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Arnold Mathematical Journal

, Volume 5, Issue 2–3, pp 197–283 | Cite as

Shifted Quantum Affine Algebras: Integral Forms in Type A

  • Michael Finkelberg
  • Alexander TsymbaliukEmail author
Research Contribution

Abstract

We define an integral form of shifted quantum affine algebras of type A and construct Poincaré–Birkhoff–Witt–Drinfeld bases for them. When the shift is trivial, our integral form coincides with the RTT integral form. We prove that these integral forms are closed with respect to the coproduct and shift homomorphisms. We prove that the homomorphism from our integral form to the corresponding quantized K-theoretic Coulomb branch of a quiver gauge theory is always surjective. In one particular case we identify this Coulomb branch with the extended quantum universal enveloping algebra of type A. Finally, we obtain the rational (homological) analogues of the above results [proved earlier in Kamnitzer et al. (Proc Am Math Soc 146(2):861–874, 2018a; On category \(\mathcal {O}\) for affine Grassmannian slices and categorified tensor products. arXiv:1806.07519, 2018b) via different techniques].

Keywords

Shifted Yangians Shifted quantum affine algebras Coulomb branch Evaluation homomorphism Drinfeld-Gavarini duality PBWD bases 

Notes

Acknowledgements

We are deeply grateful to A. Braverman, P. Etingof, B. Feigin, A. Molev, and A. Weekes. Special thanks go to R. Kodera and C. Wendlandt for pointing out two inaccuracies in the earlier version of this paper. Alexander Tsymbaliuk gratefully acknowledges support from Yale University, and is extremely grateful to MPIM (Bonn, Germany), IPMU (Kashiwa, Japan), RIMS (Kyoto, Japan) for the hospitality and wonderful conditions in the summer 2018 during the work on this project. The final version of this paper was prepared during Alexander Tsymbaliuk’s visit to IHES (Bures-sur-Yvette, France) in the summer 2019, sponsored under the ERC QUASIFT grant agreement 677368. Alexander Tsymbaliuk is indebted to T. Arakawa (RIMS), T. Milanov (IPMU), V. Pestun (IHES) for their invitations. Michael Finkelberg was partially funded within the framework of the HSE University Basic Research Program and the Russian Academic Excellence Project ‘5-100’. Alexander Tsymbaliuk was partially supported by the NSF Grant DMS-1821185.

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Copyright information

© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Research University Higher School of EconomicsMoscowRussian Federation
  2. 2.Skolkovo Institute of Science and TechnologyInstitute for Information Transmission ProblemsMoscowRussian Federation
  3. 3.Department of MathematicsYale UniversityNew HavenUSA

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