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Isomorphiesätze von Frobenius, Hopf und Gelfand-Mazur

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Abstract

In der zweiten Hälfte des 19. Jahrhunderts wurden neben den Quaternionen viele weitere hyperkomplexe Systeme entdeckt und erforscht. Vor allem in England stand diese Kunst in hohem Ansehen. Kurz nach Entdeckung der Quaternionen und vor Einführung von Matrizen erfanden John T. Graves und Arthur Cayley die nichtassoziative Divisionsalgebra der Oktaven. Hamilton führte 1853 in seinen „Lectures on Quaternions“Biquaternionen, das sind Quaternionen mit komplexen Koeffizienten, ein und bemerkte, daß sie keine Divisionsalgebra bilden. William Kingdon Clifford (1845–1879) schuf 1878 die nach ihm benannten assoziativen Algebren.

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... Given any quadratic algebra A, Frobenius' lemma states that the set V = {v ∈ A | v 2 ∈ k1} \ (k1 \ {0}) of purely imaginary elements of A forms a linear subspace of A which is supplementary to k1 (cf. [9], [3], [11]). Accordingly, each x ∈ A has unique decomposition x = λ(x)1 + ι(x), with λ(x) ∈ k and ι(x) ∈ V . ...
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In his remarkable article “Quadratic division algebras” (Trans. Amer. Math. Soc. 105 (1962), 202–221), J. M. Osborn claims to solve ‘the problem of determining all quadratic division algebras of order 4 over an arbitrary field F F of characteristic not two … \ldots modulo the theory of quadratic forms over F F ’ (cf. p. 206). While we shall explain in which respect he has not achieved this goal, we shall on the other hand complete Osborn’s basic results (by a reasoning which is finer than his) to derive in the real ground field case a classification of all 4-dimensional quadratic division algebras and the construction of a 49-parameter family of pairwise nonisomorphic 8-dimensional quadratic division algebras. To make these points clear, we begin by reformulating Osborn’s fundamental observations on quadratic algebras in categorical terms.
... In any quadratic algebra B over R, the subset ImB = {b ∈ B R1 | b 2 ∈ R1} ∪ {0} ⊂ B of purely imaginary elements is a linear subspace of B, and B = R1 ⊕ ImB (Frobe- nius [11]). We shall write α + v instead of α1 + v when referring to elements in this decomposition. ...
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We classify the real commutative division algebras, completing an app-roach by Althoen and Kugler. We solve the isomorphism problem for scalar isotopes of real quadratic division algebras, and classify the generalized pseudo-octonion algebras. In view of earlier results by Benkart, Britten and Osborn and Cuenca Mira et al., this reduces the problem of classifying the real flexi-ble division algebras to the normal form problem for the natural action of the group G2 on the set of positive definite symmetric linear endomorphisms of R 7 . In addition, the automorphism groups of the real flexible division algebras are described.
... ,[8] and 'Osborns Theorem' [10] are predecessors of the following result. ...
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Given a euclidean vector space V , a linear map η : V ∧ V → V is called dissident in case v, w, η(v ∧w) are linearly independent whenever so are v, w ∈ V . The problem of classifying all real quadratic division algebras is reduced to the problem of classifying all eight-dimensional real quadratic division algebras, and further to the problem of clas-sifying all dissident maps η : R 7 ∧ R 7 → R 7 . Should all of these satisfy η = επ for a vector product π on R 7 and a positive-definite endomorphism ε of R 7 , then the latter problem would be solved. This strong factorization property however, even though it does hold for all dissident maps in lower dimensions, is shown to fail in dimension 7. It is replaced by a weak factorization property for which a proof is announced. Evidence is given for the conjecture that even weak fac-torization will suffice to accomplish the complete classification of all eight-dimensional real quadratic division algebras.
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With every finite-dimensional algebra A over any field k we associate an 8-tuple of linear or bilinear forms on A, all of which are defined in terms of traces. For every groupoid 𝒞 formed by a class of k-algebras of fixed finite dimension, this passage is functorial and, when composed with any map that is constant on isoclasses, gives rise to an abundance of maps f: 𝒞 → I such that the fibres of f form a block decomposition of 𝒞. We study this decomposition for specific choices of 𝒞 and f, thereby putting established results from diverse algebraic theories into a unifying perspective, but also gaining new insight into classical groupoids of algebras such as associative unital algebras, division algebras, or composition algebras over general ground fields.
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A celebrated theorem of Hopf (1940) [11], Bott and Milnor (1958) [1], and Kervaire (1958) [12] states that every finite-dimensional real division algebra has dimension 1, 2, 4, or 8. While the real division algebras of dimension 1 or 2 and the real quadratic division algebras of dimension 4 have been classified (Dieterich (2005) [6], Dieterich (1998) [3], Dieterich and Öhman (2002) [9]), the problem of classifying all 8-dimensional real quadratic division algebras is still open. We contribute to a solution of that problem by proving that every 8-dimensional real quadratic division algebra has degree 1, 3, or 5. This statement is sharp. It was conjectured in Dieterich et al. (2006) [7].
Article
We study dissident maps η on Rm for m∈{3,7} by investigating liftings Φ:Rm→Rm of the selfbijection induced by η. Our main result (Theorem 2.4) asserts the existence and uniqueness, up to a non-zero scalar multiple, of a lifting Φ whose component functions are homogeneous polynomials of degree d, relatively prime and without non-trivial common zero. We prove that 1⩽d⩽m-2.We achieve a complete description of all dissident maps of degree one and we solve their isomorphism problem (Theorems 4.8 and 4.13). As a consequence, we achieve a complete description of all real quadratic division algebras of degree one and we solve their isomorphism problem (Theorems 5.1 and 5.3). Moreover we present examples of eight-dimensional real quadratic division algebras of degree 3 and 5 (Proposition 6.3). This extends earlier results of Osborn [Trans. Amer. Math. Soc. 105 (1962) 202–221], Hefendehl [Geometriae Dedicata 9 (1980) 129–152], Hefendehl-Hebeker [Arch. Math. 40 (1983) 50–60], Cuenca Mira et al. [Lin. Alg. Appl. 290 (1999) 1–22], Dieterich [Proc. Amer. Math. Soc. 128 (2000) 3159–3166] and Dieterich and Lindberg [Colloq. Math. 97 (2003) 251–276] on the classification of real quadratic division algebras.
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A square-ordered field, also called a Hilbert field of type (A), is understood to be an ordered field all of whose positive elements are squares. The problem of classifying, up to isomorphism, all 4-dimensional quadratic division algebras over a square-ordered field k is shown to be equivalent to the problem of finding normal forms for all pairs (X, Y) of 3 x 3 matrices over k, X being antisymmetric and Y being positive definite, under simultaneous conjugation by SO3(k). A solution is derived for the subproblem of this matrix pair problem defined by requiring Y + Y-t to be orthogonally diagonalizable. The classifying list is given in terms of a 9-parameter family of configurations in k(3), formed by a pair of points and an ellipsoid in normal position. Each 4-dimensional quadratic division algebra A over a square-ordered field k is shown to determine, uniquely up to sign, a self-adjoint linear endomorphism alpha of its purely imaginary hyperplane. Calling A diagonalizable in case alpha is orthogonally diagonalizable, the achieved solution of the matrix pair subproblem yields a full classification of all diagonalizable 4-dimensional quadratic division k-algebras. This generalizes earlier results of both Hefendehl-Hebeker who classified, over Hilbert fields, those 4-dimensional quadratic division algebras having infinite automorphism group, and Dieterich, who achieved a full classification of all real 4-dimensional quadratic division algebras. Finally, the paper describes explicitly how Hefendehl-Hebeker's classifying list, given in terms of a 4-parameter family of pairs of definite 3 x 3 matrices over k, embeds into the classifying list of configurations. The image of this embedding turns out to coincide with the sublist of the list formed by all non-generic configurations.
Weiter verweisen wir auf den Übersichtsartikel Norm and spectral Characterization in Banach algebras von V. A. Belfi und R. S. Doran in L’Enseign
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