Categorification of tensor powers of the vector representation of $$U_q({\mathfrak {gl}}(1|1))$$ U q ( gl ( 1 | 1 ) )

Abstract

We consider the monoidal subcategory of finite dimensional representations of Uq(gl(1|1)) generated by the vector representation, and we provide a diagram calculus for the intertwining operators, which allows to compute explicitly the canonical basis. We construct then a categorification of these representations and of the action of both Uq(gl(1|1)) and the intertwining operators using subquotient categories of the BGG category O(gl n ).

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

“…A similar description of Rep was obtained independently in the work of Sartori [21,Theorem 3.3.12], instead exploiting Schur-Weyl duality. It is nevertheless useful to have a skew Howe duality description of Rep, since skew Howe duality has been fruitfully exploited in many recent papers and seems to be closely related to foam-based descriptions of Khovanov-Rozansky homology.…”

“…In another form, this relation also appears in [20,Definition 5.5] for the case k D l D 2, and arose in that context from idempotents projecting onto simple representations of the Hecke algebra. This seems to be a quantisation of a relation well-known to experts, although it is hard to find a source.…”

“…However, we observe that the only time the moves (Move 1) and (Move 4) appear in Reidemeister moves is when there are strands pointing downwards: therefore if we restrict to braid diagrams and braid-like Reidemeister moves, it becomes possible to categorify the remaining MOY moves. A categorification of our category Rep of U q .gl.1j1//-modules was obtained in [20], using methods from the BGG category O . However, the categorification of the non-MOY relation (see Section 3.6) is still conjectural in that setting (see [20,Conjecture 7.7]).…”

“…A categorification of our category Rep of U q .gl.1j1//-modules was obtained in [20], using methods from the BGG category O . However, the categorification of the non-MOY relation (see Section 3.6) is still conjectural in that setting (see [20,Conjecture 7.7]). …”

Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians

“…Here, a completely new phenomenon appears. Besides their natural categorifications of Hecke algebras, translation functors were usually used in type A to categorify actions of (super) quantum groups, see, for instance, [8,17,35] for specific examples in this context, or [33] for an overview. In particular, they give standard examples of 2-categorifications of Kac-Moody algebras in the sense of [34] and [27], and the combinatorics of the quantum group and the Hecke algebras are directly related, see, e.g., [8,18] and [35].…”