Hall polynomials for the representation-finite hereditary algebras
Hall polynomials for the representation-finite hereditary algebras
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DOI：
10.1016/0001-8708(90)90043-m
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发表时间：
1990-12
影响因子：
1.7
通讯作者：
C. Ringel
中科院分区：
文献类型：
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作者：
C. Ringel
设k是一个域。设R是表示有限且遗传的以k为中心的有限维k-代数，因此R是Morita等价的k-种群的张量代数，其下标图A是dykin图的不相交并，且不可分解R-模的同构类集合双射地对应于相应的半单复李代数g的正根的集合@+(见[G]和[DRL])。因此，所有模分裂正合列的有限生成R-模的Grothendieck群K(R-mod)是基为@+的自由阿贝尔群。设h是g的Cartan子代数，g=n+@h0n_对应的三角分解。值得注意的是，n+是以@+为指标的一维复向量空间的直和，因此我们可以将K(R-mod)q@和n+确定为向量空间，并讨论如何恢复n的Lie乘法的问题
Let k be a field. Let R be a finite-dimensional k-algebra with centre k which is representation-finite and hereditary; thus R is Morita equivalent to the tensor algebra of a k-species with underlying graph A a disjoint union of Dynkin diagrams, and the set of isomorphism classes of indecomposable R-modules corresponds bijectively to the set@+ of positive roots of the corresponding semisimple complex Lie algebra g (see [G] and [DRl]). Consequently, the Grothendieck group K (R-mod) of all finitely generated R-modules modulo split exact sequences is the free abelian group with basis indexed by@+. Let h be a Cartan subalgebra of g and g= n+@ h 0 n _ the corresponding triangular decomposition. Note that n+ is the direct sum of one-dimensional complex vectorspaces indexed by the elements of@+, so we may identify K (R-mod) Q@ and n+ as vectorspaces, and we deal with the problem of how to recover the Lie multiplication of n, on