A generators and relations description of a representation category of Uq(𝔤𝔩(1|1))

Abstract

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Super q–Howe duality and web categories

“…The defining relation between green and red edges is (2)(3)(4)(5)(6)(7)(8)(9) [2] which we call the dumbbell relation. 3…”

“…Note that the pitchfork lemma directly implies that (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12) could also be done by exploding the edges going underneath instead of the edges going over (or exploding both).…”

“…The braiding β • ·,· descends to the subquotients N -Web gr , N -Web g and N -Web r and we denote all induced braidings also by β • ·,· . They are all given by the formulas in Definition 2.17, but some diagrams might be zero due to (2)(3)(4)(5)(6)(7)(8)(9)(10).…”

“…Example 5.2. If we take M = 0, then box N +1,1 is a column Young diagram with N + 1 nodes and the corresponding not-a-hook relation is just the exterior relation (2)(3)(4)(5)(6)(7)(8)(9)(10). In this case we have that N |0-Web gr is N -Web gr and gl N |0 -Mod es is isomorphic to gl N -Mod es .…”

On webs in quantum type C

“…However, in breakthrough work [11], Cautis-Kamnitzer-Morrison found a web-based description of FundRep( ( )) using a quantized version of skew Howe duality. This technique was adapted to give new diagrammatic descriptions of various other categories of representations of quantum groups [41,37,15,51,8]. Nevertheless, it remains an open problem to give such a description of FundRep( ( )) for simple complex of rank ≥ 3 outside type .…”