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Representation Theory of Algebraic Groups and Quantum Groups pp 313–348Cite as

Green’s Formula with ℂ*-Action and Caldero–Keller’s Formula for Cluster Algebras

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Part of the book series: Progress in Mathematics ((PM,volume 284))

Abstract

It is known that Green’s formula over finite fields gives rise to the comultiplications of Ringel–Hall algebras and quantum groups (see [Invent. Math. 120 (1995), 361–377], see also [J. Amer. Math. Soc. 4 (1991), 365–421]). In this chapter, we prove a projective version of Green’s formula in a geometric way. Then following the method of Hubery in [Hubery, Acyclic cluster algebras via Ringel–Hall algebras (preprint)], we apply this formula to proving Caldero–Keller’s multiplication formula for acyclic cluster algebras of arbitrary type.

Mathematics Subject Classifications (2000): 14M99, 16G20, 16G70, 17B35

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Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 10084, Peoples Republic of China

    Jie Xiao

Authors
  1. Jie Xiao
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  2. Fan Xu
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Corresponding author

Correspondence to Jie Xiao .

Editor information

Editors and Affiliations

  1. Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan

    Akihiko Gyoja

  2. Research Institute for Mathematical Scie, Kyoto University, Kyoto, 606-8502, Japan

    Hiraku Nakajima

  3. Faculty of Science and Technology, Sophia University, Department of Information and Communication Sciences -1, Tokyo, 102-8554, Japan

    Ken-ichi Shinoda

  4. Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan

    Toshiaki Shoji

  5. Graduate School of Science, Osaka City University, Department of Mathematics -3-138, Osaka, 558-8585, Japan

    Toshiyuki Tanisaki

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Xiao, J., Xu, F. (2010). Green’s Formula with ℂ*-Action and Caldero–Keller’s Formula for Cluster Algebras. In: Gyoja, A., Nakajima, H., Shinoda, Ki., Shoji, T., Tanisaki, T. (eds) Representation Theory of Algebraic Groups and Quantum Groups. Progress in Mathematics, vol 284. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4697-4_13

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