Crystal base of the negative half of the quantum superalgebra U_q(\mathfrakgl(m|n))

2022/10/27 by Il-Seung Jang, Jang, Il-Seung, Jae-Hoon Kwon +3
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2210.15288

Abstract

We construct a crystal base of U_q(\mathfrakgl(m|n))^-, the negative half of the quantum superalgebra U_q(\mathfrakgl(m|n)). We give a combinatorial description of the associated crystal \mathscrB_m|n(∞), which is equal to the limit of the crystals of the (q-deformed) Kac modules K(λ). We also construct a crystal base of a parabolic Verma module X(λ) associated with the subalgebra U_q(\mathfrakgl_0|n), and show that it is compatible with the crystal base of U_q(\mathfrakgl(m|n))^- and the Kac module K(λ) under the canonical embedding and projection of X(λ) to U_q(\mathfrakgl(m|n))^- and K(λ), respectively.

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