Hall algebras, hereditary algebras and quantum groups
Hall algebras, hereditary algebras and quantum groups
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DOI：
10.1007/bf01241133
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发表时间：
1995-12
影响因子：
3.1
通讯作者：
J. Green
中科院分区：
文献类型：
--
作者：
J. Green
设R是有限域k上的结合遗传代数，R-fin是R-mod的满子范畴，其对象是作为集合有限的左R-模X，IX]< oc.还假定R在C中是有限的。Ringel的意义，即IExtl（s，s '）]< c~对于所有简单的S，S'在R-fin;这个条件是满足的，例如，如果R是n-生成为k-代数[4，pp。435，436]。设~是R-fin中所有同构类的集合。若2 E~，则U~表示类2中的R-模.所有零左R-模的类记为0。设IC_~是R-fin中单模的所有同构类的集合。因此{U~：i EI}是一个单的有限左R-模的完备集。证明了Grothendieck群K 0（R-fin）与以I为自由基的自由Abel群E1={~，v~ i：v~ E7 Z}相同，使得若XE R-fin，则K 0（R-fin）的相应元素是”维数向量”dimX= Y~，~ Ivii，其中对每个i Ei，vi是单模U在X的任意合成列中的重数.显然dimX位于子半群中
Let R be an associative, hereditary algebra over a finite field k, and let R-fin be the full subcategory of R-mod whose objects are those left R-modules X which are finite as sets, IX]< oc. Assume also that R isfinitary in C. Ringel's sense, ie that IExtl (s, s')]< c~ for all simple S, S'in R-fin; this condition is met, for example, if R is finitely generated as k-algebra [4, pp. 435,436]. Let~ be the set of all isomorphism classes in R-fin. If 2 E~, then U~ will denote an R-module in class 2. The class of all zero left R-modules is denoted 0. Let I C_~ be the set of all isomorphism classes of simple modules in R-fin. Thus {U~: i EI} is a complete set of simple, finite left R-modules. We identify the Grothendieck group K0 (R-fin) with the free Abelian group E1={~, v~ i: v~ E 7Z} having I as free basis, so that if XE R-fin then the corresponding element of K0 (R-fin) is the" dimension vector" dimX= Y~,~ I vii, where for each i E/, vi is the multiplicity of the simple module U, in any composition series of X. Clearly dimX lies in the subsemigroup