# Dissident algebras

Colloquium Mathematicae (1999)

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

top

## Abstract

top
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

top

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

top
1. [1] J. F. Adams, Vector fields on spheres, Ann. of Math. 75 (1962), 603-632. Zbl0112.38102
2. [2] M. F. Atiyah and F. Hirzebruch, Bott periodicity and the parallelizability of the spheres, Proc. Cambridge Philos. Soc. 57 (1961), 223-226. Zbl0108.35902
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

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.