Construction of Symplectic Quadratic Lie Algebras from Poisson Algebras
We 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,
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Abstract We 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.
1 Introduction In this paper, we consider finite dimensional algebras over a commutative field K of characteristic zero. Recall that the Lie algebra G of a Lie group G which admits a bi-invariant pseudoRiemannian structure is quadratic (i.e. G is endowed with a symmetric non degenerate invariant (or associative) bilinear form B). Conversely, any connected Lie group whose Lie algebra G is quadratic, is endowed with bi-invariant pseudo-Riemannian structure [14]. The semisimple Lie algebras are quadratic. Many solvable Lie algebras are also quadratic. Quadratic Lie algebras appear, in particular, in connection with Lie bialgebras and physical models based on Lie algebras. Recall that quadratic Lie algebras are precisely the Lie algebras for which a Sugawara construction exists [9]. Several papers provided interesting results on the structure of quadratic Lie algebras [4, 5, 8–10, 12, 13]. In [13], Medina and Revoy have introduced the concept of double extension in order to give an inductive description of quadratic Lie algebras. This concept is also a tool to construct a new quadratic Lie algebra from a quadratic Lie algebra (g1 , B1 ) and a Lie algebra g2 (not necessarily quadratic) which acts on g1 by skew-symmetric derivations with respect to B1 . Let us remark that the non-trivial new quadratic S. Benayadi (B) IECL, CNRS UMR 7502, Université de Lorraine, Ile du Saulcy, 57045 Metz cedex, France e-mail: [email protected] A. Makhlouf et al. (eds.), Algebra, Geometry and Mathematical Physics, Springer Proceedings in Mathematics & Statistics 85, DOI: 10.1007/978-3-642-55361-5_8, © Springer-Verlag Berlin Heidelberg 2014
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Lie algebra will be obtained if g2 acts by non-inner skew-symmetric derivations on (g1 , B1 ). In general, it is difficult to find a Lie algebra g2 of dimension upper or equal to 2. In the first part of this paper, we will show how from quadratic Poissonadmissible algebra (A , B) we can find a Lie algebra g2 of dimension upper or equal to 2 acting on a quadratic Lie algebra (g1 , B1 ) by non-inner skew-symmetric derivations. In addition, we introduce the concept of symplectic quadratic Poisson algebra and we show how one constructs interesting symplectic quadratic Lie algebras from symplectic quadratic Poisson algebras. Let us recall that the Lie algebra of a Lie group which admits a bi-invariant pseudo-Riemannian metric and also a left-invariant symplectic form is a symplectic quadratic Lie algebra. These Lie groups are nilpotent and their geometry (and, consequently, that of their associated homogeneous spaces) is very rich. In particular, they carry two left-invariant affine structures: one defined by the symplectic form and another
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