Minimal graded Lie algebras and representations of quadratic algebras

Let (g0,B0) be a quadratic Lie algebra (i.e. a Lie algebra g0 with a non-degenerate symmetric invariant bilinear form B0) and let (ρ,V) be a finite dimensional representation of g0. We define on Γ(g0,B0,V)=V⁎⊕g0⊕V a structure of local Lie algebra in the sense of Kac ([4]), where the bracket between g0 and V (resp. V⁎) is given by the representation ρ (resp. ρ⁎), and where the bracket between V and V⁎ depends on B0 and ρ. This implies the existence of two Z-graded Lie algebras gmax(Γ(g0,B0,V)) and gmin(Γ(g0,B0,V)) whose local part is Γ(g0,B0,V). We investigate these graded Lie algebras, more specifically in the case where g0 is reductive. Roughly speaking, the map (g0,B0,V)⟼gmin(Γ(g0,B0,V)) is a bijection between triplets and a class of graded Lie algebras. We show that the existence of “associated sl2-triples” is equivalent to the existence of non-trivial relative invariants on some orbit, and we define the “graded Lie algebras of polynomial type” which give rise to some dual airs.