The present paper is devoted to the description of local and 2-local automorphisms on Cayley algebras over an arbitrary field $\mathbb{F}$. Given a Cayley algebra $\mathcal{C}$ with norm $n$, let $O(\mathcal{C},n)$ be the corresponding orthogonal group. We prove that the group of all local automorphisms of $\mathcal{C}$ coincides with the group $\{φ\in O(\mathcal{C},n)\mid φ(1)=1\}.$ Further we prove that the behavior of 2-local automorphisms depends on the Cayley algebra being split or division. Every 2-local automorphism on the split Cayley algebra is an automorphism, i.e. they form the exceptional Lie group $G_2(\mathbb{F})$ if $\textrm{char}\mathbb{F}\neq 2,3$. On the other hand, on division Cayley algebras over a field $\mathbb{F}$, the groups of 2-local automorphisms and local automorphisms coincide, and they are isomorphic to the group $\{φ\in O(\mathcal{C},n)\mid φ(1)=1\}.$
The split quartic Cayley algebra is a structurable algebra which has been used to give constructions of Lie algebras of type D4. Here, we calculate its group of automorphisms, its algebra of derivations and its gradings.
In this paper we introduce a new species of evolution algebras that we call Cayley evolution algebras. We show that if a field $k$ contains sufficiently many elements (for example if $k$ is infinite) then every finite group $G$ is isomorphic to $Aut(X)$ where $X$ is a finite-dimensional absolutely simple Cayley evolution $k$-algebra.
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…
We develop the general Theory of Cayley Hamilton algebras and we compare this with the theory of pseudocharacters. We finally characterize prime $T$--ideals for Cayley Hamilton algebras and discuss some of their geometry. To the memory of T. A. Springer
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.
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.
The present paper is devoted to the description of local and $2$-local derivations on Cayley algebras over an arbitrary field $\mathbb{F}$. Given a Cayley algebra $\mathcal{C}$ with norm $\mathfrak{n}$, let $\mathcal{C}_0$ be its subspace of trace $0$ elements. We prove that the space of all local derivations of $\mathcal{C}$ coincides with the Lie algebra $\{d\in (\mathcal{C},\mathfrak{n}) | d(1)=0\}$ which is isomorphic to the orthogonal Lie algebra $(\mathcal{C}_0,\mathfrak{n})$. Further we prove that, surprisingly, the behavior of $2$-local derivations depends on the Cayley algebra being s…
On locally complex algebras and low-dimensional Cayley-Dickson algebras
本文从Frobenius和（分别）的经典定理的简短证明开始。Zorn on associative and（resp.）交错实除代数这些定理表征了前三个（分别为）。4）Cayley-Dickson代数。然后引入并研究了由任意非纯量元生成的子代数同构于C的实单位元非结合代数类。我们称之为局部复代数。特别是，我们描述所有这样的代数的维数最多为4。我们的主要动机，但是，引进局部复代数是，这个概念使我们有可能扩大弗罗贝纽斯'和佐恩定理的方式，它也涉及第五凯莱-迪克森代数，sedenions。
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…
We develop the general Theory of Cayley Hamilton algebras using norms and compare with the approach, valid only in characteristic 0, using traces and presented in a previous paper $T$-ideals of Cayley Hamilton algebras, 2020, arXiv:2008.02222
Cayley-Dickson代数是从实数、复数、四元数和八元数开始的非结合代数的无穷序列。研究了高阶Cayley-Dickson代数中的零因子。特别地，我们证明了2^n维Cayley-Dickson代数中任意元素的零化子的维数至多为2 ^n-4 n +4.此外，在0和这个上界之间的每一个4的倍数实际上都是某个零化子的维数（莫雷诺的一个定理说，只有4的倍数才能出现）。我们完整地描述了零化子维数为2^n-4 n +4的所有零因子。
Dariush Kiani et al.\cite{kiani2015unitary} claim to have found the unitary Cayley graph $Cay(M_{n}(F),GL_{n}(F))$ of matrix algebras over finite field $F$ is strongly regular only when $n=2$, but they have only considered two special cases, namely when $n = 2$ and $3$ and have failed to cover the general cases. In this paper, we prove that the unitary Cayley graph of matrix algebras over finite field $F$ is strongly regular if and only if $n=2$.
We develop a theory of semialgebra Grassmann triples via Hasse-Schmidt derivations, which formally generalizes results such as the Cayley-Hamilton theorem in linear algebra, thereby providing a unified approach to classical linear algebra and tropical algebra.
We prove that large classes of algebras in the framework of root of unity quantum cluster algebras have the structures of maximal orders in central simple algebras and Cayley-Hamilton algebras in the sense of Procesi. We show that every root of unity upper quantum cluster algebra is a maximal order and obtain an explicit formula for its reduced trace. Under mild assumptions, inside each such algebra we construct a canonical central subalgebra isomorphic to the underlying upper cluster algebra, such that the pair is a Cayley-Hamilton algebra; its fully Azumaya locus is shown to contain a copy o…
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…
