## Stable maps, Q-operators and category $\mathcal {O}$

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- by David Hernandez
- Represent. Theory
**26**(2022), 179-210 - DOI: https://doi.org/10.1090/ert/604
- Published electronically: March 17, 2022
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## Abstract:

Motivated by Maulik-Okounkov stable maps associated to quiver varieties, we define and construct algebraic stable maps on tensor products of representations in the category $\mathcal {O}$ of the Borel subalgebra of an arbitrary untwisted quantum affine algebra. Our representation-theoretical construction is based on the study of the action of Cartan-Drinfeld subalgebras. We prove the algebraic stable maps are invertible and depend rationally on the spectral parameter. As an application, we obtain new $R$-matrices in the category $\mathcal {O}$ and we establish that a large family of simple modules, including the prefundamental representations associated to $Q$-operators, generically commute as representations of the Cartan-Drinfeld subalgebra. We also establish categorified $QQ^*$-systems in terms of the $R$-matrices we construct.## References

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## Bibliographic Information

**David Hernandez**- Affiliation: Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, IUF, F-75006 Paris, France
- MR Author ID: 707094
- Email: david.hernandez@u-paris.fr
- Received by editor(s): April 20, 2021
- Received by editor(s) in revised form: September 17, 2021
- Published electronically: March 17, 2022
- Additional Notes: The author was supported by the European Research Council under the European Union’s Framework Programme H2020 with ERC Grant Agreement number 647353 Qaffine
- © Copyright 2022 American Mathematical Society
- Journal: Represent. Theory
**26**(2022), 179-210 - MSC (2020): Primary 17B37, 17B10, 82B23
- DOI: https://doi.org/10.1090/ert/604
- MathSciNet review: 4396040