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Ringel-Hall Algebras Beyond their Quantum Groups I: Restriction Functor and Green Formula

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In this paper, we generalize the categorifical construction of a quantum group and its canonical basis introduced by Lusztig to the generic form of the whole Ringel-Hall algebra. We clarify the explicit relation between the Green formula and the restriction functor. By a geometric way to prove the Green formula, we show that the compatibility of multiplication and comultiplication of a Ringel-Hall algebra can be categorified under Lusztig’s framework.

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Acknowledgements

The authors are very grateful to Hiraku Nakajima for telling us that Lusztig’s restriction is a hyperbolic localization and Reference [4]. The second named author thanks Sheng-Hao Sun for explaining the contents in Reference [10] and many helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, Tsinghua University, Beijing, 100084, People’s Republic of China

    Jie Xiao & Fan Xu

  2. College of Science, Beijing Forestry University, Beijing, 100083, People’s Republic of China

    Minghui Zhao

Authors
  1. Jie Xiao
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  2. Fan Xu
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  3. Minghui Zhao
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Corresponding author

Correspondence to Minghui Zhao.

Additional information

Presented by: Michel Van den Bergh

To the memory of Professor J. A. Green.

Jie Xiao was supported by NSF of China (No. 11671221), Fan Xu was supported by NSF of China (No. 11471177, 11771217) and Minghui Zhao was supported by NSF of China (No. 11701028, 11771445).

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Xiao, J., Xu, F. & Zhao, M. Ringel-Hall Algebras Beyond their Quantum Groups I: Restriction Functor and Green Formula. Algebr Represent Theor 22, 1299–1329 (2019). https://doi.org/10.1007/s10468-018-9821-5

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  • DOI: https://doi.org/10.1007/s10468-018-9821-5

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