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## Colloquium Mathematicum

1999 | 82 | 1 | 13-23
Tytuł artykułu

### Dissident algebras

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EN
Abstrakty
EN
Given a euclidean vector space V = (V,〈〉) and a linear map η: V ∧ V → V, the anti-commutative algebra (V,η) is called dissident in case η(v ∧ w) ∉ ℝv ⊕ ℝw for each pair of non-proportional vectors (v,w) ∈ $V^2$. For any dissident algebra (V,η) and any linear form ξ: V ∧ V → ℝ, the vector space ℝ × V, endowed with the multiplication (α,v)(β,w) = (αβ -〈v,w〉+ ξ(v ∧ w), αw + βv + η(v ∧ w)), is a quadratic division algebra. Up to isomorphism, each real quadratic division algebra arises in this way. Vector product algebras are classical special cases of dissident algebras. Via composition with definite endomorphisms they produce new dissident algebras, thus initiating a construction of dissident algebras in all possible dimensions m ∈ {0,1,3,7} and of real quadratic division algebras in all possible dimensions n ∈ {1,2,4,8}. For m ≤ 3 and n ≤ 4, this construction yields complete classifications. For m=7 it produces a 28-parameter family of pairwise non-isomorphic dissident algebras. For n=8 it produces a 49-parameter family of pairwise non-isomorphic real quadratic division algebras.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
13-23
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-11-23
poprawiono
1999-03-12
Twórcy
autor
• Matematiska institutionen, Uppsala universitet, Box 480, S-751 06 Uppsala, Sweden
Bibliografia
• [1] J. F. Adams, Vector fields on spheres, Ann. of Math. 75 (1962), 603-632.
• [2] M. F. Atiyah and F. Hirzebruch, Bott periodicity and the parallelizability of the spheres, Proc. Cambridge Philos. Soc. 57 (1961), 223-226.
• [3] E. Dieterich, Zur Klassifikation vierdimensionaler reeller Divisionsalgebren, Math. Nachr. 194 (1998), 13-22.
• [4] E. Dieterich, Real quadratic division algebras, Comm. Algebra, to appear.
• [5] B. Eckmann, Stetige Lösungen linearer Gleichungssysteme, Comm. Math. Helv. 15 (1942/43), 318-339.
• [6] H. Hopf, Ein topologischer Beitrag zur reellen Algebra, ibid. 13 (1940/41), 219-239.
• [7] M. Koecher and R. Remmert, Isomorphiesätze von Frobenius\rm, \it Hopf und Gelfand-Mazur, in: Zahlen, Springer-Lehrbuch, 3. Auflage, 1992, 182-204.
• [8] M. Koecher and R. Remmert, Cayley-Zahlen oder alternative Divisionsalgebren, ibid., 205-218.
• [9] M. Koecher and R. Remmert, Kompositionsalgebren. Satz von Hurwitz. Vektorprodukt-Algebren, ibid., 219-232.
• [10] J. Milnor, Some consequences of a theorem of Bott, Ann. of Math. 68 (1958), 444-449.
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