Similarity and Consimilarity of Elements in Real Cayley-dickson Algebras
نویسنده
چکیده
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 equations ax = xb and ax = xb in these algebras. Some consequences are also presented. AMS mathematics subject classifications: 17A05, 17A35.
منابع مشابه
Automorphism groups of real Cayley-Dickson loops
The Cayley-Dickson loop Cn 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 will discuss properties of the Cayley-Dickson loops, show that these loops are Hamiltonian and describe the structure of th...
متن کاملLarge Annihilators in Cayley-dickson Algebras
Cayley-Dickson algebras are non-associative R-algebras that generalize the well-known algebras R, C, H, and O. We study zero-divisors in these algebras. In particular, we show that the annihilator of any element of the 2n-dimensional Cayley-Dickson algebra has dimension at most 2n−4n+4. Moreover, every multiple of 4 between 0 and this upper bound occurs as the dimension of some annihilator. Alt...
متن کاملThe zero divisors of the Cayley–Dickson algebras over the real numbers
In this paper we describe algebraically the zero divisors of the Cayley Dickson algebras An = R n for n ≥ 4 over the real numbers. Introduction. The Cayley–Dickson algebra An over R is an algebra structure on R2n given inductively by the formulae: Let x = (x1, x2) and y = (y1, y2) in R n = R2 × R2 then xy = (x1y1 − y2x2, y2x1 + x2y1) where x = (x1,−x2). Therefore A0 = R,A1 = C complex numbers, ...
متن کاملMultiplication 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 2 −2 and describe the multiplicati...
متن کاملCodes over subsets of algebras obtained by the Cayley-Dickson process
In this paper, we define binary block codes over subsets of real algebras obtained by the Cayley-Dickson process and we provide an algorithm to obtain codes with a better rate. This algorithm offers more flexibility than other methods known until now, similar to Lenstra's algorithm on elliptic curves compared with p − 1 Pollard's algorithm.
متن کامل