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5-dimensional indecomposable contact Lie algebras as double extensions.pdf

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5-dimensional indecomposable contact Lie algebras as double extensions.pdf

Journal of Geometry and Physics 100 (2016) 20–32 Contents lists available at ScienceDirect Journal of Geometry and Physics journal homepage: /locate/jgp 5-dimensional indecomposable contact Lie algebras as double extensions M.C. Rodríguez-Vallarte, G. Salgado ? Fac. de Ciencias, UASLP, Av. Salvador Nava s/n, Zona Universitaria, CP 78290, San Luis Potosí, S.L.P., México article info Article history: Received 21 July 2015 Received in revised form 13 October 2015 Accepted 31 October 2015 Available online 12 November 2015 MSC: primary 17Bxx secondary 53D10 Keywords: Contact Lie algebras Double extension of Lie algebras Indecomposable Lie algebras abstract In this work we shall show that a suitable double extension of a finite dimensional indecomposable contact Lie algebra is a contact Lie algebra again. In particular, with exception of the family of 5-dimensional indecomposable contact solvable Lie algebras A5,35, any 5-dimensional indecomposable contact solvable Lie algebra can be obtained as a double extension of a 3-dimensional Lie algebra. The family A5,35 can be generalized to a family of (4n + 1)-dimensional indecomposable contact solvable Lie algebras that cannot be ob- tained neither as a suspension of a symplectic Lie algebra of codimension 1 or as a double extension of a contact Lie subalgebra of codimension 2. ? 2015 Elsevier B.V. All rights reserved. 0. Introduction The notion of double extension plays a significant role providing an inductive construction of finite dimensional Lie algebras endowed with a geometric structure. This concept appears for first time in the book of V. Kac, Infinite dimensional Lie algebras (see [1], §2, exercise 2.10), in order to build finite dimensional quadratic Lie algebras, i.e., finite dimensional Lie algebras endowed with a non-degenerate invariant symmetric bilinear form. From [1] it follows that a solvable n-dimensional quadratic Lie algebra g is either an orthogonal direct sum of a quadratic solvable ideal of codimension