Symplectic structures on quadratic Lie algebras

Abstract

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 Lie algebra admitting an invertible derivation 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 description in terms of symplectic quadratic double extensions.

Polynomial Poisson structures on affine solvmanifolds

“…To get examples in dimension greater or equal to 6, one cane use the results given in [1] or [12]. Note that Theorem 1.1 asserts in part 3 that Ω + is of degree at most n − 1.…”

Leibniz algebras with pseudo-Riemannian bilinear forms

“…In Ref. [2], T * -extension is one of the main tools for their investigation. Furthermore, Keith's notion of bi-extension [6] of Lie algebras contains the T * -extension as a special case where B = B ⊥ and M := B/B ⊥ = 0.…”

Quadratic symplectic Lie superalgebras and Lie bi-superalgebras

“…These descriptions are based on some techniques of double extensions of Lie superalgebras. Recently, in [4], I. Bajo et al gave, in particular, inductive descriptions of quadratic symplectic Lie algebras by using some concepts of double extensions of Lie algebras.…”

“…We continue the work on quadratic symplectic Lie algebras done in [4]. In particular, in this work, we generalize to Lie superalgebras the concepts of double extension used in [4] in order to describe the quadratic symplectic Lie algebras.…”

“…We continue the work on quadratic symplectic Lie algebras done in [4]. In particular, in this work, we generalize to Lie superalgebras the concepts of double extension used in [4] in order to describe the quadratic symplectic Lie algebras. Since this generalization is not sufficient to give a complete inductive description of quadratic symplectic Lie superalgebras, we introduce a new concepts of double extensions of this type of Lie superalgebras.…”