Crystal base of the negative half of the quantum superalgebra $U_q(\mathfrak{gl}(m|n))$
Abstract
We construct a crystal base of $U_q(\mathfrak{gl}(m|n))^-$, the negative half of the quantum superalgebra $U_q(\mathfrak{gl}(m|n))$. We give a combinatorial description of the associated crystal $\mathscr{B}_{m|n}(\infty)$, which is equal to the limit of the crystals of the ($q$-deformed) Kac modules $K(\lambda)$. We also construct a crystal base of a parabolic Verma module $X(\lambda)$ associated with the subalgebra $U_q(\mathfrak{gl}_{0|n})$, and show that it is compatible with the crystal base of $U_q(\mathfrak{gl}(m|n))^-$ and the Kac module $K(\lambda)$ under the canonical embedding and projection of $X(\lambda)$ to $U_q(\mathfrak{gl}(m|n))^-$ and $K(\lambda)$, respectively.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2022
- DOI:
- 10.48550/arXiv.2210.15288
- arXiv:
- arXiv:2210.15288
- Bibcode:
- 2022arXiv221015288J
- Keywords:
-
- Mathematics - Quantum Algebra;
- Mathematics - Representation Theory;
- 17B37;
- 17B10
- E-Print:
- 43 pages