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Selecta Mathematica

, Volume 24, Issue 2, pp 1093–1119 | Cite as

Cohomological Hall algebras and affine quantum groups

  • Yaping Yang
  • Gufang Zhao
Article

Abstract

We study the preprojective cohomological Hall algebra (CoHA) introduced by the authors in Yang and Zhao (The cohomological Hall algebra of a preprojective algebra. arXiv: 1407.7994v5, 2015) for any quiver Q and any one-parameter formal group \({\mathbb {G}}\). In this paper, we construct a comultiplication on the CoHA, making it a bialgebra. We also construct the Drinfeld double of the CoHA. The Drinfeld double is a quantum affine algebra of the Lie algebra \(\mathfrak {g}_Q\) associated to Q, whose quantization comes from the formal group \({\mathbb G}\). We prove, when the group \({\mathbb G}\) is the additive group, the Drinfeld double of the CoHA is isomorphic to the Yangian.

Keywords

Quantum group Shuffle algebra Hall algebra Yangian Drinfeld double 

Mathematics Subject Classification

Primary 17B37 Secondary 14F43 55N22 

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Notes

Acknowledgements

We thank the anonymous referee for helpful comments. Most of the work was done when both authors were temporary faculty members at the University of Massachusetts, Amherst.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsThe University of MelbourneParkvilleAustralia
  2. 2.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA

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