Geometric structures on Lie algebras and double extensions

Rodríguez-Vallarte, M.; Salgado, G.

doi:10.1090/proc/14127

Geometric structures on Lie algebras and double extensions
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by M. C. Rodríguez-Vallarte and G. Salgado PDF

Proc. Amer. Math. Soc. 146 (2018), 4199-4209 Request permission

Abstract:

Given a finite-dimensional real or complex Lie algebra ${\frak g}$ equipped with a geometric structure (i.e., either an invariant metric, a symplectic or contact structure), the aim of this work is to show that the double extension process introduced by V. Kac allows one to generate Lie algebras equipped with the same type of geometric structure. In particular, for an exact symplectic Lie algebra, through a double extension process it is possible to construct new exact symplectic Lie algebras.

References

M. A. Alvarez, M. C. Rodríguez-Vallarte, and G. Salgado, Contact nilpotent Lie algebras, Proc. Amer. Math. Soc. 145 (2017), no. 4, 1467–1474. MR 3601539, DOI 10.1090/proc/13341
Hedi Benamor and Saïd Benayadi, Double extension of quadratic Lie superalgebras, Comm. Algebra 27 (1999), no. 1, 67–88. MR 1668212, DOI 10.1080/00927879908826421
Rutwig Campoamor-Stursberg, A characterization of exact symplectic Lie algebras $\mathfrak {g}$ in terms of the generalized Casimir invariants, New developments in mathematical physics research, Nova Sci. Publ., Hauppauge, NY, 2004, pp. 55–84. MR 2076274
Claude Chevalley and Samuel Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124. MR 24908, DOI 10.1090/S0002-9947-1948-0024908-8
Jean-Michel Dardié and Alberto Médina, Algèbres de Lie kählériennes et double extension, J. Algebra 185 (1996), no. 3, 774–795 (French, with English summary). MR 1419723, DOI 10.1006/jabr.1996.0350
G. Favre and L. J. Santharoubane, Symmetric, invariant, nondegenerate bilinear form on a Lie algebra, J. Algebra 105 (1987), no. 2, 451–464. MR 873679, DOI 10.1016/0021-8693(87)90209-2
Michel Goze and Elisabeth Remm, Contact and Frobeniusian forms on Lie groups, Differential Geom. Appl. 35 (2014), 74–94. MR 3231748, DOI 10.1016/j.difgeo.2014.05.008
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Yu. Khakimdjanov, M. Goze, and A. Medina, Symplectic or contact structures on Lie groups, Differential Geom. Appl. 21 (2004), no. 1, 41–54. MR 2067457, DOI 10.1016/j.difgeo.2003.12.006
Alberto Medina and Philippe Revoy, Groupes de Lie Poisson et double extension, Séminaire Gaston Darboux de Géométrie et Topologie Différentielle, 1990–1991 (Montpellier, 1990–1991) Univ. Montpellier II, Montpellier, 1992, pp. iv, 87–105 (French, with English summary). MR 1186222
Alberto Medina and Philippe Revoy, Algèbres de Lie et produit scalaire invariant, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 3, 553–561 (French). MR 826103, DOI 10.24033/asens.1496
Alberto Medina and Philippe Revoy, Groupes de Lie à structure symplectique invariante, Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 20, Springer, New York, 1991, pp. 247–266 (French). MR 1104932, DOI 10.1007/978-1-4613-9719-9_{1}7
M. C. Rodríguez-Vallarte and G. Salgado, 5-dimensional indecomposable contact Lie algebras as double extensions, J. Geom. Phys. 100 (2016), 20–32. MR 3435759, DOI 10.1016/j.geomphys.2015.10.014
M. C. Rodríguez-Vallarte, G. Salgado, and O. A. Sánchez-Valenzuela, Heisenberg Lie superalgebras and their invariant superorthogonal and supersymplectic forms, J. Algebra 332 (2011), 71–86. MR 2774679, DOI 10.1016/j.jalgebra.2011.02.003