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MAXIMAL NILPOTENT COMPLEX STRUCTURES

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Abstract

Let the pair (𝔤, J) be a nilpotent Lie algebra 𝔤 (NLA for short) endowed with a nilpotent complex structure J. In this paper, motivated by a question in the work of Cordero, Fernández, Gray and Ugarte [6], we prove that 2 ≤ v(J) ≤ 3 for (𝔤, J) when v(𝔤) = 2, where v(𝔤) is the step of 𝔤 and v(J) is the unique smallest integer such that 𝔞(J)v(J) = 𝔤 as in the [6, Def. 1, 8]. When v(𝔤) = 3, for arbitrary n ≥ 3, there exists a pair (𝔤, J) such that v(J) = dimℂ 𝔤 = n, for which we call the J in the pair (𝔤, J), satisfying v(J) = dimℂ 𝔤 = n, a maximal nilpotent (MaxN for short) complex structure. The algebraic dimension of a nilmanifold endowed with a left invariant MaxN complex structure is discussed. Furthermore, a structure theorem is proved for the pair (𝔤, J), where v(𝔤) = 3 and J is a MaxN complex structure.

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Authors and Affiliations

  1. School of Mathematics and Economics, Hubei University of Education, Wuhan, 430205, China

    QIN GAO

  2. School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China

    QUANTING ZHAO

  3. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, China

    FANGYANG ZHENG

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  1. QIN GAO
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Correspondence to QUANTING ZHAO.

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Gao is partially supported by National Natural Science Foundations of China with the grant No.11901176.

The corresponding author Zhao is partially supported by National Natural Science Foundations of China with the grant No.11801205 and China Scholarship Council to Ohio State University.

Zheng is partially supported by National Natural Science Foundations of China with the grant No.12071050.

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GAO, Q., ZHAO, Q. & ZHENG, F. MAXIMAL NILPOTENT COMPLEX STRUCTURES. Transformation Groups 28, 241–284 (2023). https://doi.org/10.1007/s00031-021-09688-3

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  • DOI: https://doi.org/10.1007/s00031-021-09688-3

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