Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extending $\pi$-systems to bases of root systems   Let $R$ be an indecomposable root system. It is well known that any root is
part of a basis $B$ of $R$. But when can you extend a set of two or more roots
to a basis $B$ of $R$? A $\pi$-system is a linearly independent set of roots,
$C$, such that if $\alpha$ and $\beta$ are in $C$, then $\alpha - \beta$ is not
a root. We will use results of Dynkin and Bourbaki to show that with two
exceptions, $A_3 \subset B_n$ and $A_7 \subset E_8$, an indecomposable
$\pi$-system whose Dynkin diagram is a subdiagram of the Dynkin diagram of $R$
can always be extended to a basis of $R$.

14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph
8
The categories of this paper are: hep-ex
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol
16
The categories of this paper are: stat
31
The categories of this paper are: ao-sci
9
The categories of this paper are: nlin
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs