Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Generalized derivations and Hom-Lie algebra structures on
  $\mathfrak{sl}_2$   The purpose of this paper is to show that there are Hom-Lie algebra
structures on $\mathfrak{sl}_2(\mathbb{F}) \oplus \mathbb{F}D$, where $D$ is a
special type of generalized derivation of $\mathfrak{sl}_2(\mathbb{F})$, and
$\mathbb{F}$ is an algebraically closed field of characteristic zero. It is
shown that the generalized derivations $D$ of $\mathfrak{sl}_2(\mathbb{F})$
that we study in this work, satisfy the Hom-Lie Jacobi identity for the Lie
bracket of $\mathfrak{sl}_2(\mathbb{F})$. We study the representation theory of
Hom-Lie algebras within the appropriate category and prove that any finite
dimensional representation of a Hom-Lie algebra of the form
$\mathfrak{sl}_2(\mathbb{F}) \oplus \mathbb{F}D$, is completely reducible, in
analogy to the well known Theorem of Weyl from the classical Lie theory. We
apply this result to characterize the non-solvable Lie algebras having an
invertible generalized derivation of the type of $D$. Finally, using root space
decomposition techniques we provide an intrinsic proof of the fact that
$\mathfrak{sl}_2(\mathbb{F})$ is the only simple Lie algebra admitting
non-trivial Hom-Lie structures.

13
The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
16
The categories of this paper are: stat
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
19
The categories of this paper are: eess
32
The categories of this paper are: atom-ph
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
30
The categories of this paper are: q-alg
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The categories of this paper are: cmp-lg
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The categories of this paper are: astro-ph
9
The categories of this paper are: nlin