Publication: Quaternary binary trilinear forms of norm unity
No Thumbnail Available
Official URL
Full text at PDC
Publication Date
2006
Authors
Bernhardsson, Bo
Cobos, Fernando
Kühn, Tomas
Mondoc, Daniel
Peetre, Jaak
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Estonian Academy of Sciences
Citations
Exportar
Abstract
So far trilinear forms have mostly been considered in low dimensions, in particular the dimension two (binary) case, and when the ring of scalars K is either the real numbers R
or the complex ones C. The main aim in both situations has been to decide when a normalized form has norm unity. Here we consider the case of quaternions, K = H. This note is rather preliminary, and somewhat experimental, where the computer program Mathematica plays a certain role. A preliminary result obtained is that the form has norm unity if and only if the discriminant of a certain 5-dimensional quadratic form has all its principal minors nonnegative.
We found also a rather unexpected similarity between the noncommutative case of Hnand the commutative one of R and C.
Description
UCM subjects
Unesco subjects
Keywords
Citation
Gel’fand, I. M., Kapranov, M. M. and Zelevinsky, A. V. Discriminants, Resultants, and Multilinear Determinants. Birkhäuser–Boston, MA, 1994.
Arazy, J. and Friedman, Y. Contractive projections in C1 and C1. Mem. Amer. Math. Soc., 1978, 13, No. 200.
Peetre, J. Quaternary binary trilinear forms. 2005 (http://www.maths.lth.se/matematiklu/personal/jaak/engJP.html).
Cobos, F., Kühn, T. and Peetre, J. Extreme points of the complex binary trilinear ball. Studia Math., 2000, 138, 81–92.
Bernhardsson, B. and Peetre, J. Singular values of trilinear forms. Experiment. Math., 2001, 10, 509–517.
Cobos, F., Kühn, T. and Peetre, J. Supplement to “Generalized non-commutative integration– trilinear forms and Jordan quintuples”. Technical report, Lund, 1995.
Gantmacher, F. R. Theory of Matrices. 2nd edition, Nauka, Moscow, 1966 (in Russian).
Second German translation: Matrizentheorie. Springer-Verlag, Berlin, 1986.
Graslewicz, R. and John, K. Extreme elements of the unit ball of bilinear operators on l2
2. Arch. Math. (Basel), 1988, 50, 264–269.