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On cluster theory and quantum dilogarithm identities

By Bernhard Keller


These are expanded notes from three survey lectures given at the 14th International Conference on Representations of Algebras (ICRA XIV) held in Tokyo in August 2010. We first study identities between products of quantum dilogarithm series associated with Dynkin quivers following Reineke. We then examine similar identities for quivers with potential and link them to Fomin-Zelevinsky's theory of cluster algebras. Here we mainly follow ideas due to Bridgeland, Fock-Goncharov, Kontsevich-Soibelman and Nagao.Comment: 32 pages, 5 figures; v2: reference added; v3: section 4.5 corrected, references updated; v4: end of section 1.7 correcte

Topics: Mathematics - Representation Theory, Mathematics - Combinatorics, Mathematics - Quantum Algebra, 16G20 (Primary) 18E30 (Secondary) 17B37, 14N35
Year: 2011
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