Geometric Construction of Crystal Bases

Details Geometric Construction of Crystal Bases Geometric Construction of Crystal Bases

1996-06-08

arxiv.org/abs/q-alg/9606009v1

Mathematics

Quantum Algebra

AmS-LaTeX v1.1, 26 pages

Scientific paper

We realize the crystal associated to the quantized enveloping algebras with a symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety. As a by-product, we give a counterexample to the conjecture of Kazhdan--Lusztig on the irreducibility of the characteristic variety of the intersection cohomology sheaves associated with the Schubert cells of type A and also to the similar problem asked by Lusztig on the characteristic variety of the perverse sheaves corresponding to canonical bases.

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