A Diagrammatic Category for

A Diagrammatic Category for the Representation Theory of U q (sl n) This thesis provides a partial answer to a question posed by Greg Kuperberg in [24] and again by Justin Roberts as problem 12.18 in Problems on invariants of knots and 3-manifolds [28], 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 ≥ 0, I define a certain tensor category of trivalent graphs, modulo iso-topy, 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 for the tensor category ideal. 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 elements of the kernel , and some obstructions to further elements. It remains a conjecture that these elements really generate the kernel. The argument is essentially the following. Take some trivalent graph in the diagrammatic category for some value of n, and consider the morphism of U q (sl n) representations it is sent too. Forgetting the full action of U q (sl n), keeping only a U q (sl n−1) action, the source and target representations branch into direct sums, and the morphism becomes a matrix of maps of U q (sl n−1) representations. Arguing inductively now, we attempt to write each such matrix entry as a linear combination of diagrams for n − 1. This gives a functor dGT between diagrammatic categories, realising the forgetful functor at the representation theory level. Now, if a certain linear combination of diagrams for n is to be in the kernel of the representation functor, each matrix entry of dGT applied to that linear combination must already be in the kernel of the representation functor one level down. This allows us to perform inductive calculations, both establishing families of elements of the kernel, and finding obstructions to other linear combinations being in the kernel.