Representation theory of comtrans algebras (Q1196817)

A comtrans algebra \(E\) over a commutative ring \(R\) with 1 is an \(R\)- module \(E\) equipped with two trilinear operations \([ , , ]\) and \(\langle , ,\rangle\) from \(E^ 3\) to \(E\) such that \([x,x,z]=0\), \(\langle x,y,z\rangle+\langle y,z,x\rangle+\langle z,x,y\rangle=0\) and \([x,y,x]=\langle x,y,z\rangle\) for all \(x,y,z\in E\). These algebras first appeared from a problem in differential geometry [\textit{C. Scheiderer}, Gewebegeometrie, Tagungsbericht 27, Mathematisches Forschungsinstitut Oberwolfach (1984)]. This paper considers representations of these algebras by showing that they are equivalent to the representations of an associative universal enveloping algebra. The structure of the universal algebra of \(E\) is that of the tensor algebra of \(V=(E\wedge E)\oplus(E\otimes E)\oplus(E\otimes E)\).