Crystal base of the negative half of the quantum superalgebra \(U_q( \mathfrak{gl}(m | n))\) (Q6051018)

In the paper under review, the authors construct a crystal base of the negative half of the quantum superalgebra \(U_q(\mathfrak{gl}(m|n))\), and investigate several properties of the associated crystal \(\mathscr{B}_{m|n}(\infty)\) including combinatorial description. One of key ingredients is to consider the crystal \(\mathcal{B}(K(\lambda))\) of the \(q\)-deformed \emph{Kac-module} \(K(\lambda)\) in the setting over the \emph{generalized quantum group} \(\mathcal{U}_{m|n}\), that is isomorphic to \(U_q(\mathfrak{gl}(m|n))\) under some extension. The authors study crystal morphisms between \(\mathcal{B}(K(\lambda))\) and \(\mathcal{B}(K(\mu))\) when \(\lambda < \mu\) and consider the induced direct system, which gives the limit crystal \(\mathcal{B}(K(\infty))\). They study the lattice \(\mathcal{L}_{m|n}(\infty)\) and the associated crystal base \(\mathcal{B}_{m|n}(\infty)\) of the negative half \(\mathcal{U}^-_{m|n}\), and show that the limit crystal \(\mathcal{B}(K(\infty))\) is isomorphic to the crystal base \(\mathcal{B}_{m|n}(\infty)\). The author also construct a crystal base of a parabolic Verma module \(X(\lambda)\) and study compatibility with \(\mathcal{U}^-_{m|n}\) and \(K(\lambda)\) in the viewpoint of the crystal theory.