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.
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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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Yang, Y., Zhao, G. Cohomological Hall algebras and affine quantum groups. Sel. Math. New Ser. 24, 1093–1119 (2018). https://doi.org/10.1007/s00029-017-0366-y
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DOI: https://doi.org/10.1007/s00029-017-0366-y