The Kostant invariant and special $\epsilon$-orthogonal representations for $\epsilon$-quadratic colour Lie algebras

TL;DR

This paper explores the construction of extended colour Lie algebras from given colour Lie algebras and their representations, generalizing known algebraic structures and identities, with implications for Lie algebra theory.

Contribution

It introduces methods to construct extended colour Lie algebras incorporating both original brackets and actions, generalizing classical Lie and Lie superalgebras.

Findings

Construction methods for extended colour Lie algebras.

Identification of covariants satisfying identities similar to Mathews identities.

Connections to known results in Lie algebra and superalgebra theory.

Abstract

Let k be a field of characteristic not two or three, let $\mathfrak{g}$ be a finite-dimensional colour Lie algebra and let V be a finite-dimensional representation of $\mathfrak{g}$. In this article we give various ways of constructing a colour Lie algebra $\tilde{\mathfrak{g}}$ whose bracket in some sense extends both the bracket of $\mathfrak{g}$ and the action of $\mathfrak{g}$ on V. Colour Lie algebras, originally introduced by R. Ree ([Ree60]), generalise both Lie algebras and Lie superalgebras, and in those cases our results imply many known results ([Kos99], [Kos01], [CK15], [SS15]). For a class of representations arising in this context we show there are covariants satisfying identities analogous to Mathews identities for binary cubics.