Linear quivers, generic extensions and Kashiwara operators

作者：Deng Bangming; Du Jie; Zhang Guanglian

来源：Indagationes Mathematicae, 2007, 18(1): 3-21.

DOI：10.1016/S0019-3577(07)80003-6

In the present paper, we introduce the generic extension graph G of a Dynkin or cyclic quiver Q and then compare this graph with the crystal graph C for the quantized enveloping algebra associated to Q via two maps p(Q), kappa(Q): Omega -> Lambda(Q) induced by generic extensions and Kashiwara operators, respectively, where Lambda(Q) is the set of isoclasses of nilpotent representations of Q, and Omega is the set of all words on the alphabet I, the vertex set of Q. We prove that, if Q is a (finite or infinite) linear quiver, then the intersection of the fibres p(Q)(-1) (lambda) and kappa(-1)(Q) (lambda) is non-empty for every lambda is an element of Lambda(Q). We will also show that this non-emptyness property fails for cyclic quivers.

• 出版日期2007-3-26
• 单位盐城师范学院; 北京师范大学; 清华大学