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Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities

  • Naveed Hussain , Stephen S.-T. Yau EMAIL logo and Huaiqing Zuo
Published/Copyright: January 6, 2022
Forum Mathematicum
From the journal Forum Mathematicum Volume 34 Issue 2

Abstract

The Levi theorem tells us that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. 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. Therefore, it is important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this paper, we give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. As an application, we obtain the correspondence between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to 1. Moreover, we give a new characterization theorem for zero-dimensional simple complete intersection singularities.

Keywords: Derivation; nilpotent Lie algebra; isolated singularity
MSC 2010: 14B05; 32S05

Dedicated to Professor William Fulton on the occasion of his 82th birthday

Communicated by Jan Frahm

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11961141005

Award Identifier / Grant number: 11771231

Funding statement: Both Yau and Zuo are supported by NSFC Grant 11961141005. Zuo is supported by Tsinghua University Initiative Scientific Research Program and NSFC Grant 11771231. Yau is supported by Tsinghua University start-up fund and Tsinghua University Education Foundation fund (No. 042202008). Hussain is supported by an innovation team project of Humanities and Social Sciences in Colleges and Universities of Guangdong Province (No. 2020wcxtd008).

A Appendix

Maple code description.

Query(Alg1, Alg2, parm, “Homomorphism”) returns a 4-tuple TF, Eq, Soln, B. Here TF is true if Maple finds a set of values for the parameters for which the Matrix A is a homomorphism; Eq is the defining set of equations for the parameters parm in order that the matrix A be a homomorphism; Soln is a list of solutions to the equations Eq; and B is the list of Matrices obtained by evaluating A on the solutions in the list Soln.

A.1 Maple program of Proposition 2.1

> with(DifferentialGeometry):with(LieAlgebras):>L1:=_DG([[“LieAlgebra”,Alg1,[5]],[[[1,3,2],2],[[1,4,2],-1],[[1,5,3],-1],[[1,5,4],2]>DGsetup(L1):>L2:=_DG([[“LieAlgebra”,Alg2,[5]],[[[1,2,3],1],[[1,3,4],1],[[2,5,4],1]> DGsetup(L2):> A:= Matrix([[a11, a12, a13, a14, a15], [a21, a22, a23, a24, a25], [a31, a32, a33, a34, a35], [a41, a42, a43, a44, a45], [a51, a52, a53, a54, a55]])> TF, EQ, SOLN, B:= Query(Alg1, Alg2, A, {a11, a12, a13, a14, a15, a21, a22, a23, a24, a25, a31, a32, a33, a34, a35, a41, a42, a43, a44, a45, a51, a52, a53, a54, a55}, “Homomorphism”)

A.2 Maple program of Proposition 2.1

>with(DifferentialGeometry):with(LieAlgebras):>L1:=_DG([[“LieAlgebra”,Alg3,[5]],[[[1,3,2],2],[[1,4,2],-1],[[1,5,3],-1],[[1,5,4],2]> DGsetup(L1):>L2:=_DG([[“LieAlgebra”,Alg4,[5]],[[[1,2,3],1],[[1,3,4],1],[[2,5,4],1]> DGsetup(L2, [f], [θ])>

A:=[100001000410-2111-141110001]

>ϕ:= Transformation(Alg3, Alg4, A)> Query(Alg3, Alg4, A, “Homomorphism”)> true

Acknowledgements

The authors would like to thank Craig Seeley for offering us some useful advice when we tried to construct some isomorphisms in the proof of Theorem A.

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Received: 2021-09-04
Revised: 2021-11-25
Published Online: 2022-01-06
Published in Print: 2022-03-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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  2. On the variance of the nodal volume of arithmetic random waves
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  4. On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators
  5. Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities
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https://doi.org/10.1515/forum-2021-0227
Keywords for this article
Derivation; nilpotent Lie algebra; isolated singularity

Articles in the same Issue

  1. Frontmatter
  2. On the variance of the nodal volume of arithmetic random waves
  3. On length densities
  4. On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators
  5. Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities
  6. On cutting blocking sets and their codes
  7. Inertia groups and smooth structures on quaternionic projective spaces
  8. On the number of zeros of diagonal quartic forms over finite fields
  9. Weak martingale Hardy-type spaces associated with quasi-Banach function lattice
  10. A dual version of Huppert's ρ-σ conjecture for character codegrees
  11. A topological correspondence between partial actions of groups and inverse semigroup actions
  12. On homotopy braids
  13. Holomorphic convexity of pseudoconvex spaces in terms of the rank of structural sheaf
  14. The Shintani double zeta functions
  15. Homological dimensions relative to preresolving subcategories II
  16. Simplicity of indecomposable set-theoretic solutions of the Yang–Baxter equation
  17. A converse theorem for quasimodular forms
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