Affine quivers and canonical bases
Affine quivers and canonical bases
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DOI：
10.1007/bf02699432
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发表时间：
1992-12
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通讯作者：
G. Lusztig
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作者：
G. Lusztig
设U是Drinfeld和Jimbo赋予对称广义Cartan矩阵的量子化包络代数（见[D]），L^ L^ 00 U（x）U~是它的三角分解，在[L]中，给出了U^= U（x）U~的一个纯几何构造（作为一个霍普夫代数）给出了在模空间上的表示的一个符号的反常层;这种构造同时给出了U~+的一个具有良好性质的正则基。(This理论上相当于U的一个几何构造，因为通过DrinfekT的量子二重构造[D，§ 13]，可以用一种简单的方法从霍普夫代数U重构出霍普夫代数U^。）在[L]的一般性中，进入U~的标准基的单反常层仅以抽象的方式定义，但不知道具体的形式，除了在最简单的情况下（类型A，D，E），当它们恰好是对应于轨道的单反常层时。本文的目的之一是具体描述的简单的反常层，形成规范的基础上仿射的情况下（即一个对称仿射嘉当矩阵的情况下）。
Let U be the quantized enveloping algebra attached by Drinfeld and Jimbo to a symmetric generalized Cartan matrix (see [D]); let L^ L^ 00 U (x) U~ be its triangular decomposition.In [L], a purely geometric construction of U^= U (x) U~(as a Hopf algebra) was given in terms of perverse sheaves on the moduli space of representations of a quiver; the construction gave at the same time a canonical basis of U~ with very favourable properties.(This amounts, in principle, to a geometric construction ofU, since by DrinfekTs quantum double construction [D, § 13], the Hopf algebra U can be reconstructed in a simple way from the Hopf algebra U^.) In the generality of [L], the simple perverse sheaves which enter in the canonical basis of U~ are only defined in an abstract way, but are not known in a concrete form, except in the simplest case (type A, D, E) when they are exactly the simple perverse sheaves corresponding to orbits. One of the aims of this paper is to describe in concrete terms the simple perverse sheaves which form the canonical basis in the affme case (that is, the case of a symmetric affine Cartan matrix).