Hall Algebras and Quantum Symmetric Pairs II: Reflection Functors

Abstract

Recently the authors initiated an \(\imath \)Hall algebra approach to (universal) \(\imath \)quantum groups arising from quantum symmetric pairs. In this paper we construct and study BGP type reflection functors which lead to isomorphisms of the \(\imath \)Hall algebras associated to acyclic \(\imath \)quivers. For Dynkin quivers, these symmetries on \(\imath \)Hall algebras induce automorphisms of universal \(\imath \)quantum groups, which are shown to satisfy the braid group relations associated to the restricted Weyl group of a symmetric pair. This leads to a conceptual construction of q-root vectors and PBW bases for (universal) quasi-split \(\imath \)quantum groups of ADE type.

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Acknowledgements

We thank Institute of Mathematics at Academia Sinica (Taipei) and East China Normal University (Shanghai) for hospitality and support where part of this work was carried out. WW is partially supported by NSF Grant DMS-1702254 and DMS-2001351. We thank 2 anonymous referees for their careful readings and suggestions which help make this paper more readable.

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Correspondence to Weiqiang Wang.

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Communicated by Y. Kawahigashi.

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Lu, M., Wang, W. Hall Algebras and Quantum Symmetric Pairs II: Reflection Functors. Commun. Math. Phys. 381, 799–855 (2021). https://doi.org/10.1007/s00220-021-03965-8

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