Semicanonical bases and preprojective algebras
Abstract
We study the multiplicative properties of the dual of Lusztig's semicanonical basis.The elements of this basis are naturally indexed by theirreducible components of Lusztig's nilpotent varieties, whichcan be interpreted as varieties of modules over preprojective algebras.We prove that the product of two dual semicanonical basis vectorsis again a dual semicanonical basis vector provided the closure ofthe direct sum of thecorresponding two irreducible components is again an irreducible component.It follows that the semicanonical basis and the canonical basiscoincide if and only if we are in Dynkin type $A_n$ with $n \leq 4$.Finally, we provide a detailed study of the varieties of modules over the preprojectivealgebra of type $A_5$.We show that in this case the multiplicative properties ofthe dual semicanonical basis are controlled by the Ringel form of a certain tubular algebra of type (6,3,2) and by thecorresponding elliptic root system of type $E_8^{(1,1)}$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 February 2004
 arXiv:
 arXiv:math/0402448
 Bibcode:
 2004math......2448G
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Quantum Algebra;
 14M99;
 16D70;
 16E20;
 16G20;
 16G70;
 17B37;
 20G42
 EPrint:
 Minor corrections. Final version to appear in Annales Scientifiques de l'ENS