Some Hall Polynomials for Representation-Finite Trivial Extension Algebras☆
Some Hall Polynomials for Representation-Finite Trivial Extension Algebras☆
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DOI：
10.1006/jabr.1997.7113
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发表时间：
1997-11
影响因子：
0.9
通讯作者：
L. Peng
中科院分区：
文献类型：
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作者：
L. Peng
设k是一个有限域，并假定Λ是一个有限维结合代数，1。用mod Λ表示所有可分解生成的（右）Λ-模的范畴，用ind Λ表示全子范畴，其中每个对象都是不可分解的（右）Λ-模的同类的代表。我们感兴趣的是对于L，M，N ∈ mod Λ的Hall多项式MNL的存在性（定义见977113或下面的第1节）。在Λ是有向的情况下，7证明了Λ具有Hall多项式，而在Λ是循环序列的情况下，4也得到了同样的结果。在[8]中证明了任何表示有限代数都有Hall多项式。本文证明了：若Λ是一个表示有限平凡扩张代数，则对任意L，M，N ∈ mod Λ且N不可分解，Λ有Hall多项式<$ML N和<$MNL.利用这些霍尔多项式，我们可以自然地将基为ind Λ的自由阿贝尔群（记为K（mod Λ））构造成李代数，并且K（mod Λ）<$ZQ的泛包络代数正好是H（Λ）1 <$ZQ，其中H（Λ）1是Λ的退化霍尔代数（参见下面的第5节）。
Letkbe a finite field and assume that Λ is a finite dimensional associativek-algebra with 1. Denote by mod Λ the category of all finitely generated (right) Λ-modules and by ind Λ the full subcategory in which every object is a representative of the isoclass of an indecomposable (right) Λ-module. We are interested in the existance of the Hall polynomial ϕMNLfor anL,M,N ∈ mod Λ (for the definition, see977113or Section 1 below). In case Λ is directed,7has shown that Λ has Hall polynomials, and in case Λ is cyclic serial, the same result has also been obtained by4. It has been conjectured in8that any representation-finitek-algebra has Hall polynomials. In this investigation, we shall show that if Λ is a representation-finite trivial extension algebra, then, for anyL,M,N ∈ mod Λ withNindecomposable, Λ has the Hall polynomials ϕMLNand ϕMNL. Using these Hall polynomials, we can naturally structure the free abelian group with a basis ind Λ, denoted byK(mod Λ), into a Lie algebra and the universal enveloping algebra ofK(mod Λ) ⊗ Z Qis just H(Λ)1 ⊗ Z Q, where H(Λ)1is the degenerated Hall algebra of Λ (see Section 5 below).