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C-vectors of 2-Calabi–Yau categories and Borel subalgebras of \(\mathfrak {sl}_\infty \)

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Abstract

We develop a general framework for c-vectors of 2-Calabi–Yau categories, which deals with cluster tilting subcategories that are not reachable from each other and contain infinitely many indecomposable objects. It does not rely on iterative sequences of mutations. We prove a categorical (infinite-rank) generalization of the Nakanishi–Zelevinsky duality for c-vectors and establish two formulae for the effective computation of c-vectors—one in terms of indices and the other in terms of dimension vectors for cluster tilted algebras. In this framework, we construct a correspondence between the c-vectors of the cluster categories \({\mathscr {C}}(A_\infty )\) of type \(A_\infty \) due to Igusa–Todorov and the roots of the Borel subalgebras of \(\mathfrak {sl}_\infty \). Contrary to the finite dimensional case, the Borel subalgebras of \(\mathfrak {sl}_\infty \) are not conjugate to each other. On the categorical side, the cluster tilting subcategories of \({\mathscr {C}}(A_\infty )\) exhibit different homological properties. The correspondence builds a bridge between the two classes of objects.

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Acknowledgements

PJ was supported by EPSRC grant EP/P016014/1. MY was supported by NSF Grants DMS-1303038 and DMS-1601862, and by a Scheme 2 Grant from the London Mathematical Society. PJ thanks Louisiana State University and MY thanks Newcastle University for hospitality during visits. We thank Ken Goodearl and Jon McCammond for helpful comments on ordered sets. We are grateful to the anonymous referee for numerous valuable suggestions.

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  1. School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom

    Peter Jørgensen

  2. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Milen Yakimov

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  1. Peter Jørgensen
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  2. Milen Yakimov
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Correspondence to Peter Jørgensen.

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Jørgensen, P., Yakimov, M. C-vectors of 2-Calabi–Yau categories and Borel subalgebras of \(\mathfrak {sl}_\infty \). Sel. Math. New Ser. 26, 1 (2020). https://doi.org/10.1007/s00029-019-0525-4

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