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COMPLEX STRUCTURES ON STRATIFIED LIE ALGEBRAS

Part of: Linear algebraic groups and related topics Lie groups

Published online by Cambridge University Press:  19 April 2022

JUNZE ZHANG Open the ORCID record for JUNZE ZHANG [Opens in a new window]
JUNZE ZHANG*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales 2052, Australia
*
e-mail: z5055984@ad.unsw.edu.au
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Abstract

We investigate some properties of complex structures on Lie algebras. In particular, we focus on nilpotent complex structures that are characterised by suitable J-invariant ascending or descending central series, $\mathfrak {d}^{\,j}$ and $\mathfrak {d}_j$ , respectively. We introduce a new descending series $\mathfrak {p}_j$ and use it to prove a new characterisation of nilpotent complex structures. We also examine whether nilpotent complex structures on stratified Lie algebras preserve the strata. We find that there exists a J-invariant stratification on a step $2$ nilpotent Lie algebra with a complex structure.

Keywords

complex structurestratified Lie algebranilpotent Lie algebraJ-invariant central series

MSC classification

Primary: 22E25: Nilpotent and solvable Lie groups
Secondary: 20G07: Structure theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 2 , October 2022 , pp. 320 - 332
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

Andrada, A., Barberis, M. L. and Dotti, I., ‘Classification of abelian complex structures on 6-dimensional Lie algebras’, J. Lond. Math. Soc. (2) 83(1) (2011), 232–255.CrossRefGoogle Scholar
Barberis, M. L. and Dotti, I., ‘Abelian complex structures on solvable Lie algebras’, J. Lie Theory 14(1) (2004), 25–34.Google Scholar
Cordero, L. A., Fernández, M., Gray, A. and Ugarte, L., ‘Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology’, Trans. Amer. Math. Soc. 352(12) (2000), 5405–5433.CrossRefGoogle Scholar
Cordero, L. A., Fernández, M., Gray, A. and Ugarte, L., ‘Nilpotent complex structures’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 95(1) (2001), 45–55.Google Scholar
Cowling, M. G., Li, J., Ottazzi, A. and Wu, Q., ‘Conformal and CR mappings on Carnot groups’, Proc. Amer. Math. Soc. Ser. B 7 (2000), 67–81.CrossRefGoogle Scholar
Cowling, M. G. and Ottazzi, A., ‘Structure of stratified groups I. Product decompositions’, J. Lie Theory 27(1) (2017), 177–183.Google Scholar
Gao, Q., Zhao, Q. and Zheng, F., ‘Maximal nilpotent complex structures’, Transformation Groups, to appear. https://doi.org/10.1007/s00031-021-09688-3.CrossRefGoogle Scholar
Hilgert, J. and Neeb, K.-H., Structure and Geometry of Lie Groups, Springer Monographs in Mathematics (Springer, New York, 2012).CrossRefGoogle Scholar
Knapp, A. W., Lie Groups Beyond an Introduction, 2nd edn, Progress in Mathematics, 140 (Birkhäuser, Boston, MA, 2002).Google Scholar
Latorre, A., Ugarte, L. and Villacampa, R., ‘The ascending central series of nilpotent Lie algebras with complex structure’, Trans. Amer. Math. Soc. 372(6) (2019), 3867–3903.CrossRefGoogle Scholar
Latorre, A., Ugarte, L. and Villacampa, R., ‘Complex structures on nilpotent Lie algebras with one-dimensional center’, Preprint, 2020, arXiv:2011.09916.Google Scholar
Le Donne, E., ‘A primer on Carnot groups: homogenous groups, Carnot–Carathéodory spaces, and regularity of their isometries’, Anal. Geom. Metr. Spaces 5(1) (2017), 116–137.CrossRefGoogle Scholar
Newlander, A. and Nirenberg, L., ‘Complex analytic coordinates in almost complex manifolds’, Ann. of Math. (2) 65 (1957), 391–404.CrossRefGoogle Scholar
Remm, E., ‘Non-existence of complex structures on filiform Lie algebras’, An. Univ. Vest Timiş. Ser. Mat.-Inform. 39 (Special Issue: Mathematics) (2001), 391–399.Google Scholar
Rollenske, S., ‘Geometry of nilmanifolds with left-invariant complex structure and deformations in the large’, Proc. Lond. Math. Soc. (3) 99(2) (2009), 425–460.CrossRefGoogle Scholar