## Real vector product algebras

Citations: | 1 - 1 self |

### BibTeX

@MISC{Darpö_realvector,

author = {Erik Darpö and Erik Darpö},

title = {Real vector product algebras},

year = {}

}

### OpenURL

### Abstract

We give a short and elementary proof for the classification of the real vector product algebras. A real vector product algebra is a Euclidean space V = (V, 〈 〉) together with an anti-symmetric bilinear map V × V → V, (u, v) ↦ → uv having the property that the set {u, v, uv} is orthonormal whenever {u, v} is. A morphism between vector product algebras V and W is a linear map ϕ: V → W such that ϕ(uv) = ϕ(u)ϕ(v) and 〈ϕ(u), ϕ(v) 〉 = 〈u, v 〉 for all u, v ∈ V. Two vector product algebras V and W are isomorphic if there exists a bijective morphism ϕ: V → W. Note that in our definition, vector product algebras are not required to be finite-dimensional. This, however, follows from the theory. In the classification theorem, our Theorem 3, it is established that real vector product algebras exist in dimension 0, 1, 3 and 7 only, and form one isomorphism class in each

### Citations

88 |
On indecomposable quadratic forms
- O’Meara
- 1980
(Show Context)
Citation Context ...’s theorem, as well as the standard result that every regular isotropic quadratic space is universal. The relevant theory about Hilbert fields is found in [3], for the other prerequisites we refer to =-=[9]-=-. A composition algebra is a quadratic space A together with a bilinear multiplication A × A → A, (x, y) ↦→ xy such that N(xy) = N(x)N(y) for all x, y ∈ A. A morphism between composition algebras A an... |

19 |
On quaternions and their generalization and the history of the eight square theorem
- Dickson
- 1919
(Show Context)
Citation Context ...ce between the respective categories of vector product algebras and composition algebras with identity element (c.f. [2]). Unital composition algebras have been extensively studied (some examples are =-=[1, 4, 6, 7]-=-). They occur only in dimensions 1, 2, 4 and 8 (so in particular they are all finite-dimensional), and can be constructed from the ground field via application of the Cayley-Dickson process. Two unita... |

15 |
Composition Algebras and Their Automorphisms
- Jacobson
(Show Context)
Citation Context ...5], it is proven that every unital composition algebra has dimension 1, 2, 4 or 8 (this corresponds to our Theorem 6:1). Results equivalent to Theorem 6:2, Proposition 7 and Corollary 14 are given in =-=[4]-=-. Nevertheless, developing the theory directly for vector product algebra has certain advantages. First, it illuminates the structure of these algebras more clearly than what is possible with the appr... |

14 |
Vector cross products
- Brown, Gray
- 1967
(Show Context)
Citation Context ...he bilinear form induced from A. These construction actually determine an equivalence between the respective categories of vector product algebras and composition algebras with identity element (c.f. =-=[2]-=-). Unital composition algebras have been extensively studied (some examples are [1, 4, 6, 7]). They occur only in dimensions 1, 2, 4 and 8 (so in particular they are all finite-dimensional), and can b... |

12 |
On the dimension of a composition algebra
- Rost
- 1996
(Show Context)
Citation Context ...n [1] using topological methods. A treatment based on the correspondence with composition algebras is found in [3], as well as an alternative approach relying on results from algebraic topology. Rost =-=[4]-=- has considered vector product algebras over general fields k with char k �= 2, and by elementary means proven that the equation d(d − 1)(d − 3)(d − 7) = 0 must hold in the ground field, if d is the d... |

12 |
Quadratic forms permiting composition
- Albert
- 1942
(Show Context)
Citation Context ...ce between the respective categories of vector product algebras and composition algebras with identity element (c.f. [2]). Unital composition algebras have been extensively studied (some examples are =-=[1, 4, 6, 7]-=-). They occur only in dimensions 1, 2, 4 and 8 (so in particular they are all finite-dimensional), and can be constructed from the ground field via application of the Cayley-Dickson process. Two unita... |

8 |
products of vectors in higher dimensional Euclidean spaces
- Massey, Cross
- 1983
(Show Context)
Citation Context ...or example [2]). Vector product algebras were first considered, and classified, by Eckmann [1] using topological methods. A treatment based on the correspondence with composition algebras is found in =-=[3]-=-, as well as an alternative approach relying on results from algebraic topology. Rost [4] has considered vector product algebras over general fields k with char k �= 2, and by elementary means proven ... |

6 |
Stetige Lösungen linearer Gleichungssysteme
- Eckmann
- 1943
(Show Context)
Citation Context ..., w〉, αv + βw + uv) is a composition algebra. Conversely, every composition algebra arises in this way (see for example [2]). Vector product algebras were first considered, and classified, by Eckmann =-=[1]-=- using topological methods. A treatment based on the correspondence with composition algebras is found in [3], as well as an alternative approach relying on results from algebraic topology. Rost [4] h... |

4 |
Quadratic forms ‘à la’ local theory
- Fröhlich
- 1967
(Show Context)
Citation Context ...scription of the morphisms between vector 1By a classification we mean a list of pairwise non-isomorphic algebras, exhausting all isomorphism classes. 2The Hilbert fields were defined and examined in =-=[3]-=-. A recapitulation is given in our Section 3.3. 2sproduct algebras, and provides means to determine weather two given algebras are isomorphic. Finally, the reduction to quadratic forms and the formula... |

4 |
Infinite-dimensional quadratic forms admitting composition
- Kaplansky
- 1953
(Show Context)
Citation Context ... question of which quadratic spaces that admit a vector product algebra structure. For many of the results in the present article, equivalent statements are already known for composition algebras. In =-=[5]-=-, it is proven that every unital composition algebra has dimension 1, 2, 4 or 8 (this corresponds to our Theorem 6:1). Results equivalent to Theorem 6:2, Proposition 7 and Corollary 14 are given in [4... |

3 |
Trace formulas in vector product algebras
- Meyberg
- 2002
(Show Context)
Citation Context ...sition algebras (see below), which implies that every vector product algebra must have dimension 0, 1, 3 or 7. An alternative proof for the dimension relation d(d−1)(d−3)(d−7) = 0 is given by Meyberg =-=[8]-=-. The correspondence with composition algebras, and thereby with the Hurwitz’ problem is an important argument for the significance of vector product algebras. In the rest of the present section we ex... |

3 |
Composition algebras and their
- Jacobson
- 1958
(Show Context)
Citation Context ...ce between the respective categories of vector product algebras and composition algebras with identity element (c.f. [2]). Unital composition algebras have been extensively studied (some examples are =-=[1, 4, 6, 7]-=-). They occur only in dimensions 1, 2, 4 and 8 (so in particular they are all finite-dimensional), and can be constructed from the ground field via application of the Cayley-Dickson process. Two unita... |

2 |
Composition algebras. Hurwitz’s theorem— Vector-product algebras
- Koecher, Remmert
- 1995
(Show Context)
Citation Context ...lgebra V , the space R×V with multiplication given by (α, v) · (β, w) = (αβ − 〈v, w〉, αv + βw + uv) is a composition algebra. Conversely, every composition algebra arises in this way (see for example =-=[2]-=-). Vector product algebras were first considered, and classified, by Eckmann [1] using topological methods. A treatment based on the correspondence with composition algebras is found in [3], as well a... |

1 |
Reella vektorproduktalgebror
- Darpö
(Show Context)
Citation Context ...roduct were first considered, and classified, by Eckmann [2] in 1942, using topological methods. Other treatments are found in for example [7] and [6]. The technique of the present article is used in =-=[1]-=- for a comprehensive proof of the classification theorem in this special case. Vector product algebras over general ground fields of characteristic different from 2 have been studied by Rost [10]. He ... |

1 |
Reella vektorprodukter
- Darpö
(Show Context)
Citation Context ...oducts were first considered, and classified, by Eckmann [5] in 1942, using topological methods. Other treatments are found in for example [9] and [8]. The technique of the present article is used in =-=[3]-=- for a comprehensive proof of the classification theorem in this special case. In this article, we present a direct proof of Theorem 1, avoiding the detour via composition algebras. In spite of its si... |