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On the Generalized Cartan Matrices Arising from k -th Yau Algebras of Isolated Hypersurface Singularities
Algebras and Representation Theory ( IF 0.5 ) Pub Date : 2020-07-18 , DOI: 10.1007/s10468-020-09981-x
Naveed Hussain , Stephen S.-T. Yau , Huaiqing Zuo

Let (V,0) be an isolated hypersurface singularity defined by the holomorphic function \(f: (\mathbb {C}^{n}, 0)\rightarrow (\mathbb {C}, 0)\). The k-th Yau algebra Lk(V ) is defined to be the Lie algebra of derivations of the k-th moduli algebra \(A^{k}(V) := \mathcal {O}_{n}/(f, m^{k}J(f))\), where k ≥ 0, m is the maximal ideal of \(\mathcal {O}_{n}\). I.e., Lk(V ) := Der(Ak(V ),Ak(V )). These new series of derivation Lie algebras are quite subtle invariants since they capture enough information about the complexity of singularities. In this paper we formulate a conjecture for the complete characterization of ADE singularities by using generalized Cartan matrix Ck(V ) associated to k-th Yau algebras Lk(V ), k ≥ 1. In this paper, we provide evidence for the conjecture and give a new complete characterization for ADE singularities. Furthermore, we compute their other various invariants that arising from the 1-st Yau algebra L1(V ).



中文翻译:

由孤立超曲面奇点的第k个Yau代数引起的广义Cartan矩阵

令(V,0)是由全纯函数\(f:(\ mathbb {C} ^ {n},0)\ rightarrow(\ mathbb {C},0)\)定义的孤立的超曲面奇点。第k个Yau代数L kV)被定义为第k个模数代数\(A ^ {k}(V):= \ mathcal {O} _ {n} /( F,M ^ {K}Ĵ(F))\) ,其中ķ ≥0,是的最大理想\(\ mathcal {ö} _ {N} \) 。即,L kV):= Der(A kV),A kV))。这些新的导出Lie代数系列是非常微妙的不变量,因为它们捕获了有关奇点复杂性的足够信息。在本文中,我们通过使用广义的Cartan矩阵制定ADE奇点的完整表征猜想Ç ķV相关联于)ķ第攸代数大号ķV),ķ ≥1。在本文中,我们为提供证据猜想并给出ADE奇异点的新完整表征。此外,我们计算它们的其他各种不变量,这些不变量是由第1个Yau代数L 1V)引起的。

更新日期:2020-07-18
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