Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Positive Systems of Kostant Roots   Let $\mathfrak{g}$ be a simple complex Lie algebra and let $\mathfrak{t}
\subset \mathfrak{g}$ be a toral subalgebra of $\mathfrak{g}$. As a
$\mathfrak{t}$-module $\mathfrak{g}$ decomposes as \[\mathfrak{g} =
\mathfrak{s} \oplus \big(\oplus_{\nu \in \mathcal{R}} \mathfrak{g}^\nu\big)\]
where $\mathfrak{s} \subset \mathfrak{g}$ is the reductive part of a parabolic
subalgebra of $\mathfrak{g}$ and $\mathcal{R}$ is the Kostant root system
associated to $\mathfrak{t}$. When $\mathfrak{t}$ is a Cartan subalgebra of
$\mathfrak{g}$ the decomposition above is nothing but the root decomposition of
$\mathfrak{g}$ with respect to $\mathfrak{t}$; in general the properties of
$\mathcal{R}$ resemble the properties of usual root systems. In this note we
study the following problem: "Given a subset $\mathcal{S} \subset \mathcal{R}$,
is there a parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{g}$ containing
$\mathcal{M} = \oplus_{\nu \in \mathcal{S}} \mathfrak{g}^\nu$ and whose
reductive part equals $\mathfrak{s}$?". Our main results is that, for a
classical simple Lie algebra $\mathfrak{g}$ and a saturated $\mathcal{S}
\subset \mathcal{R}$, the condition
$(\operatorname{Sym}^\cdot(\mathcal{M}))^{\mathfrak{s}} = \mathbf{C}$ is
necessary and sufficient for the existence of such a $\mathfrak{p}$. In
contrast, we show that this statement is no longer true for the exceptional Lie
algebras $\mathrm{F}_4, \mathrm{E}_6, \mathrm{E}_7$, and $\mathrm{E}_8$.
Finally, we discuss the problem in the case when $\mathcal{S}$ is not
saturated.

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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: adap-org
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The categories of this paper are: eess
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The categories of this paper are: hep-lat
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
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The categories of this paper are: physics
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The categories of this paper are: stat
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The categories of this paper are: alg-geom
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ph