Research Article
Tensor products of infinite dimensional modules in the BGG category of a quantized simple Lie algebra of type ADE
Abstract
We consider the BGG category $\O$ of a quantized universal enveloping algebra $U_q(\mathfrak{g})$. It is well-known that $M\otimes N\in \O$ if $M$ or $N$ is finite dimensional. When $\mathfrak{g}$ is simple and of type ADE, we prove in this paper that $M\otimes N\notin \O$ if $M$ and $N$ are both infinite dimensional.
References
- H. H. Andersen and V. Mazorchuk, Category $\O$ for quantum groups, J. Eur. Math. Soc. (JEMS), 17(2) (2015), 405-431.
- J. E. Humphreys, Representations of semisimple Lie algebras in the BGG category $\O$, Grad. Stud. Math., American Math. Soc., 94, 2008.
- R. Kane, Reflection Groups and Invariant Theory, CMS Books Math./Ouvrages Math. SMC, 5, Springer-Verlag, New York, 2001.
- C. Voigt and R. Yuncken, Complex semisimple quantum groups and representation theory, Lecture Notes in Math., 2264, Springer, Cham, 2020.
- Z. Wei, Tensor-closed objects in the BGG category of a quantized semisimple Lie algebra, Int. Electron. J. Algebra, 29 (2021), 175-186.
Year 2024,
Volume: 35 Issue: 35, 108 - 120, 09.01.2024
Abstract
References
- H. H. Andersen and V. Mazorchuk, Category $\O$ for quantum groups, J. Eur. Math. Soc. (JEMS), 17(2) (2015), 405-431.
- J. E. Humphreys, Representations of semisimple Lie algebras in the BGG category $\O$, Grad. Stud. Math., American Math. Soc., 94, 2008.
- R. Kane, Reflection Groups and Invariant Theory, CMS Books Math./Ouvrages Math. SMC, 5, Springer-Verlag, New York, 2001.
- C. Voigt and R. Yuncken, Complex semisimple quantum groups and representation theory, Lecture Notes in Math., 2264, Springer, Cham, 2020.
- Z. Wei, Tensor-closed objects in the BGG category of a quantized semisimple Lie algebra, Int. Electron. J. Algebra, 29 (2021), 175-186.
There are 5 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Algebra and Number Theory, Group Theory and Generalisations, Category Theory, K Theory, Homological Algebra, Pure Mathematics (Other) |
| Journal Section | Research Article |
| Authors | |
| Early Pub Date | September 8, 2023 |
| Publication Date | January 9, 2024 |
| DOI | https://doi.org/10.24330/ieja.1357059 |
| IZ | https://izlik.org/JA92GS23KR |
| Published in Issue | Year 2024 Volume: 35 Issue: 35 |