Trilinear products and comtrans algebra representations

Abstract

Comtrans algebras are modules over a commutative ring R equipped with two trilinear operations: a left alternative commutator and a translator satisfying the Jacobi identity, the commutator and translator being connected by the so-called comtrans identity. The standard construction of a comtrans algebra uses the ternary commutator and translator of a trilinear product. If 6 is invertible in R, then each comtrans algebra arises in this standard way from the so-called bogus product. Consider a vector space E of dimension n over a field R. While the dimension of the space of all trilinear products on E is n 4 , the dimension of the space of all comtrans algebras on E is less, namely 5 6 n 4 - 1 2 n 3 - 1 3 n 2 . The paper determines which trilinear products may be represented as linear combinations of the commutator and translator of a comtrans algebra. For R not of characteristic 3, the necessary and sufficient condition for such a representation is the strong alternativity xxy + xyx + yxx = 0 of the trilinear product xyz. For R also not of characteristic 2, it is shown that the representation may be given by the bogus product. A suitable representation for the characteristic 2 case is also obtained.