Embeddings among quantum affine $\mathfrak{sl}_n$
Abstract
We establish an explicit embedding of a quantum affine $\mathfrak{sl}_n$ into a quantum affine $\mathfrak{sl}_{n+1}$. This embedding serves as a common generalization of two natural, but seemingly unrelated, embeddings, one on the quantum affine Schur algebra level and the other on the non-quantum level. The embedding on the quantum affine Schur algebras is used extensively in the analysis of canonical bases of quantum affine $\mathfrak{sl}_n$ and $\mathfrak{gl}_n$. The embedding on the non-quantum level is used crucially in a work of Riche and Williamson on the study of modular representation theory of general linear groups over a finite field. The same embedding is also used in a work of Maksimau on the categorical representations of affine general linear algebras. We further provide a more natural compatibility statement of the embedding on the idempotent version with that on the quantum affine Schur algebra level. A $\mathfrak{gl}_n$-variant of the embedding is also established.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2022
- DOI:
- 10.48550/arXiv.2208.07803
- arXiv:
- arXiv:2208.07803
- Bibcode:
- 2022arXiv220807803L
- Keywords:
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- Mathematics - Quantum Algebra;
- Mathematics - Representation Theory;
- 17B37
- E-Print:
- Acta Mathematica Sinica, English Series, to appear