Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The elements in crystal bases corresponding to exceptional modules   According to the Ringel-Green Theorem([G],[R1]), the generic composition
algebra of the Hall algebra provides a realization of the positive part of the
quantum group. Furthermore, its Drinfeld double can be identified with the
whole quantum group([X],[XY]), in which the BGP-reflection functors coincide
with Lusztig's symmetries. We first assert the elements corresponding to
exceptional modules lie in the integral generic composition algebra, hence in
the integral form of the quantum group. Then we prove that these elements lie
in the crystal basis up to a sign. Eventually we show that the sign can be
removed by the geometric method. Our results hold for any type of Cartan datum.

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The categories of this paper are: gr-qc
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The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
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The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
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The categories of this paper are: quant-ph
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
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The categories of this paper are: solv-int
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The categories of this paper are: comp-gas
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The categories of this paper are: chao-dyn
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The categories of this paper are: bayes-an