The Composition Algebra of a Cyclic Quiver
The Composition Algebra of a Cyclic Quiver
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DOI：
10.1112/plms/s3-66.3.507
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发表时间：
1993-05
影响因子：
1.8
通讯作者：
C. Ringel
中科院分区：
文献类型：
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作者：
C. Ringel
请注意，A也称为具有循环取向的An_x型箭图。我们用Ao表示A的顶点集合(通常我们用Z/Nz或也用集合{xx，x2，·-，xn)或仅用{1，2，…，n}来表示A的顶点集合，用箭头xt->；xi+L或i->；i+L来表示。有n个一维表示，对应于A的顶点；这些都是(直到同构)T的所有简单对象。对应于A的顶点a的简单表示用5(A)表示，如果S‘同构于S(A)，我们记为[5’]=a。给定一个简单表示5，并且(直到同构)有一个唯一的不可分解表示S[L]，长度为i，顶5，这样我们就得到了所有不可分解的表示(同样是直到同构)。因此，我们可以通过n元组划分的集合IT来索引T中的同构类；对应于n e U的A的表示将由M{jz)表示；见§3.3。
Note that A is also called the quiver of type An_x with cyclic orientation. We denote by Ao the set of vertices of A (and often we will identify Ao with Z/nZ, or also with the set {xx, x2, •-, xn) or just with {1, 2, ..., n}, with arrows xt—>xi+l or i->i + l). There are n one-dimensional representations, corresponding to the vertices of A; these are (up to isomorphism) all the simple objects of T. The simple representation corresponding to the vertex a of A will be denoted by 5(a), and if S' is isomorphic to S(a), we will write [5'] = a. Given a simple representation 5, and I eNu there is (up to isomorphism) a unique indecomposable representation S[l] of length / with top 5, and we obtain in this way all indecomposable representations (again up to isomorphism). It follows that we can index the isomorphism classes in T by the set IT of n-tuples of partitions; the representation of A corresponding to n e U. will be denoted by M{JZ); see § 3.3.