Cayley-Dickson Doubling Process Research Articles

How to obtain division algebras from a generalized Cayley–Dickson doubling process

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 quaternion algebra over a field F and allowing the element used in the doubling to be an invertible element in the algebra. The resulting unital algebras are not third power-associative, hence not quadratic. Starting with a quaternion division algebra D, we obtain division algebras A for all elements chosen in D outside of F. This is independent of where the element is placed inside the product. Thus three pairwise non-isomorphic families of eight-dimensional division algebras are obtained. Their Albert isotopes yield more division algebras with large derivation algebra.

MULTIPLICATION GROUPS AND INNER MAPPING GROUPS OF CAYLEY–DICKSON LOOPS

The Cayley–Dickson loop Qn 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 establish that the inner mapping group Inn (Qn) is an elementary abelian 2-group of order 22n-2 and describe the multiplication group Mlt (Qn) as a semidirect product of Inn (Qn) × ℤ2 and an elementary abelian 2-group of order 2n. We prove that one-sided inner mapping groups Inn l(Qn) and Inn r(Qn) are equal, elementary abelian 2-groups of order 22n-1-1. We establish that one-sided multiplication groups Mlt l(Qn) and Mlt r(Qn) are isomorphic, and show that Mlt l(Qn) is a semidirect product of Inn l(Qn) × ℤ2 and an elementary abelian 2-group of order 2n.

The Cayley–Dickson process for 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.

How to Obtain Division Algebras from Twisted Cayley–Dickson Doublings

We change the order of the factors in the classical Cayley–Dickson doubling process and investigate the eight-dimensional algebras obtained when doubling a quaternion algebra using these new multiplications, allowing the element c used in the doubling process to be an invertible element in the quaternion algebra. By changing the place of c inside the multiplication we obtain different families of algebras and conditions when these algebras are division.

Automorphism Groups of Cayley-Dickson Loops

The Cayley-Dickson loop Qn 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, and sedenions).We discuss properties of the Cayley-Dickson loops, show that these loops are Hamiltonian, and describe the structure of their automorphism groups.

Nonassociative quaternion algebras over rings

Non-split nonassociative quaternion algebras over fields were first discovered over the real numbers independently by Dickson and Albert. They were later classified over arbitrary fields by Waterhouse. These algebras naturally appeared as the most interesting case in the classification of the four-dimensional nonassociative algebras which contain a separable field extension of the base field in their nucleus. We investigate algebras of constant rank 4 over an arbitrary ringR which contain a quadratic etale subalgebraS overR in their nucleus. A generalized Cayley-Dickson doubling process is introduced to construct a special class of these algebras.

Matrix representation of octonions and generalizations

We define a special matrix multiplication among a special subset of 2N×2N matrices, and study the resulting (nonassociative) algebras and their subalgebras. We derive the conditions under which these algebras become alternative nonassociative, and when they become associative. In particular, these algebras yield special matrix representations of octonions and complex numbers; they naturally lead to the Cayley–Dickson doubling process. Our matrix representation of octonions also yields elegant insights into Dirac’s equation for a free particle. A few other results and remarks arise as byproducts.

Quaternion algebras over elliptic curves

All composition algebras of rank 2 and 4 over elliptic curves are enumerated and partly classified, and examples of octonion algebras are constructed using the generalized Cayley-Dickson doubling process. The underlying field is assumed to be perfect, and of characteristic not two. Some applications are given.

The cayley-dickson doubling process revisited

We introduce the category of composition triples over locally ringed spaces. In this category we find the category of composition algebras over locally ringed spaces as subcategory. We describe the Cay ley-Dickson doubling process in the category of composition triples.

Third power associative composition algebras

Classical composition algebras, with a unit element, are well-known and can be obtained by means of the Cayley-Dickson doubling process. If the condition on the existence of unit element is dropped, many new algebras arise. However, it is shown in this paper that if such a weak condition as the associativity of third powers of any element is imposed, only the known flexible composition algebras appear.

Composition algebras over algebraic curves of genus zero

We rephrase the classical theory of composition algebras over fields, particularly the Cayley-Dickson Doubling Process and Zorn's Vector Matrices, in the setting of locally ringed spaces. Fixing an arbitrary base field, we use these constructions to classify composition algebras over (complete smooth) curves of genus zero. Applications are given to composition algebras over function fields of genus zero and polynomial rings.