A Diagrammatic Category for the Representation Theory of Uq (slₙ)

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 Uq (slₙ) (thought of as a spherical category) via generators and relations? " For each n 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 Uq (slₙ). 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 Uq (slₙ). 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 Uq (sl₍-₁) in Uq (slₙ}. 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.