Dissident algebras

Ernst Dieterich

Colloquium Mathematicae (1999)

  • Volume: 82, Issue: 1, page 13-23
  • ISSN: 0010-1354

Abstract

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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.

How to cite

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Dieterich, Ernst. "Dissident algebras." Colloquium Mathematicae 82.1 (1999): 13-23. <http://eudml.org/doc/210747>.

@article{Dieterich1999,
abstract = {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.},
author = {Dieterich, Ernst},
journal = {Colloquium Mathematicae},
keywords = {anti-commutative algebra; quadratic division algebra; vector product algebras; dissident algebras},
language = {eng},
number = {1},
pages = {13-23},
title = {Dissident algebras},
url = {http://eudml.org/doc/210747},
volume = {82},
year = {1999},
}

TY - JOUR
AU - Dieterich, Ernst
TI - Dissident algebras
JO - Colloquium Mathematicae
PY - 1999
VL - 82
IS - 1
SP - 13
EP - 23
AB - 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.
LA - eng
KW - anti-commutative algebra; quadratic division algebra; vector product algebras; dissident algebras
UR - http://eudml.org/doc/210747
ER -

References

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  3. [3] E. Dieterich, Zur Klassifikation vierdimensionaler reeller Divisionsalgebren, Math. Nachr. 194 (1998), 13-22. 
  4. [4] E. Dieterich, Real quadratic division algebras, Comm. Algebra, to appear. 
  5. [5] B. Eckmann, Stetige Lösungen linearer Gleichungssysteme, Comm. Math. Helv. 15 (1942/43), 318-339. Zbl0028.32001
  6. [6] H. Hopf, Ein topologischer Beitrag zur reellen Algebra, ibid. 13 (1940/41), 219-239. Zbl0024.36002
  7. [7] M. Koecher and R. Remmert, Isomorphiesätze von Frobenius, Hopf und Gelfand-Mazur, in: Zahlen, Springer-Lehrbuch, 3. Auflage, 1992, 182-204. 
  8. [8] M. Koecher and R. Remmert, Cayley-Zahlen oder alternative Divisionsalgebren, ibid., 205-218. 
  9. [9] M. Koecher and R. Remmert, Kompositionsalgebren. Satz von Hurwitz. Vektorprodukt-Algebren, ibid., 219-232. 
  10. [10] J. Milnor, Some consequences of a theorem of Bott, Ann. of Math. 68 (1958), 444-449. Zbl0085.17301

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