryptographic systems. We focus on enumeration properties rather than the classification and comparison questions which are extensively studied elsewhere, at least for algebras over fields. The results presented here are technical but not inherently difficult.
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