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On some embeddings between the cyclotomic quiver Hecke algebras


Authors: Kai Zhou and Jun Hu
Journal: Proc. Amer. Math. Soc. 148 (2020), 495-511
MSC (2010): Primary 20C08, 16G99, 06B15
DOI: https://doi.org/10.1090/proc/14733
Published electronically: August 7, 2019
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Abstract: Let $ I$ be a finite index set and let $ A=(a_{ij})_{i,j\in I}$ be an arbitrary indecomposable symmetrizable generalized Cartan matrix. Let $ Q^+$ be the positive root lattice and $ P^+$ the set of dominant weights. For any $ \beta \in Q^+$ and $ \Lambda \in P^+$, let $ \mathscr {R}_{\beta }^{\Lambda }$ be the corresponding cyclotomic quiver Hecke algebra over a field $ K$. For each $ i\in I$, there is a natural unital algebra homomorphism $ \iota _{\beta ,i}$ from $ \mathscr {R}_{\beta }^{\Lambda }$ to $ e(\beta ,i)\mathscr {R}_{\beta +\alpha _i}^{\Lambda }e(\beta ,i)$. In this paper we show that the homomorphism $ \iota _\beta :=\bigoplus _{i\in I}\iota _{\beta ,i}: \mathscr {R}_{\beta }^{\L... ...oplus _{i\in I}e(\beta ,i)\mathscr {R}_{\beta +\alpha _i}^{\Lambda }e(\beta ,i)$ is always injective unless $ \beta =0$ and $ \ell (\Lambda )=0$ or $ A$ is of finite type and $ \beta =\Lambda -w_0\Lambda $, where $ w_0$ is the unique longest element in the finite Weyl group associated to the finite Cartan matrix $ A$, and $ \ell (\Lambda )$ is the level of $ \Lambda $.


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Additional Information

Kai Zhou
Affiliation: School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, People’s Republic of China
Email: 1083864334@qq.com

Jun Hu
Affiliation: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China
Email: junhu404@bit.edu.cn

DOI: https://doi.org/10.1090/proc/14733
Keywords: Cyclotomic quiver Hecke algebras, quantum groups, integrable highest weight modules
Received by editor(s): December 25, 2018
Received by editor(s) in revised form: May 22, 2019
Published electronically: August 7, 2019
Additional Notes: The authors’ research was supported by the National Natural Science Foundation of China (No. 11525102).
Communicated by: Kailash C. Misra
Article copyright: © Copyright 2019 American Mathematical Society