The irreducible Components of Lusztig's Nilpotent Variety and Crystal Graphs
Let Q be a quiver without loops. We denote the quantized version of the enveloping algebra of the negative part of the corresponding Kac-Moody-Lie algebra by U-. Lusztig has defined varieties R(P(Q);d)0, also called Lusztig's nilpotent varieties, consisting of nilpotent representations of the preprojective algebra P(Q) of Q of dimension vector d. It was shown by Kashiwara and Saito ([1]) that the irreducible components of the various R(P(Q);d)0, where d runs through all possible dimension vectors of Q, form the crystal of U-.The principal aim of this talk is to describe the irreducible components of R(P(Q);d)0 using so-called nilpotent class representations (nc-representation) of Q with dimensions vector d. Informally a nc-representation assigns to Q and d certain nilpotent classes, so that the generic nc-representations are in natural bijection with the irreducible components of R(P(Q);d)0.
This construction leeds naturally to certain similar varieties R(Fn;l0,¼,ln)0 associated to the free associative algebra Fn and n+1 nilpotent classes li. Then we can define for any n an associated local crystal graph Cn. We show, how one can reconstruct the global crystal graph from the local crystal graphs in a purely combinatorial way.
[1] M. Kashiwara, Y. Saito: Geometric construction of crystal bases. Duke Mat. J. 89 (1997), no. 1, 9 - 36