Algebras and Representation Theory

, Volume 22, Issue 5, pp 1299–1329 | Cite as

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

  • Jie Xiao
  • Fan Xu
  • Minghui ZhaoEmail author


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.


Hall algebra Green formula Induction functor Restriction functor Simple perverse sheaf 

Mathematics Subject Classification (2010)

16G20 17B37 


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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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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsTsinghua UniversityBeijingPeople’s Republic of China
  2. 2.College of ScienceBeijing Forestry UniversityBeijingPeople’s Republic of China

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