Webs and quantum skew Howe duality

Abstract

We give a diagrammatic presentation in terms of generators mod relations of the representation category of $U_q(\mathfrak{sl}_n)$. More precisely, we produce all the relations among $\rm{SL}_n$-webs, thus describing the full subcategory tensor-generated by fundamental representations $\bigwedge^k \mathbb{C}^n$ (this subcategory can be idempotent completed to recover the entire representation category). Our result answers a question posed by Kuperberg [arXiv:q-alg/9712003] and affirms conjectures of Kim [arXiv:math.QA/0310143] and Morrison [arXiv:0704.1503]. Our main tool is an application of quantum skew Howe duality.Comment: 32 pages, added a missing relation which had been implicitly used; this version has the same content as the published versio

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

“…There are a few qualitative differences between this result and the one established in [4]. One difference is that due to the superalgebra structure, the i th exterior power of the standard representation is never the 0 representation for i > 0, which means that trivalent graphs of all colours are necessary, whereas in the case of U q .sl.n// a strand with high enough colours would cause the diagram to evaluate identically to 0.…”

“…Much of this section follows the proof of the corresponding result obtained in [4]. More information about braided exterior algebras can be found in [1].…”

“…Ladders were introduced in [4] to give a diagrammatic description of certain subcategories of P U q .gl.m//. We use them here to give a similar description of P U 1 q .gl.m//.…”

“…In a recent paper, Cautis, Kamnitzer and Morrison [4] show that in fact the diagram calculus given by Murakami, Ohtsuki and Yamada (MOY) gives a complete description of a certain category of representations of U q .sl n /. That is, if one defines a category Rep.sl n / to be the category of tensor products of exterior powers of the standard representation of sl n , then by interpreting certain trivalent graphs to be maps in Rep.sl n /, one can actually describe every morphism in Rep.sl n / by MOY diagrams.…”

“…Defining the category Rep to be the category with objects given by tensor products of exterior powers of the standard representation, and morphisms given by morphisms of U q .gl.1j1//-modules, we establish a diagram calculus for Rep similar to the one above. The diagrams take the form of ladders, introduced by Cautis, Kamnitzer and Morrison [4], which are very similar to MOY diagrams but slightly more 'rigid' in the sense that two ladders differing only by an isotopy are not always identified. Forming a certain category Lad Theorem There is an equivalence of categories…”