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July 2020 Variation of complex structures and variation of Lie algebras II: new Lie algebras arising from singularities
Bingyi Chen, Naveed Hussain, Stephen S.-T. Yau, Huaiqing Zuo
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J. Differential Geom. 115(3): 437-473 (July 2020). DOI: 10.4310/jdg/1594260016

Abstract

Finite dimensional Lie algebras are semi-direct product of the semi-simple Lie algebras and solvable Lie algebras. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. It is extremely important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this article, a new natural connection between the set of complex analytic isolated hypersurface singularities and the set of finite dimensional solvable (nilpotent) Lie algebras has been constructed. We construct finite dimensional solvable (nilpotent) Lie algebras naturally from isolated hypersurface singularities. These constructions help us to understand the solvable (nilpotent) Lie algebras from the geometric point of view. Moreover, it is known that the classification of nilpotent Lie algebras in higher dimensions ($\gt 7$) remains to be a vast open area. There are one-parameter families of non-isomorphic nilpotent Lie algebras (but no two-parameter families) in dimension seven. Dimension seven is the watershed of the existence of such families. It is well-known that no such family exists in dimension less than seven, while it is hard to construct one-parameter family in dimension greater than seven. In this article, we construct an one-parameter family of solvable (resp. nilpotent) Lie algebras of dimension $11$ (resp. $10$) from $\tilde{E}_7$ singularities and show that the weak Torelli-type theorem holds. We shall also construct an one-parameter family of solvable (resp. nilpotent) Lie algebras of dimension $12$ (resp. $11$) from $\tilde{E}_8$ singularities and show that the Torelli-type theorem holds. Moreover, we investigate the numerical relation between the dimensions of the new Lie algebras and Yau algebras. Finally, the new Lie algebras arising from fewnomial isolated singularities are also computed.

Funding Statement

The authors Yau and Zuo were supported by NSFC Grants 11531007 and 11961141005. Zuo was supported by NSFC Grant 11771231. Yau was supported by Tsinghua university start-up fund and Tsinghua university education foundation fund (042202008).

Dedication

Dedicated to Professor Heisuke Hironaka on the occasion of his 87th birthday

Citation

Download Citation

Bingyi Chen. Naveed Hussain. Stephen S.-T. Yau. Huaiqing Zuo. "Variation of complex structures and variation of Lie algebras II: new Lie algebras arising from singularities." J. Differential Geom. 115 (3) 437 - 473, July 2020. https://doi.org/10.4310/jdg/1594260016

Information

Received: 14 November 2018; Published: July 2020
First available in Project Euclid: 9 July 2020

zbMATH: 07225028
MathSciNet: MR4120816
Digital Object Identifier: 10.4310/jdg/1594260016

Subjects:
Primary: 14B05, 32S05

Rights: Copyright © 2020 Lehigh University

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Vol.115 • No. 3 • July 2020
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