The Kostant invariant and special ϵ-orthogonal representations for ϵ-quadratic colour Lie algebras

Let k be a field of characteristic not two or three, let g be a finite-dimensional colour Lie algebra and let V be a finite-dimensional representation of g. In this article we give various ways of constructing a colour Lie algebra g˜ whose bracket in some sense extends both the bracket of g and the action of g on V. Colour Lie algebras, originally introduced by R. Ree ([18]), generalise both Lie algebras and Lie superalgebras, and in those cases our results imply many known results ([13], [14], [5], [23]). For a class of representations arising in this context we show there are covariants satisfying identities analogous to Mathews identities for binary cubics.