A Diagrammatic Category for the Representation Theory of U_q(sl_n)
| dc.creator | Morrison, Scott | |
| dc.date | 2007-04-12 | |
| dc.date.accessioned | 2026-07-07T07:56:18Z | |
| dc.date.available | 2026-07-07T07:56:18Z | |
| dc.description | 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. | |
| dc.description | 91 pages, UC Berkeley Ph.D. thesis | |
| dc.identifier | https://arxiv.org/abs/0704.1503 | |
| dc.identifier | http://arxiv.org/abs/0704.1503 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/127259 | |
| dc.subject | Quantum Algebra | |
| dc.subject | Representation Theory | |
| dc.title | A Diagrammatic Category for the Representation Theory of U_q(sl_n) | |
| dc.type | text |