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 non- negative. We found also a rather unexpected similarity between the noncommutative case of H and the commutative one of R and C.
1. Gel’fand, I. M., Kapranov, M. M. and Zelevinsky, A. V. Discriminants, Resultants, and Multilinear Determinants. Birkhäuser–Boston, MA, 1994.
https://doi.org/10.1007/978-0-8176-4771-1
2. Arazy, J. and Friedman, Y. Contractive projections in C1 and C∞. Mem. Amer. Math. Soc., 1978, 13, No. 200.
https://doi.org/10.1090/memo/0200
3. Peetre, J. Quaternary binary trilinear forms. 2005.
http://www.maths.lth.se/matematiklu/ personal/jaak/engJP.html
4. Cobos, F., Kühn, T. and Peetre, J. Extreme points of the complex binary trilinear ball. Studia Math., 2000, 138, 81–92.
5. Bernhardsson, B. and Peetre, J. Singular values of trilinear forms. Experiment. Math., 2001, 10, 509–517.
https://doi.org/10.1080/10586458.2001.10504670
6. Cobos, F., Kühn, T. and Peetre, J. Supplement to “Generalized non-commutative integra- tion – trilinear forms and Jordan quintuples”. Technical report, Lund, 1995.
7. Gantmacher, F. R. Theory of Matrices. 2nd edition, Nauka, Moscow, 1966 (in Russian). Second German translation: Matrizentheorie. Springer-Verlag, Berlin, 1986.
https://doi.org/10.1007/978-3-642-71243-2
8. Grạślewicz, R. and John, K. Extreme elements of the unit ball of bilinear operators on l22. Arch. Math. (Basel), 1988, 50, 264–269.
https://doi.org/10.1007/BF01187744
