Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: It\^o's theorem and metabelian Leibniz algebras   We prove that the celebrated It\^{o}'s theorem for groups remains valid at
the level of Leibniz algebras: if $\mathfrak{g}$ is a Leibniz algebra such that
$\mathfrak{g} = A + B$, for two abelian subalgebras $A$ and $B$, then
$\mathfrak{g}$ is metabelian, i.e. $[ \, [\mathfrak{g}, \, \mathfrak{g}], \, [
\mathfrak{g}, \, \mathfrak{g} ] \, ] = 0$. A structure type theorem for
metabelian Leibniz/Lie algebras is proved. All metabelian Leibniz algebras
having the derived algebra of dimension $1$ are described, classified and their
automorphisms groups are explicitly determined as subgroups of a semidirect
product of groups $P^* \ltimes \bigl(k^* \times {\rm Aut}_{k} (P) \bigl)$
associated to any vector space $P$.

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The categories of this paper are: nlin
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The categories of this paper are: adap-org
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
11
The categories of this paper are: hep-lat
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
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The categories of this paper are: hep-th
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
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The categories of this paper are: alg-geom
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The categories of this paper are: cs
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The categories of this paper are: mtrl-th
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The categories of this paper are: astro-ph