Geometric and Combinatorial Realizations of Crystals of Enveloping Algebras
Abstract
Kashiwara and Saito have defined a crystal structure on the set of irreducible components of Lusztig's quiver varieties. This gives a geometric realization of the crystal graph of the lower half of the quantum group associated to a simplylaced KacMoody algebra. Using an enumeration of the irreducible components of Lusztig's quiver varieties in finite and affine type A by combinatorial data, we compute the geometrically defined crystal structure in terms of this combinatorics. We conclude by comparing the combinatorial realization of the crystal graph thus obtained with other combinatorial models involving Young tableaux and Young walls.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 2006
 arXiv:
 arXiv:math/0601511
 Bibcode:
 2006math......1511S
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Algebraic Geometry;
 Mathematics  Representation Theory;
 16G20;
 17B37
 EPrint:
 12 pages