2020
DOI: 10.1016/j.jalgebra.2020.03.005
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On 3-dimensional complex Hom-Lie algebras

Abstract: In this paper, we study and classify the 3-dimensional Hom-Lie algebras over 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 g. The well known Lie brackets for 3-dimensional Lie algebras arise within the appropriate representatives. Then, for each product representative, we provide a complete set of canonical forms for the linear maps g → g that turn g into a Hom-Lie algebra. As by-products, Ho… Show more

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Cited by 14 publications
(8 citation statements)
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“…Now observe that (11) yields, a j−1,j−1 = a j+1,j+1 , for all j. If in (7) we make the changes j → j − 1 and j → j + 1, we get,…”
Section: 1mentioning
confidence: 95%
See 2 more Smart Citations
“…Now observe that (11) yields, a j−1,j−1 = a j+1,j+1 , for all j. If in (7) we make the changes j → j − 1 and j → j + 1, we get,…”
Section: 1mentioning
confidence: 95%
“…This also proves that a j+1,j+1 = a j−1,j−1 = −ζ, if and only if a j,j = 2ζ, for 1 ≤ j ≤ m. On the other hand, if m is odd, it follows from ( 7) that, a j,j = −ζ for all j, which implies ζ = 0. Thus, let us assume that m = 2ℓ, with ℓ ∈ N. From (7), we know that (m − 2j)a j,j = −(m − 2j)ζ, for all j. Whence, a j,j = −ζ for all j = ℓ, and thus…”
Section: 1mentioning
confidence: 99%
See 1 more Smart Citation
“…To explicitly calculate G-derivations, we take a point of view of affine varieties to regard the set of G-derivations of g throughout this closing section. This viewpoint has been shown to be useful to doing specific computations; see [CZ19] and [GDSSV20]. Suppose dim(g) = n ∈ N + .…”
Section: Computations Of G-derivations On Sl 2 (C)mentioning
confidence: 99%
“…To explicitly calculate G-derivations, we take a point of view of affine varieties to regard the set of G-derivations of g throughout this closing section. This viewpoint has been shown to be useful to doing specific computations; see [9] and [14]. Suppose dim(g) = n ∈ N + .…”
mentioning
confidence: 99%