A Diagrammatic Category for the Representation Theory of U_q(sl_n)

TL;DR

This thesis develops a diagrammatic framework to describe the representation category of quantum groups U_q(sl_n), providing generators for the kernel of a functor to understand the category's structure, advancing the categorification of quantum group representations.

Contribution

It introduces a tensor category of trivalent graphs and constructs a functor to the quantum group representations, identifying generators of the kernel and making progress towards a complete description.

Findings

Constructed a diagrammatic category for U_q(sl_n) representations

Identified certain generators of the kernel of the functor

Established inductive methods for kernel analysis

Abstract

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.