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Generalized Cartan Matrices Associated to k-th Yau Algebras of Singularities and Characterization Theorem

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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 \(\mathcal {O}_{n}\). The Generalized Cartan matrix Ck(V ) is an object associated to Lk(V ). We previously proposed a conjecture that ADE singularities can be completely characterized by Ck(V ), and verified it for k = 1 in our previous work. In this paper, we continue this work and verify this conjecture for k = 2.

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Authors and Affiliations

  1. School of Data Sciences, Guangzhou Huashang College, Guangzhou, 511300, People’s Republic of China

    Naveed Hussain

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

    Stephen S.-T. Yau

  3. Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Huairou, 101400, People’s Republic of China

    Stephen S.-T. Yau

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

    Huaiqing Zuo

Authors
  1. Naveed Hussain
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  2. Stephen S.-T. Yau
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  3. Huaiqing Zuo
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Correspondence to Stephen S.-T. Yau or Huaiqing Zuo.

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Presented by: Peter Littelmann

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Both Yau and Zuo are supported by NSFC Grants 11961141005. Zuo is supported by NSFC Grant 11771231. Yau is supported by Tsinghua University start-up fund and Tsinghua University Education Foundation fund (042202008). Naveed is spported by innovation team project of Humanities and Social Sciences in Colleges and universities of Guangdong Province (No.: 2020wcxtd008).

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Hussain, N., Yau, S.ST. & Zuo, H. Generalized Cartan Matrices Associated to k-th Yau Algebras of Singularities and Characterization Theorem. Algebr Represent Theor 25, 1461–1492 (2022). https://doi.org/10.1007/s10468-021-10074-6

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  • DOI: https://doi.org/10.1007/s10468-021-10074-6

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