Theorie der alternativen ringe

“…We have thus proved \0 0/ for all aG93; it is known that there is only one nonassociative algebra with unity over g carrying a nonsingular quadratic form admitting composition and having zero-divisors [15]. Thus we have an abstract characterization of E; another may be given as a simple non-associative alternative algebra with zero-divisors [27]. The specific form used here will be advantageous in calculations with the derivation algebra, as well as in that it gives us a starting point independent of the other literature in this area.…”

Section: -£41 -£42 -£43mentioning

confidence: 81%

On Automorphisms of Lie Algebras of Classical Type. II

Seligman¹

1960

Transactions of the American Mathematical Society

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“…[26]). Moreover, the famous theorems of Frobenius [19] and Zorn [32] assert that A is classified by {R, C, H, O} (cf. [24], [25]).…”

Section: Proof (I)⇒(ii) If η P Is Collinear Then Y(κ) P Is Collineamentioning

confidence: 99%

“…The problem of constructing and, ultimately, classifying all real division algebras originated in the discovery of the quaternion algebra H (Hamilton 1843) and the octonion algebra O (Graves 1843, Cayley 1845). The once vivid interest in this problem was severely inhibited by the theorems of Frobenius [19] and Zorn [32], asserting that the associative real division algebras are classified by {R, C, H} and the alternative real division algebras are classified by {R, C, H, O}. Hopf's contribution [22] awoke the interest of topologists and launched a new phase in this subject, culminating in Bott, Milnor and Kervaire's (1, 2, 4, 8)-Theorem [8], [23] and Adams's Formula [1] for the span of S n−1 .…”

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confidence: 99%

Dissident maps on the seven-dimensional Euclidean space

Dieterich

Lindberg

2003

Colloq. Math.

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Abstract. Our article contributes to the classification of dissident maps on R 7 , which in turn contributes to the classification of 8-dimensional real division algebras.We study two large classes of dissident maps on R 7 . The first class is formed by all composed dissident maps, obtained from a vector product on R 7 by composition with a definite endomorphism. The second class is formed by all doubled dissident maps, obtained as the purely imaginary parts of the structures of those 8-dimensional real quadratic division algebras which arise from a 4-dimensional real quadratic division algebra by doubling. For each of these two classes we exhibit a complete (but redundant) classification, given by a 49-parameter family of composed dissident maps and a 9-parameter family of doubled dissident maps respectively. The intersection of these two classes forms one isoclass of dissident maps only, namely the isoclass consisting of all vector products on R 7 . Introduction.A dissident map on a finite-dimensional Euclidean vector space V is understood to be a linear map η : V ∧ V → V such that v, w, η(v ∧ w) are linearly independent whenever v, w ∈ V are. The notion of a dissident map provides a link between seemingly diverse aspects of real geometric algebra, thereby revealing its shifting significance. While it generalizes on the one hand the classical notion of a vector product, it specializes on the other hand the structure of a real division algebra. Moreover it yields naturally a large class of selfbijections of the projective space P(V ), many of which are collineations, but some of which, surprisingly, are not.Dissident maps are known to exist in dimensions 0, 1, 3 and 7 only. In dimensions 0 and 1 they are trivial. In dimension 3 they are classified completely and irredundantly. But in dimension 7 they are still far from fully understood.Our article investigates dissident maps on a 7-dimensional Euclidean space by studying and separating two classes of them, namely the composed dissident maps and the doubled dissident maps. Once an exhaustive 49-parameter family of composed dissident maps on R 7 and an exhaustive 9-parameter family of doubled dissident maps on R 7 are obtained, the problem

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Theorie der alternativen ringe

Cited by 134 publications

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Finite semifields and nonsingular tensors

Finite semifields and nonsingular tensors

On Automorphisms of Lie Algebras of Classical Type. II

Dissident maps on the seven-dimensional Euclidean space

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