The ascending central series of nilpotent Lie algebras with complex structure

Latorre, Adela; Ugarte, Luis; Villacampa, Raquel

doi:10.1090/tran/7512

The ascending central series of nilpotent Lie algebras with complex structure
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by Adela Latorre, Luis Ugarte and Raquel Villacampa PDF

Trans. Amer. Math. Soc. 372 (2019), 3867-3903 Request permission

Abstract:

We obtain several restrictions on the terms of the ascending central series of a nilpotent Lie algebra $\mathfrak {g}$ under the presence of a complex structure $J$. In particular, we find a bound for the dimension of the center of $\mathfrak {g}$ when it does not contain any non-trivial $J$-invariant ideal. Thanks to these results, we provide a structural theorem describing the ascending central series of 8-dimensional nilpotent Lie algebras $\mathfrak {g}$ admitting this particular type of complex structure $J$. Since our method is constructive, it allows us to describe the complex structure equations that parametrize all such pairs $(\mathfrak {g}, J)$.

References

A. Andrada, M. L. Barberis, and I. Dotti, Corrigendum: Classification of abelian complex structures on six-dimensional Lie algebras [MR2763953], J. Lond. Math. Soc. (2) 87 (2013), no. 1, 319–320. MR 3022719, DOI 10.1112/jlms/jds044
Adrián Andrada and Raquel Villacampa, Abelian balanced Hermitian structures on unimodular Lie algebras, Transform. Groups 21 (2016), no. 4, 903–927. MR 3569562, DOI 10.1007/s00031-015-9352-7
Daniele Angella, Antonio Otal, Luis Ugarte, and Raquel Villacampa, Complex structures of splitting type, Rev. Mat. Iberoam. 33 (2017), no. 4, 1309–1350. MR 3729601, DOI 10.4171/RMI/973
Rutwig Campoamor Stursberg, Isolda E. Cardoso, and Gabriela P. Ovando, Extending invariant complex structures, Internat. J. Math. 26 (2015), no. 11, 1550096, 25. MR 3413991, DOI 10.1142/S0129167X15500962
Gil R. Cavalcanti and Marco Gualtieri, Generalized complex structures on nilmanifolds, J. Symplectic Geom. 2 (2004), no. 3, 393–410. MR 2131642, DOI 10.4310/JSG.2004.v2.n3.a5
M. Ceballos, A. Otal, L. Ugarte, and R. Villacampa, Invariant complex structures on 6-nilmanifolds: classification, Frölicher spectral sequence and special Hermitian metrics, J. Geom. Anal. 26 (2016), no. 1, 252–286. MR 3441513, DOI 10.1007/s12220-014-9548-4
Luis A. Cordero, Marisa Fernández, Alfred Gray, and Luis Ugarte, Nilpotent complex structures on compact nilmanifolds, Proceedings of the Workshop on Differential Geometry and Topology (Palermo, 1996), 1997, pp. 83–100. MR 1602971
Luis A. Cordero, Marisa Fernández, Alfred Gray, and Luis Ugarte, The holomorphic fiber bundle structure of some compact complex nilmanifolds, Proceedings of the 1st International Meeting on Geometry and Topology (Braga, 1997) Cent. Mat. Univ. Minho, Braga, 1998, pp. 207–221. MR 1694956
Luis A. Cordero, Marisa Fernández, Alfred Gray, and Luis Ugarte, Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5405–5433. MR 1665327, DOI 10.1090/S0002-9947-00-02486-7
Andrzej Czarnecki and Marcin Sroka, Six-dimensional product Lie algebras admitting integrable complex structures, J. Pure Appl. Algebra 222 (2018), no. 5, 1111–1125. MR 3742221, DOI 10.1016/j.jpaa.2017.06.010
Isabel G. Dotti and Anna Fino, Hypercomplex eight-dimensional nilpotent Lie groups, J. Pure Appl. Algebra 184 (2003), no. 1, 41–57. MR 1997309, DOI 10.1016/S0022-4049(03)00104-X
Nicola Enrietti, Anna Fino, and Luigi Vezzoni, Tamed symplectic forms and strong Kähler with torsion metrics, J. Symplectic Geom. 10 (2012), no. 2, 203–223. MR 2926995, DOI 10.4310/JSG.2012.v10.n2.a3
Anna Fino, Antonio Otal, and Luis Ugarte, Six-dimensional solvmanifolds with holomorphically trivial canonical bundle, Int. Math. Res. Not. IMRN 24 (2015), 13757–13799. MR 3436163, DOI 10.1093/imrn/rnv112
Lucia García Vergnolle and Elisabeth Remm, Complex structures on quasi-filiform Lie algebras, J. Lie Theory 19 (2009), no. 2, 251–265. MR 2572128
Ming-Peng Gong, Classification of nilpotent Lie algebras of dimension 7 (over algebraically closed fields and R), ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–University of Waterloo (Canada). MR 2698220
Michel Goze and Elisabeth Remm, Non existence of complex structures on filiform Lie algebras, Comm. Algebra 30 (2002), no. 8, 3777–3788. MR 1922311, DOI 10.1081/AGB-120005819
D. V. Millionshchikov, Complex structures on nilpotent Lie algebras and descending central series, Rend. Semin. Mat. Univ. Politec. Torino 74 (2016), no. 1, 163–182. MR 3772585
Iku Nakamura, Complex parallelisable manifolds and their small deformations, J. Differential Geometry 10 (1975), 85–112. MR 393580
Gabriela Ovando, Invariant complex structures on solvable real Lie groups, Manuscripta Math. 103 (2000), no. 1, 19–30. MR 1794329, DOI 10.1007/s002290070026
S. M. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), no. 2-3, 311–333. MR 1812058, DOI 10.1016/S0022-4049(00)00033-5
Luis Ugarte, Hermitian structures on six-dimensional nilmanifolds, Transform. Groups 12 (2007), no. 1, 175–202. MR 2308035, DOI 10.1007/s00031-005-1134-1
Luis Ugarte and Raquel Villacampa, Non-nilpotent complex geometry of nilmanifolds and heterotic supersymmetry, Asian J. Math. 18 (2014), no. 2, 229–246. MR 3217635, DOI 10.4310/AJM.2014.v18.n2.a3