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The crystal of the categorified quantum group
Abstract
A categorification of the crystal B(Ũq(g)) of the modified quantized enveloping algebra associated to a symmetrizable Kac-Moody algebra g is given. A crystal is constructed from a categorical g-action on abelian categories without finiteness conditions, and is used to construct the bicrystal structure on the set of self-dual indecomposable objects of the categorified quantum group. Crystals from Webster's generalized tensor product categorification are identified with tensor products of lowest and highest weight crystals. As an application, a categorification of the extremal weight module is obtained.
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BibTeX
@misc{lee2026crystal,
title = {{The crystal of the categorified quantum group}},
author = {Soo-Hong Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30073},
eprint = {2609.30073},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30073},
}
RIS
TY - UNPB
TI - The crystal of the categorified quantum group
AU - Soo-Hong Lee
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30073
UR - https://arxiv.org/abs/2609.30073
AB - A categorification of the crystal B(Ũ_q(g)) of the modified quantized enveloping algebra associated to a symmetrizable Kac-Moody algebra g is given. A crystal is constructed from a categorical g-action on abelian categories without finiteness conditions, and is used to construct the bicrystal structure on the set of self-dual indecomposable objects of the categorified quantum group. Crystals from Webster's generalized tensor product categorification are identified with tensor products of lowest and highest weight crystals. As an application, a categorification of the extremal weight module is obtained.
ER -