Symplectic structures on quadratic Lie algebras

We study quadratic Lie algebras over a field K of null characteristic which admit, at the same time, a symplectic structure. We see that if K is algebraically closed every such Lie algebra may be constructed as the T*-extension of a nilpotent algebra admitting an invertiblederivation and also as the double extension of another quadratic symplectic Lie algebra by the one-dimensional Lie algebra. Finally, we prove that every symplectic quadratic Lie algebra is a special symplectic Manin algebra and we give an inductive classification in terms of symplectic quadratic double extensions

Dirac operators and the Very Strange Formula for Lie superalgebras

Using a super-affine version of Kostant's cubic Dirac operator, we prove a very strange formula for quadratic finite-dimensional Lie superalgebras with a reductive even subalgebra.Comment: Latex file, 25 pages. A few misprints corrected. To appear in the forthcoming volume "Advances in Lie Superalgebras", Springer INdAM Serie

The Harish-Chandra isomorphism for reductive symmetric superpairs

We consider symmetric pairs of Lie superalgebras which are strongly reductive and of even type, and introduce a graded Harish-Chandra homomorphism. We prove that its image is a certain explicit filtered subalgebra of the Weyl invariants on a Cartan subspace whose associated graded is the image of Chevalley's restriction map on symmetric invariants. This generalises results of Harish-Chandra and V. Kac, M. Gorelik.Comment: 43 pages; v2: substantially improved versio

Construction of symplectic quadratic Lie algebras from Poisson algebras

International audienceWe introduce the notion of quadratic (resp. symplectic quadratic) Poisson algebras and we show how one can construct new interesting quadratic (resp. symplectic quadratic) Lie algebras from quadratic (resp. symplectic quadratic) Poisson algebras. Finally, we give inductive descriptions of symplectic quadratic Poisson algebras