Kac–Moody groups and cluster algebras
Kac–Moody groups and cluster algebras
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DOI：
10.1016/j.aim.2011.05.011
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发表时间：
2010-01
影响因子：
1.7
通讯作者：
C. Geiss;B. Leclerc;J. Schroer
中科院分区：
文献类型：
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作者：
C. Geiss;B. Leclerc;J. Schroer
设Q为无有向环的有限振子，设Λ为相关的预投影代数，设g为Weyl群W的相关Kac-Moody李代数，设n为g的正部。对于每个Weyl群元素W， Buan, Iyama， Reiten和Scott引入了一个子范畴Cwof mod（Λ）。已知C是一个Frobenius范畴，它的稳定范畴C是一个二维的Calabi-Yau范畴。我们证明了cw在单幂群N(w)的坐标环C[N(w)]上得到一个聚类代数结构：=N∩(w−1N−w)。这里N是李代数N(N)的补全N(N)的亲单幂亲群。我们可以用U(N)的子代数U(N)的渐变对偶来识别C[N(w)]，即U(N)的泛包膜代数U(N)的渐变对偶。设S (N)为Lusztig的半正则基S （U(N)）的对偶。我们证明了C[N(w)]的所有聚类单项都属于S，并且S∩C[N(w)]是C[N(w)]的C基。此外，通过对系数环的生成元进行形式反演得到的C[N(w)]簇代数与Kac-Moody群的单幂元nw上正则函数的代数C[Nw]同构，并得到C[Nw]对应的对偶半规范基。作为一个应用，我们得到了每个非循环聚类代数的一个基，它以一种自然的方式包含了所有的聚类单项式。
Let Q be a finite quiver without oriented cycles, let Λ be the associated preprojective algebra, let g be the associated Kac–Moody Lie algebra with Weyl group W, and let n be the positive part of g. For each Weyl group element w, a subcategory Cwof mod(Λ) was introduced by Buan, Iyama, Reiten and Scott. It is known that Cwis a Frobenius category and that its stable category C̲wis a Calabi–Yau category of dimension two. We show that Cwyields a cluster algebra structure on the coordinate ring C[N(w)] of the unipotent group N(w):=N∩(w−1N−w). Here N is the pro-unipotent pro-group with Lie algebra the completion nˆ of n. One can identify C[N(w)] with a subalgebra of U(n)gr⁎, the graded dual of the universal enveloping algebra U(n) of n. Let S⁎be the dual of Lusztigʼs semicanonical basis S of U(n). We show that all cluster monomials of C[N(w)] belong to S⁎, and that S⁎∩C[N(w)] is a C-basis of C[N(w)]. Moreover, we show that the cluster algebra obtained from C[N(w)] by formally inverting the generators of the coefficient ring is isomorphic to the algebra C[Nw] of regular functions on the unipotent cell Nwof the Kac–Moody group with Lie algebra g. We obtain a corresponding dual semicanonical basis of C[Nw]. As one application we obtain a basis for each acyclic cluster algebra, which contains all cluster monomials in a natural way.