Back to Search
Start Over
On 3-dimensional complex Hom-Lie algebras
- Source :
- Journal of Algebra. 555:361-385
- Publication Year :
- 2020
- Publisher :
- Elsevier BV, 2020.
-
Abstract
- We study and classify the 3-dimensional Hom-Lie algebras over $\mathbb{C}$. We provide first a complete set of representatives for the isomorphism classes of skew-symmetric bilinear products defined on a 3-dimensional complex vector space $\mathfrak{g}$. The well known Lie brackets for the 3-dimensional Lie algebras are included into appropriate isomorphism classes of such products representatives. For each product representative, we provide a complete set of canonical forms for the linear maps $\mathfrak{g} \to \mathfrak{g}$ that turn $g$ into a Hom-Lie algebra, thus characterizing the corresponding isomorphism classes. As by-products, Hom-Lie algebras for which the linear maps $\mathfrak{g} \to \mathfrak{g}$ are not homomorphisms for their products, are exhibited. Examples also arise of non-isomorphic families of HomLie algebras which share, however, a fixed Lie-algebra product on $\mathfrak{g}$. In particular, this is the case for the complex simple Lie algebra $\mathfrak{sl}_2(\mathbb{C})$. Similarly, there are isomorphism classes for which their skew-symmetric bilinear products can never be Lie algebra brackets on $\mathfrak{g}$.
- Subjects :
- Pure mathematics
Algebra and Number Theory
010102 general mathematics
Mathematics - Rings and Algebras
Space (mathematics)
01 natural sciences
Set (abstract data type)
Rings and Algebras (math.RA)
Simple (abstract algebra)
Product (mathematics)
0103 physical sciences
Lie algebra
FOS: Mathematics
Homomorphism
Canonical form
010307 mathematical physics
Isomorphism
Primary: 17-XX, Secondary: 17A30, 17A36, 17BXX, 17B60
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 00218693
- Volume :
- 555
- Database :
- OpenAIRE
- Journal :
- Journal of Algebra
- Accession number :
- edsair.doi.dedup.....60843253fbd6bbe69c5bc5708f3f06de