Proceedings of the Japan Academy, Series A, Mathematical Sciences

Symplectic structures on free nilpotent Lie algebras

Viviana del Barco

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In this short note we show a necessary and sufficient condition for the existence of symplectic structures on free nilpotent Lie algebras and their one-dimensional trivial extensions.

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Proc. Japan Acad. Ser. A Math. Sci., Volume 95, Number 8 (2019), 88-90.

First available in Project Euclid: 2 October 2019

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Mathematical Reviews number (MathSciNet)

Primary: 53D05: Symplectic manifolds, general
Secondary: 17B01: Identities, free Lie (super)algebras 17B30: Solvable, nilpotent (super)algebras 22E25: Nilpotent and solvable Lie groups

Free nilpotent Lie algebras symplectic structures


del Barco, Viviana. Symplectic structures on free nilpotent Lie algebras. Proc. Japan Acad. Ser. A Math. Sci. 95 (2019), no. 8, 88--90. doi:10.3792/pjaa.95.88.

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  • O. Baues and V. Cortés, Symplectic Lie groups I–III, arXiv:1307.1629.
  • C. Benson and C. S. Gordon, Kähler and symplectic structures on nilmanifolds, Topology 27 (1988), no. 4, 513–518.
  • M. Goze and A. Bouyakoub, Sur les algèbres de Lie munies d'une forme symplectique, Rend. Sem. Fac. Sci. Univ. Cagliari 57 (1987), no. 1, 85–97.
  • L. Cagliero and V. del Barco, Nilradicals of parabolic subalgebras admitting symplectic structures, Differential Geom. Appl. 46 (2016), 1–13.
  • I. Dotti and P. Tirao, Symplectic structures on Heisenberg-type nilmanifolds, Manuscripta Math. 102 (2000), no. 3, 383–401.
  • M. Goze and E. Remm, Symplectic structures on 2-step nilpotent Lie algebras, arXiv:1510.08212.
  • M. Grayson and R. Grossman, Models for free nilpotent Lie algebras, J. Algebra 135 (1990), no. 1, 177–191.
  • Z.-D. Guan, Toward a classification of compact nilmanifolds with symplectic structures, Int. Math. Res. Not. IMRN 2010, no. 22, 4377–4384.
  • M. Hall, A basis for free Lie rings and higher commutators in free groups, Proc. Amer. Math. Soc. 1 (1950), 575–581.
  • A. I. Malcev, On a class of homogeneous spaces, Amer. Math. Soc. Translation 1951 (1951), no. 39.
  • D. V. Millionschikov, Graded filiform Lie algebras and symplectic nilmanifolds, in Geometry, topology, and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, 212, Adv. Math. Sci., 55, Amer. Math. Soc., Providence, RI, 2004, pp. 259–279.
  • V. V. Morozov, Classification of nilpotent Lie algebras of sixth order, Izv. Vysš. Učebn. Zaved. Matematika 1958, no. 4 (5), 161–171.
  • K. Nomizu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. (2) 59 (1954), 531–538.
  • H. Pouseele and P. Tirao, Compact symplectic nilmanifolds associated with graphs, J. Pure Appl. Algebra 213 (2009), no. 9, 1788–1794.
  • C. Reutenauer, Free Lie algebras, London Mathematical Society Monographs. New Series, 7, The Clarendon Press, Oxford University Press, New York, 1993.
  • J. P. Serre, Lie algebras and Lie groups, 1964 lectures given at Harvard University, 2nd ed., Lecture Notes in Mathematics, 1500, Springer-Verlag, Berlin, 1992.