A diagrammatic category for the representation theory of ( Uqsln )

By S. Morrison. 2007

72 citations recorded by the source corpus.

This thesis provides a partial answer to a question posed by Greg Kuperberg in q-alg/9712003 and again by Justin Roberts as problem 12.18 in "Problems on invariants of knots and 3-manifolds", math.GT/0406190, essentially: "Can one describe the category of representations of the quantum group U_q(sl_n) (thought of as a spherical category) via generators and relations?" For each n \geq 0, I define a certain tensor category of trivalent graphs, modulo isotopy, and construct a functor from this category onto (a full subcategory of) the category of representations of the quantum group U_q(sl_n). One would like to describe completely the kernel of this functor, by providing generators. The resulting quotient of the diagrammatic category would then be a category equivalent to the representation category of U_q(sl_n). I make significant progress towards this, describing certain generators of the kernel, and some obstructions to further elements. It remains a conjecture that these relations generate the kernel. My results extend those of q-alg/9712003, MR1659228, math.QA/0310143 and math.GT/0506403. The argument is essentially by constructing a diagrammatic version of the forgetful functor coming from the inclusion of U_q(sl_{n-1}) in U_q(sl_n}. We know this functor is faithful, so a diagram is in the kernel for n exactly if its image under the diagrammatic forgetful functor is in the kernel for n-1. This allows us to perform inductive calculations, both establishing families of elements of the kernel, and finding obstructions.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b4c3d1a0c241523d3ed0f9af850848c11290b1ca

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In Mathematical Physics

Citations in the corpus

Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, result): “Clark-Morrison-Walker sl3 foams.”
Light ladders and clasp conjectures (2015, background): “Morrison [23] attempted to describe webs for sln in a similar fashion, giving generators and a correct list of relations (together with some redundant ones), but he was unable to prove at that time that the list of relations was sufficient.”
Skein theory for the D_2pi planar algebras (2008, background): “Some progress has been made on the representation categories of Uq(sln) for n ≥ 4 [29, 20, 27].”
sl.N/-link homology (N 4) using foams and the Kapustin-Li formula (2007, background): “There are more general web moves [13], which allow for the evaluation of arbitrary webs, but we do not need them here.”
Quantum picturalism (2009, methodology): “Related graphical methods have been around for a bit more than a decade now in mathematical physics and pure mathematics, for example in [24, 71, 72, 73, 74] and references therein.”