Trilinear alternating forms on a vector space of dimension 7. (English) Zbl 0646.15019

Cohen, Arjeh M.; Helminck, Aloysius G.

Commun. Algebra 16, No. 1, 1-25 (1988).

If E is a vector space of dimension n over the field F, then a trilinear form f on E is called alternating, if for all \(x_ 1,x_ 2,x_ 3\in E\) with \(x_ i=x_ j\) for at least two distinct i,j (1\(\leq i,j\leq 3)\) we have \(f(x_ 1,x_ 2,x_ 3)=0\). The classification of these forms was carried out by J. A. Schouten [Rend. Circ. Mat. Palermo 55, 137-156 (1931; Zbl 0001.35401)] for \(F={\mathbb{C}}\) (complex numbers) and by J. Cresp in his Thesis (1976) for F algebraically closed of characteristic \(\neq 2,3\) and \(n=7.\)
In the underlying paper a short proof is given of the classification for F algebraically closed of arbitrary characteristic and \(n=7\). Moreover a classification for \(n\leq 7\) follows over certain non-algebraically closed fields, including all finite fields by use of Galois cohomology.

Reviewer: A.H.Boers

MSC:

15A63	Quadratic and bilinear forms, inner products
11E76	Forms of degree higher than two
20G15	Linear algebraic groups over arbitrary fields
17D05	Alternative rings

Keywords:

alternating forms; dimension 7; trilinear forms on a vector space; classification; non-algebraically closed fields; finite fields; Galois cohomology

Citations:

Zbl 0001.35401

References:

[1]	Borel A., Linear Algebraic Groups (1969) · Zbl 0186.33201
[2]	Bourbaki N., Elements de Mathématique Algèbre (1971)
[3]	Cresp J., Orbits in , Kummer manifolds and the cohomology of a hyperplane section of a Grassmannian, Thesis (1976)
[4]	Dieudonné J., La géométrie des groupes classiques (1955) · Zbl 0067.26104
[5]	Gurevich G.B., Foundations of the Theory of Algebraic Invariants (1964) · Zbl 0128.24601
[6]	Lang S., Algebra (1966)
[7]	DOI: 10.1007/BF03016791 · Zbl 0001.35401 · doi:10.1007/BF03016791
[8]	Serre J.P., Lecture Notes in Math 5 (1964)
[9]	Springer T.A., Oktaven, Jordan-Algebren und Ausnahmegruppen (1963)
[10]	Migliore, Phd. Thesis (1982)