Classification of trilinear operations and their minimal polynomial identities
We use the representation theory of the symmetric group to classify
all multilinear operations over the field of rational numbers up to
equivalence. In the case n = 3, we obtain explicit representatives
of the equivalence classes of trilinear operations:
[a,b,c] =
x1 abc + x2 acb + x3 bac +
x4 bca + x5 cab + x6 cba.
From these results we obtain one-parameter families of deformations
of the classical Lie, Jordan and anti-Jordan triple products and the
corresponding varieties of triple systems. We use computational
algebra to study the nonassociative polynomial identities satisfied
by these operations in every totally associative ternary algebra.
We obtain 18 new trilinear operations for which the corresponding
varieties of triple systems are defined by identities of degrees 3
and 5. For 10 of these operations we classify their obvious
identities in degree 3 and their minimal identities in degree 5.
M. R. Bremner < bremner@math.usask.ca >
L. A. Peresi < peresi@ime.usp.br >