· 2016· Preprint
0 HOM-LIE ALGEBRAS WITH SYMMETRIC INVARIANT NONDEGENERATE BILINEAR FORMS
- 172citations
- 2016year
Short summary
This paper introduces and studies quadratic Hom–Lie algebras, defined as Hom–Lie algebras with symmetric invariant nondegenerate bilinear forms, and extends the Double Extension Theory to these algebras.
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Key points
- Introduces and defines quadratic Hom–Lie algebras.
- Extends the Double Extension Theory to quadratic Hom–Lie algebras.
- Analyzes adjoint and coadjoint representations for applications to quadratic Hom–Lie algebras.
- Characterizes centerless involutive quadratic Hom–Lie algebras.
- Establishes a correspondence between involutive quadratic Hom–Lie algebras and quadratic simple Lie algebras with symmetric involution.
AI-generated from the title and abstract; the full text is not read.
Abstract
The aim of this paper is to introduce and study quadratic Hom–Lie algebras, which are Hom–Lie algebras equipped with symmetric invariant nondegenerate bilinear forms. We provide several constructions leading to examples and extend the Double Extension Theory to this class of nonassociative algebras. Elements of Representation Theory for Hom–Lie algebras, including adjoint and coadjoint representations, are supplied with application to quadratic Hom–Lie algebras. Centerless involutive quadratic Hom–Lie algebras are characterized. We reduce the case where the twist map is invertible to the study of involutive quadratic Lie algebras. Also, we establish a correspondence between the class of involutive quadratic Hom–Lie algebras and quadratic simple Lie algebras with symmetric involution.
The authors' abstract, as published at the source., 2016 · DOI ↗
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Field: Algebra and Number Theory
Algebra and Number TheoryMathematics