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 ${\mathcal{B}}_{m|n}(\infty )$, 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.

43 pages

Cited by in corpus (1)

• Crystal bases of parabolic Verma modules over the quantum orthosymplectic superalgebras