Acyclic quantum cluster algebras via Hall algebras of morphisms
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Mathematische Zeitschrift
Acyclic quantum cluster algebras via Hall algebras of morphisms Ming Ding1 · Fan Xu2 · Haicheng Zhang3 Received: 10 April 2019 / Accepted: 21 November 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Let A be the path algebra of a finite acyclic quiver Q over a finite field. We realize the quantum cluster algebra with principal coefficients associated to Q as a sub-quotient of a certain Hall algebra involving the category of morphisms between projective A-modules. Keywords Quantum cluster algebra · Hall algebra · Morphism category Mathematics Subject Classification 17B37 · 16G20 · 17B20
1 Introduction The Hall algebra of a finite dimensional algebra A over a finite field was introduced by Ringel [25] in 1990. Ringel [25,26] proved that if A is a representation-finite hereditary algebra, the Ringel–Hall algebra of A provides a realization of the positive part of the corresponding quantum group. Ringel’s approach establishes a relation between the representation theory of algebras and Lie theory, and provides an algebraic framework for studying the Lie theory resulting from Hall algebras associated to various abelian categories. Toën [32] generalized Ringel’s construction to define the derived Hall algebra for a DG-enhanced triangulated category satisfying certain finiteness conditions. Later on, for a triangulated category satisfying the left homological finiteness condition, Xiao and Xu [34] showed that Toën’s construction
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Haicheng Zhang [email protected] Ming Ding [email protected] Fan Xu [email protected]
1
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, People’s Republic of China
2
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
3
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, People’s Republic of China
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M. Ding et al.
still provides an associative unital algebra. It was expected but so far not successful to realize an entire quantum group through derived Hall algebra over triangulated category. In 2013, Bridgeland [5] provided a realization of the whole quantum group via the Hall algebra of 2-cyclic complexes of projective modules over a hereditary algebra. Lusztig [18,19] invented the geometric version of Ringel–Hall algebra constructions and obtained the canonical basis of the positive part of a quantum group as the direct summands of some semisimple constructible complexes over module varieties of a quiver. Kashiwara [16] applied an algebraic approach to define the crystal basis of the positive part of a quantum group. It is noteworthy that the canonical basis of a quantum group coincides with its crystal basis. In [20], Lusztig has also introduced the semicanonical basis of the positive part of the enveloping algebra associated to a quiver Q as certain constructible functions over the varieties of nilpotent representations of the preprojective algebra of Q, and the bas
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