Representations associated to gradations of Lie algebras and colour Lie algebras

Let $k$ be a field of characteristic not 2 or 3. Colour Lie algebras generalise both Lie algebras and Lie superalgebras. In this thesis we study representations $V$ of colour Lie algebras $\mathfrak{g}$ arising from colour Lie algebras structures on the vector space $\mathfrak{g}\oplus V$. Firstly, we study the general structure of simple three-dimensional Lie algebras over $k$. Then, we classify up to isomorphism finite-dimensional Lie superalgebras whose even part is a simple three-dimensional Lie algebra. Next, to an abelian group $\Gamma$ and a commutation factor $\epsilon$ of $\Gamma$, we develop the multilinear algebra associated to $\Gamma$-graded vector spaces. In this context, colour Lie algebras play the role of Lie algebras. This language allows us to state and prove a theorem reconstructing an $\epsilon$-quadratic colour Lie algebra $\mathfrak{g}\oplus V$ from an $\epsilon$-orthogonal representation $V$ of an $\epsilon$-quadratic colour Lie algebra $\mathfrak{g}$. This theorem involves an invariant taking its values in the $\epsilon$-exterior algebra of $V$ and generalises results of Kostant and Chen-Kang. We then introduce the notion of a special $\epsilon$-orthogonal representation $V$ of an $\epsilon$-quadratic colour Lie algebra $\mathfrak{g}$ and show that it allows us to define an $\epsilon$-quadratic colour Lie algebra structure on the vector space $\mathfrak{g}\oplus \mathfrak{sl}(2,k)\oplus V\otimes k^2$. Finally we give examples of special $\epsilon$-orthogonal representations and in particular examples of special orthogonal representations of Lie algebras amongst which are: a one-parameter family of representations of $\mathfrak{sl}(2,k)\times \mathfrak{sl}(2,k)$ ; the 7-dimensional fundamental representation of a Lie algebra of type $G_2$ ; the 8-dimensional spinor representation of a Lie algebra of type $\mathfrak{so}(7)$.