Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A note on the structure of underlying Lie algebras   Every Lie algebra over a field $E$ gives rise to new Lie algebras over any
subfield $F \subseteq E$ by restricting the scalar multiplication. This paper
studies the structure of these underlying Lie algebra in relation to the
structure of the original Lie algebra, in particular the question how much of
the original Lie algebra can be recovered from its underlying Lie algebra over
subfields $F$. By introducing the conjugate of a Lie algebra we show that in
some specific cases the Lie algebra is completely determined by its underlying
Lie algebra. Furthermore we construct examples showing that these assumptions
are necessary.
  As an application, we give for every positive $n$ an example of a real
$2$-step nilpotent Lie algebra which has exactly $n$ different bi-invariant
complex structures. This answers an open question by Di Scala, Lauret and
Vezzoni motivated by their work on quasi-K\"ahler Chern-flat manifolds in
differential geometry. The methods we develop work for general Lie algebras and
for general Galois extensions $F \subseteq E$, in contrast to the original
question which only considered nilpotent Lie algebras of nilpotency class $2$
and the field extension $\mathbb{R} \subseteq \mathbb{C}$. We demonstrate this
increased generality by characterizing the complex Lie algebras of dimension
$\leq 4$ which are defined over $\mathbb{R}$ and over $\mathbb{Q}$.

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The categories of this paper are: hep-th
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The categories of this paper are: nlin
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The categories of this paper are: econ
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The categories of this paper are: eess
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The categories of this paper are: adap-org
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The categories of this paper are: supr-con
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The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat
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The categories of this paper are: physics
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph
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The categories of this paper are: stat
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The categories of this paper are: q-alg