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On the Generalized Cartan Matrices Arising from k-th Yau Algebras of Isolated Hypersurface Singularities

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Abstract

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 ).

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Author notes

  1. Naveed Hussain

    Present address: School of Data Sciences, Huashang College Guangdong University of Finance and Economics, Guangzhou, Guangdong, 511300, People’s Republic of China

Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

    Naveed Hussain, Stephen S.-T. Yau & Huaiqing Zuo

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  1. Naveed Hussain
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  2. Stephen S.-T. Yau
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Correspondence to Stephen S.-T. Yau.

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Presented by: Peter Littelmann

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Dedicated to Professor Stephen Halperin on the occasion of his 78th birthday

Both Yau and Zuo are supported by NSFC Grants 11961141005 and 11531007. Zuo is supported by NSFC Grant 11771231. Yau is supported by Tsinghua University start-up fund and Tsinghua University Education Foundation fund (042202008).

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Hussain, N., Yau, S.ST. & Zuo, H. On the Generalized Cartan Matrices Arising from k-th Yau Algebras of Isolated Hypersurface Singularities. Algebr Represent Theor 24, 1101–1124 (2021). https://doi.org/10.1007/s10468-020-09981-x

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