本文部分回答了Greg Kuperberg在q-alg/9712003中提出的一个问题，以及Justin Roberts在"Problems on invariants of knots and 3-manifold"，math. GT/0406190中提出的问题12.18："能否通过生成元和关系描述量子群U_q（sl_n）的表示范畴？"对于每n个\geq 0，我定义一个三价图的张量范畴，模合痕，并从这个范畴构造一个函子到量子群U_q（sl_n）的表示范畴（的一个满子范畴）。人们希望通过提供生成器来完整地描述这个函子的内核。由此得到的图解范畴的商则是等价于U_q（sl_n）的表示范畴的范畴。我在这方面取得了重大进展，描述了内核的某些生成器，以及对进一步元素的一些障碍。这仍然是一个猜想，这些关系产生的内核。本文的结果推广了q-alg/9712003、MR1659228、math. QA/0310143和math. GT/0506403的结果。这个论证实质上是通过构造一个由U_q（sl_{n-1}）包含在U_q（sl_n}）中而得到的健忘函子的图解形式。我们知道这个函子是忠实的，所以一个图在n的核中，如果它在图遗忘函子下的像在n-1的核中。这允许我们执行归纳计算，既建立内核的元素族，又找到障碍物。
英文摘要：
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.
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