共查询到20条相似文献,搜索用时 32 毫秒
1.
S. V. Ludkovsky 《Journal of Mathematical Sciences》2009,157(2):252-310
This paper is devoted to the noncommutative version of the Laplace transformation. New types of direct and inverse transformations
of the Laplace type over general Cayley-Dickson algebras, in particular, the skew field of quaternions and the octonion algebra
are investigated. Examples are given. Theorems about properties of such transformations and also theorems about images and
originals in conjunction with the operations of multiplication, differentiation, integration, convolution, shift, and homothety
are proved. Applications are given to a solution of differential equations over the Cayley-Dickson algebras.
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 52, Functional
Analysis, 2008. 相似文献
2.
Let A be a *-algebra. An additive mapping E : A → A is called a Jordan *-derivation if E(X2) = E(x)x*+xE(x) holds, for all x 6 A. These mappings have been extensively studied in the last 6 years by Bresar, Semrl, Vukman and Zalar because they are closely connected with the problem of representability of quadratic functionals by sesquilinear forms. This study was, however, always in the setting of associative rings. In the present paper we study Jordan *-derivations on the Cayley-Dickson algebra of octonions, which is not associative. Our first main result is that every Jordan *-derivation on the octonion algebra is of the form E(x)=ax*-xa. In the terminology of earlier papers this means that every Jordan *-derivation on the octonion algebra is inner. This generalizes the known fact that Jordan *-derivations on complex and quaternion algebras are inner. Our second main result is a representation theorem for quadratic functionals on octonion modules. Its proof uses the result mentioned above on Jordan *-derivations. 相似文献
3.
The article is devoted to spectral theory of super-differential operators over the quaternion skew field and the octonion
algebra. An existence of their resolvent functions is proved, their spectra are investigated. It is shown, that spectra are
contained in general in the quaternion skew field or the octonion algebra and can not be reduced to the field of complex numbers. 相似文献
4.
Susanne Pumplün 《manuscripta mathematica》2010,132(3-4):307-333
Let ${(X,\mathcal{O}_X)}$ be a locally ringed space. We investigate the structure of symmetric composition algebras over X obtained from cubic alternative algebras ${\mathcal{A}}$ over X generalizing a method first presented by J. R. Faulkner. We find examples of Okubo algebras over elliptic curves which do not have any isotopes which are octonion algebras and of an octonion algebra which is a Cayley-Dickson doubling of a quaternion algebra but does not contain any quadratic étale algebras. 相似文献
5.
《代数通讯》2013,41(8):3523-3534
The Cayley-Dickson process gives a recursive method of constructing a nonassociative algebra of dimension 2 n for all n ≥ 0, beginning with any ring of scalars. The algebras in this sequence are known to be flexible quadratic algebras; it follows that they are noncommutative Jordan algebras: they satisfy the flexible identity in degree 3 and the Jordan identity in degree 4. For the integral sedenion algebra (the double of the octonions) we determine a complete set of generators for the multilinear identities in degrees ≤ 5. Since these identities are satisfied by all flexible quadratic algebras, it follows that a multilinear identity of degree ≤ 5 is satisfied by all the algebras obtained from the Cayley-Dickson process if and only if it is satisfied by the sedenions. 相似文献
6.
S. V. Ludkovsky 《Journal of Mathematical Sciences》2007,141(3):1299-1330
Functions of several octonion variables are investigated and integral representation theorems for them are proved. By using
these theorems, solutions of
are studied. More generally, functions of several Cayley-Dickson variables are considered. Integral formulas of the Martinelli-Bochner,
Leray, and Koppelman type used in complex analysis are proved in a new generalized form for functions of Cayley-Dickson variables
instead of complex variables. Moreover, analogues of Stein manifolds over Cayley-Dickson graded algebras are defined and investigated.
__________
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 28, Algebra
and Analysis, 2005. 相似文献
7.
8.
S.V. Lüdkovsky 《Bulletin des Sciences Mathématiques》2003,127(9):755-796
We investigate differentiability of functions defined on regions of the real quaternion field and obtain a noncommutative version of the Cauchy-Riemann conditions. Then we study the noncommutative analog of the Cauchy integral as well as criteria for functions of a quternion variable to be analytic. In particular, the quaternionic exponential and logarithmic functions are being considered. Main results include quaternion versions of Hurwitz' theorem, Mittag-Leffler's theorem and Weierstrass' theorem. 相似文献
9.
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. 相似文献
10.
Yongxin Zhou 《代数通讯》2013,41(1):523-539
In this paper we establish a relationship between alternative loop algebras and Cayley-Dickson algebras: any Cayley-Dickson algebra is a quotient algebra of an alternative loop algebra. Thus we give a new way of representing a Cayley-Dickson algebra. As an application, a result of Luiz G. X. de Barros is generalized. 相似文献
11.
Michiel Hazewinkel 《Acta Appl Math》2003,75(1-3):55-83
This paper is concerned with two generalizations of the Hopf algebra of symmetric functions that have more or less recently appeared. The Hopf algebra of noncommutative symmetric functions and its dual, the Hopf algebra of quasisymmetric functions. The focus is on the incredibly rich structure of the Hopf algebra of symmetric functions and the question of which structures and properties have good analogues for the noncommutative symmetric functions and/or the quasisymmetric functions. This paper attempts to survey the ongoing investigations in this topic as dictated by the knowledge and interests of its author. There are many open questions that are discussed. 相似文献
12.
Yongge Tian 《Advances in Applied Clifford Algebras》1999,9(1):61-76
The similarity and consimilarity of elements in the real quaternion, octonion and sedenion algebras, as well as in the general real Cayley-Dickson algebras are considered by solving the two fundamental equationsax=xb andax = [`(x)]bax = \bar xb in these algebras. Some consequences are also presented. 相似文献
13.
Yongge Tian 《Advances in Applied Clifford Algebras》2000,10(1):61-90
As is well-known, the real quaternion division algebra ℍ is algebraically isomorphic to a 4-by-4 real matrix algebra. But
the real division octonion algebra
can not be algebraically isomorphic to any matrix algebras over the real number field ℝ, because
is a non-associative algebra over ℝ. However since
is an extension of ℍ by the Cayley-Dickson process and is also finite-dimensional, some pseudo real matrix representations
of octonions can still be introduced through real matrix representations of quaternions. In this paper we give a complete
investigation to real matrix representations of octonions, and consider their various applications to octonions as well as
matrices of octonions. 相似文献
14.
We characterize the forms that occur as restrictions of norm forms of octonion fields. The results are applied to forms of types E\(_6\), E\(_7\), and E\(_8\) and to positive definite forms over fields that allow a unique non-split octonion algebra, e.g., the field of rational numbers. 相似文献
15.
Transform methods are used to establish algebra homomorphisms related to convoluted semigroups and convoluted cosine functions. Such families are now basic in the study of the abstract Cauchy problem. The framework they provide is flexible enough to encompass most of the concepts used up to now to treat Cauchy problems of the first- and second-order in general Banach spaces. Starting with the study of the classical Laplace convolution and a cosine convolution, along with associated dual transforms, natural algebra homomorphisms are introduced which capture the convoluted semigroup and cosine function properties. These correspond to extensions of the Cauchy functional equation for semigroups and the abstract d'Alembert equation for the case of cosine operator functions. The algebra homomorphisms obtained provide a way to prove Hille-Yosida type generation theorems for the operator families under consideration. 相似文献
16.
I. P. Shestakov 《Algebra and Logic》2011,49(6):561-565
We work to find a basis of identities for an octonion algebra modulo an associator ideal of a free alternative algebra, or,
in other words, a basis for an associative replica of an ideal of identities of an octonion algebra. 相似文献
17.
Polynomial identities in algebras are the central objects of Polynomial Identities Theory. They play an important role in learning of algebras properties. In particular, the Hall identity is fulfilled in the quaternion algebra and does not hold in other non-commutative associative algebras. For this reason, the Hall identity is important for the quaternion algebra. The idea of this work is to generalize the Hall identity to algebras obtained by the Cayley-Dickson process. Starting from the above remarks, in this paper, we prove that the Hall identity is true in all algebras obtained by the Cayley-Dickson process and, in some conditions, the converse of this statement is also true for split quaternion algebras. From Hall identity, we will find some new properties and identities in algebras obtained by the Cayley-Dickson process. 相似文献
18.
19.
Noncommutative Galois Extension and Graded <Emphasis Type="Italic">q</Emphasis>-Differential Algebra
Viktor Abramov 《Advances in Applied Clifford Algebras》2016,26(1):1-11
We show that a semi-commutative Galois extension of a unital associative algebra can be endowed with the structure of a graded q-differential algebra. We study the first and higher order noncommutative differential calculus of semi-commutative Galois extension induced by the graded q-differential algebra. As an example we consider the quaternions which can be viewed as the semi-commutative Galois extension of complex numbers. 相似文献
20.
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 ‘modulus’ and ‘argument’ from an arbitrary quaternion in Cartesian form. 相似文献