Annals of Global Analysis and Geometry

, Volume 53, Issue 4, pp 543–559 | Cite as

Hyper-para-Kähler Lie algebras with abelian complex structures and their classification up to dimension 8

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Abstract

Hyper-para-Kähler structures on Lie algebras where the complex structure is abelian are studied. We show that there is a one-to-one correspondence between such hyper-para-Kähler Lie algebras and complex commutative (hence, associative) symplectic left-symmetric algebras admitting a semilinear map \(K_s\) verifying certain algebraic properties. Such equivalence allows us to give a complete classification, up to holomorphic isomorphism, of pairs \(({\mathfrak g},J)\) of 8-dimensional Lie algebras endowed with abelian complex structures which admit hyper-para-Kähler structures.

Keywords

Hyper-para-Kähler manifold Abelian (para)complex structure Symplectic left-symmetric algebra 

Mathematics Subject Classification

53C55 53D05 17B30 

References

  1. 1.
    Andrada, A.: Hypersymplectic Lie algebras. J. Geom. Phys. 56, 2039–2067 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Andrada, A., Barberis, M.L., Dotti, I.G.: Abelian Hermitian geometry. Differ. Geom. Appl. 30(5), 509–519 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Andrada, A., Dotti, I.G.: Double products and hypersymplectic structures on \({\mathbb{R}^{4n}}\). Commun. Math. Phys. 262, 1–16 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Andrada, A., Salamon, S.: Complex product structures on Lie algebras. Forum Math. 17(2), 261–295 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bai, C., Xiang, N.: Special symplectic Lie groups and hypersymplectic Lie groups. Manuscripta Math. 133, 373–408 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bajo, I., Benayadi, S.: Abelian para-Kähler structures on Lie algebras. Differ. Geom. Appl. 29(2), 160–173 (2011)CrossRefMATHGoogle Scholar
  7. 7.
    Bajo, I., Sanmartín, E., Pseudo-Kähler Lie algebras with abelian complex structures. J. Phys. A Math. Theor. 45, Art. ID 465205 (21 p) (2012)Google Scholar
  8. 8.
    Benayadi, S., Boucetta, M.: On para-Kähler and hyper-para-Kähler Lie algebras. J. Algebra 436, 61–101 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dancer, A., Swann, A.: Toric hypersymplectic quotients. Trans. Am. Math. Soc. 359(3), 1265–1284 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dotti, I.G., Fino, A.: Hypercomplex eight-dimensional nilpotent Lie groups. J. Pure Appl. Algebra 184, 41–57 (2003)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fino, A., Pedersen, H., Poon, Y.-S., Sørensen, M.W.: Neutral Calabi-Yau structures on Kodaira manifolds. Commun. Math. Phys. 248, 255–268 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Helmstetter, J.: Radical d’une algèbre symétrique à gauche. Ann. Inst. Fourier 29(4), 17–35 (1979)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hitchin, N.: Hypersymplectic quotients. Acta Acad. Sci. Tauriensis 124, 169–180 (1990)Google Scholar
  14. 14.
    Kruse, R.L., Price, D.T.: Nilpotent Rings. Gordon and Breach, New York (1969)MATHGoogle Scholar
  15. 15.
    Magnin, L.: Sur les algèbres de Lie nilpotentes de dimension \(\le \) 7. J. Geom. Phys. 3, 119–144 (1986)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Depto. Matemática Aplicada IIE.I. TelecomunicaciónVigoSpain
  2. 2.Depto. MatemáticasFacultad de CC.EEVigoSpain

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