Degenerations of 8-Dimensional 2-step Nilpotent Lie Algebras
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Degenerations of 8-Dimensional 2-step Nilpotent Lie Algebras Mar´ıa Alejandra Alvarez1 Received: 12 December 2019 / Accepted: 30 July 2020 / © Springer Nature B.V. 2020
Abstract In this work, we consider degenerations between 8-dimensional 2-step nilpotent Lie algebras over C and obtain the geometric classification of the variety N82 . Keywords Nilpotent Lie algebras · Variety of Lie algebras · Degenerations Mathematics Subject Classification (2010) 17B30 · 17B99
1 Preliminaries The algebraic classification of Lie algebras is a wild problem. Lie algebras are classified up to dimension 6 (see for instance [33] for a list of indecomposable Lie algebras of dimension ≤ 6 over C and R). In the class of nilpotent Lie algebras, there are classifications up to dimension 7 over algebraically closed fields and R (see for instance [14] or [29]). In dimension 8, there are only classifications of 2-step nilpotent and filiform Lie algebras over C (see [34] and [11] respectively). A related problem is the one concerning the geometric classification of Lie algebras, their degenerations, rigid elements and irreducible components. Regarding this problem in the variety of Lie algebras we can mention [2, 6–8, 10, 12, 17, 19, 27, 28, 31, 32, 35]. Moreover, the study of the geometric classification for varieties of different structures is an active research field, several results have been obtained recently in different directions regarding nilpotent algebras (see for instance [1, 3, 4, 13, 15, 16, 20–26]). In this work we obtain degenerations between 2-step nilpotent Lie algebras of dimension 8 over C and provide the irreducible components of the variety N82 , which turn out to be the orbit closures of three rigid Lie algebras.
Communicated by:Presented by: Michel Brion Mar´ıa Alejandra Alvarez
[email protected] 1
Departamento de Matem´aticas, Facultad de Ciencias B´asicas, Universidad de Antofagasta, Antofagasta, Chile
M.A. Alvarez
1.1 The Variety of Lie Algebras Let V be a complex n-dimensional vector space with a fixed basis {e1 , . . . , en }, and let g = (V , [·, ·]) be a Lie algebra with underlying vector space V and Lie product [·, ·]. The 3 set of Lie algebra structures on the space V is an algebraic variety in Cn in the following sense: Every Lie algebra structure on V , g, can be identified with its set of structure consn 3 k k ∈ Cn , where [ei , ej ] = ci,j ek . This set of structure constants satisfies the tants ci,j k=1 k +ck = 0 polynomial equations given by the skew-symmetry and the Jacobi identity, i.e. ci,j j,i n l r l r l r and cj,k ci,l + ck,i cj,l + ci,j ck,l = 0. We will denote by Ln the algebraic variety of l=1
Lie algebras of fixed dimension n. The group G = GL(n, C) acts on Ln via change of basis: g · [X, Y ] = g [g −1 X, g −1 Y ] ,
X, Y ∈ g, g ∈ GL(n, C).
Also, one can define the Zariski topology on Ln . Given two Lie algebras g and h, we say that g degenerates to h, and denoted by g → h, if h lies in the Zariski closure of the G-orbit O(g). A degeneration g → h is called proper
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