Graded Lie algebras associated to a representation of a quadratic algebra

Hubert Rubenthaler (1)

1 IRMA - Institut de Recherche Mathématique Avancée

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 $ \Gamma(\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. $V^*)$ is given by the representation $\rho$ (resp. $\rho^*$), and where the bracket between $V$ and $V^*$ depends on $B_{0}$ and $\rho$. This implies the existence of two $\Z$-graded Lie algebras ${\go g}_{max}(\Gamma(\go{g}_{0}, B_{0}, V))$ and ${\go g}_{min}(\Gamma(\go{g}_{0}, B_{0}, V))$ whose local part is $\Gamma(\go{g}_{0},B_{0}, V)$. We investigate these graded Lie algebras, more specifically in the case where ${\go g}_{0}$ is reductive. Roughly speaking, the map $(\go{g}_{0},B_{0}, V)\longmapsto {\go g}_{min}(\Gamma(\go{g}_{0}, B_{0}, V))$ 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.

Mots clés: quadratic Lie algebra; reductive Lie algebra; Graded Lie algebra

Domaines: Théorie des représentations [math.RT]

Hubert Rubenthaler. Graded Lie algebras associated to a representation of a quadratic algebra. 2014. ⟨hal-01070209v3⟩