Category 𝒪 for truncated shifted Yangians and the bi-infinite Bott-Samelson variety
Abstract
In this paper, we study the category 𝒪 of representations of shifted Yangians associated to a simply-laced simple Lie algebra g over ℂ. In particular, we prove that the (complexified) Grothendieck ring of this category is isomorphic to the Cox ring of the open bi-infinite Bott-Samelson variety, which is a pro-variety we construct from Bott-Samelson varieties for alternating heaps. Using work of Francone-Leclerc, we prove a conjecture of Hernandez-Zhang by identifying the above Grothendieck ring with a cluster algebra defined by Geiss-Hernandez-Leclerc. Our methods also yield an action of the Langlands dual group G∨ on this Grothendieck ring, and show that the shifted coproducts defined in work of the first and fifth authors with collaborators give rise to coproducts for truncated shifted Yangians. This machinery then allows us to prove further conjectures of Frenkel-Hernandez and Geiss-Hernandez-Leclerc on extended QQ-systems, and to obtain a generalization of a duality defined by Hernandez-Leclerc.
Related
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BibTeX
@misc{kamnitzer2026category,
title = {{Category $\mathcalO$ for truncated shifted Yangians and the bi-infinite Bott-Samelson variety}},
author = {Joel Kamnitzer and Antoine Labelle and Alexis Leroux-Lapierre and Théo Pinet and Alex Weekes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.04480},
eprint = {2607.04480},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04480},
}
RIS
TY - UNPB
TI - Category 𝒪 for truncated shifted Yangians and the bi-infinite Bott-Samelson variety
AU - Joel Kamnitzer
AU - Antoine Labelle
AU - Alexis Leroux-Lapierre
AU - Théo Pinet
AU - Alex Weekes
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2607.04480
UR - https://arxiv.org/abs/2607.04480
AB - In this paper, we study the category 𝒪 of representations of shifted Yangians associated to a simply-laced simple Lie algebra g over ℂ. In particular, we prove that the (complexified) Grothendieck ring of this category is isomorphic to the Cox ring of the open bi-infinite Bott-Samelson variety, which is a pro-variety we construct from Bott-Samelson varieties for alternating heaps. Using work of Francone-Leclerc, we prove a conjecture of Hernandez-Zhang by identifying the above Grothendieck ring with a cluster algebra defined by Geiss-Hernandez-Leclerc. Our methods also yield an action of the Langlands dual group G^∨ on this Grothendieck ring, and show that the shifted coproducts defined in work of the first and fifth authors with collaborators give rise to coproducts for truncated shifted Yangians. This machinery then allows us to prove further conjectures of Frenkel-Hernandez and Geiss-Hernandez-Leclerc on extended QQ-systems, and to obtain a generalization of a duality defined by Hernandez-Leclerc.
ER -