Minimal graded Lie algebras and representations of quadratic algebras

Abstract

Let $({\go g}\_{0},B\_{0})$ be a quadratic Lie algebra (i.e. a Lie algebra $\go{g}\_{0}$ with a non degenerate symmetric invariant bilinear form $B\_0$) and let $(\rho ,V)$ be a finite dimensional representation of ${\go g}\_{0}$. We define on $ Γ(\go{g}\_{0}, B\_{0}, V)=V^*\oplus {\go g}\_{0}\oplus V$ a structure of local Lie algebra in the sense of Kac (\cite{Kac1}), where the bracket between $\go{g}\_{0}$ and $V$ (resp. $V^{\ast })$ is given by the representation $\rho$ (resp. ${\rho }^{\ast }$), and where the bracket between $V$ and $V^{\ast }$ depends on $B\_0$ and $\rho$. This implies the existence of two $\mathbb{Z}$-graded Lie algebras ${\go g}\_{max}(Γ(\go{g}\_{0}, B\_{0}, V))$ and ${\go g}\_{min}(Γ(\go{g}\_{0}, B\_{0}, V))$ whose local part is $Γ(\go{g}\_{0},B\_{0}, V)$. We investigate these graded Lie algebras, more specifically in the case where ${\go g}\_{0}$ is reductive. Roughly speaking, the map $(\go{g}\_{0},B\_{0}, V)\longmapsto {\go g}\_{min}(Γ(\go{g}\_{0}, B\_{0}, V))$ a bijection between triplets and a class of graded Lie algebras. We show that the existence of "associated $\go {sl}\_{2}$-triples" is equivalent to the existence of non trivial relative invariants on some orbit, and we define the "graded Lie algebras of symplectic type" which give rise to some dual pairs.