Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Parametrizations of infinite biconvex sets in affine root systems   We investigate in detail relationships between the set ${\mathfrak B}^\infty$
of all infinite ``biconvex'' sets in the positive root system $\Delta_+$ of an
arbitrary untwisted affine Lie algebra ${\mathfrak g}$ and the set ${\mathcal
W}^\infty$ of all infinite ``reduced word'' of the Weyl group of ${\mathfrak
g}$. The study is applied to the classification of ``convex orders'' on
$\Delta_+$ (cf. \cite{kI}), which are indispensable to construct ``convex
bases'' of Poincar\'e-Birkhoff-Witt type of the upper triangular subalgebra
$U_q^+$ of the quantized universal enveloping algebra $U_q({\mathfrak g})$. We
construct a set $\boldsymbol{\mathcal P}$ by using data of the underlying
finite-dimensional simple Lie algebra, and bijective mappings
$\nabla\colon\boldsymbol{\mathcal P}\to{\mathfrak B}^\infty$ and
$\chi\colon\boldsymbol{\mathcal P}\to W^\infty$ such that
$\nabla=\varPhi^\infty\circ\chi$, where $W^\infty$ is an quotient set of
${\mathcal W}^\infty$ and $\varPhi^\infty\colon W^\infty\to{\mathfrak
B}^\infty$ is a natural injective mapping.

25
The categories of this paper are: patt-sol
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
28
The categories of this paper are: alg-geom
21
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: econ
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
12
The categories of this paper are: nucl-th