Spectral invariants for finite dimensional Lie algebras

Fatemeh Azari Key; Rongwei Yang

doi:10.1016/j.laa.2020.10.024

Spectral invariants for finite dimensional Lie algebras

Fatemeh Azari Key
, Rongwei Yang

Mathematics and Statistics

University at Albany

Dalian University of Technology

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

For a Lie algebra L with basis {x₁,x₂,…,x_n}, its associated characteristic polynomial Q_L(z) is the determinant of the linear pencil z₀I+z₁adx₁+⋯+z_nadx_n. This paper shows that Q_L is invariant under the automorphism group Aut(L). The zero variety and factorization of Q_L reflect the structure of L. In the case L is solvable Q_L is known to be a product of linear factors. This fact gives rise to the definition of spectral matrix and the Poincaré polynomial for solvable Lie algebras. Application is given to 1-dimensional extensions of nilpotent Lie algebras.

Original language	English
Pages (from-to)	148-170
Number of pages	23
Journal	Linear Algebra and Its Applications
Volume	611
DOIs	https://doi.org/10.1016/j.laa.2020.10.024
State	Published - Feb 15 2021

Keywords

Automorphism group
Betti number
Characteristic polynomial
Eigen-variety
Lie algebra
Poincaré polynomial
Solvable and nilpotent Lie algebras
Spectral matrix

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10.1016/j.laa.2020.10.024

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@article{edf4ca284a444e32a5f3cde0fba130b0,

title = "Spectral invariants for finite dimensional Lie algebras",

abstract = "For a Lie algebra L with basis \{x1,x2,…,xn\}, its associated characteristic polynomial QL(z) is the determinant of the linear pencil z0I+z1adx1+⋯+znadxn. This paper shows that QL is invariant under the automorphism group Aut(L). The zero variety and factorization of QL reflect the structure of L. In the case L is solvable QL is known to be a product of linear factors. This fact gives rise to the definition of spectral matrix and the Poincar{\'e} polynomial for solvable Lie algebras. Application is given to 1-dimensional extensions of nilpotent Lie algebras.",

keywords = "Automorphism group, Betti number, Characteristic polynomial, Eigen-variety, Lie algebra, Poincar{\'e} polynomial, Solvable and nilpotent Lie algebras, Spectral matrix",

author = "\{Azari Key\}, Fatemeh and Rongwei Yang",

note = "Publisher Copyright: {\textcopyright} 2020 Elsevier Inc.",

year = "2021",

month = feb,

day = "15",

doi = "10.1016/j.laa.2020.10.024",

language = "English",

volume = "611",

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journal = "Linear Algebra and Its Applications",

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TY - JOUR

T1 - Spectral invariants for finite dimensional Lie algebras

AU - Azari Key, Fatemeh

AU - Yang, Rongwei

PY - 2021/2/15

Y1 - 2021/2/15

N2 - For a Lie algebra L with basis {x1,x2,…,xn}, its associated characteristic polynomial QL(z) is the determinant of the linear pencil z0I+z1adx1+⋯+znadxn. This paper shows that QL is invariant under the automorphism group Aut(L). The zero variety and factorization of QL reflect the structure of L. In the case L is solvable QL is known to be a product of linear factors. This fact gives rise to the definition of spectral matrix and the Poincaré polynomial for solvable Lie algebras. Application is given to 1-dimensional extensions of nilpotent Lie algebras.

AB - For a Lie algebra L with basis {x1,x2,…,xn}, its associated characteristic polynomial QL(z) is the determinant of the linear pencil z0I+z1adx1+⋯+znadxn. This paper shows that QL is invariant under the automorphism group Aut(L). The zero variety and factorization of QL reflect the structure of L. In the case L is solvable QL is known to be a product of linear factors. This fact gives rise to the definition of spectral matrix and the Poincaré polynomial for solvable Lie algebras. Application is given to 1-dimensional extensions of nilpotent Lie algebras.

KW - Automorphism group

KW - Betti number

KW - Characteristic polynomial

KW - Eigen-variety

KW - Lie algebra

KW - Poincaré polynomial

KW - Solvable and nilpotent Lie algebras

KW - Spectral matrix

UR - https://www.scopus.com/pages/publications/85095809984

U2 - 10.1016/j.laa.2020.10.024

DO - 10.1016/j.laa.2020.10.024

M3 - Article

SN - 0024-3795

VL - 611

SP - 148

EP - 170

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

Spectral invariants for finite dimensional Lie algebras

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Keywords

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