Supersymmetric Solvable Lattice Models and the Kac module

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We will review the "usual paradigm" for a class of solvable lattice models including the six-vertex model and its generalizations. This attaches an element of a braided tensor category, such as the module category for a quantum group, to each edge of a grid. Examples found by Buciumas, Brubaker, Bump and Gustafsson related to quantum affine gl(r|n) seem to be slightly outside this paradigm. We will review the Kac modules for gl(r|n) and speculate how they might help with this puzzle. This line of thought leads to the following prediction: the q-Fock space of Kashiwara, Miwa and Stern, defined as a module for quantum affine gl(n), is secretly also a module for quantum affine gl(1|n).