The elements in crystal bases corresponding to exceptional modules

TL;DR

This paper proves that elements corresponding to exceptional modules are in the crystal basis of quantum groups, establishing their integrality and sign properties through algebraic and geometric methods across all Cartan types.

Contribution

It demonstrates that elements linked to exceptional modules are in the crystal basis and clarifies their sign properties using geometric techniques, generalizing to all Cartan data.

Findings

Elements for exceptional modules are in the integral form of the quantum group.

These elements are in the crystal basis up to a sign.

The sign ambiguity can be removed via geometric methods.

Abstract

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.