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

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.

arXiv comment91 pages, UC Berkeley Ph.D. thesis

References in corpus (6)

• Spiders for rank 2 Lie algebras
• Spherical Categories
• Confluence Theory for Graphs
• On Khovanov's cobordism theory for su(3) knot homology
• Fast Khovanov Homology Computations
• The quantum $\mathfrak{sl}(n,\mathbb{C})$ representation theory and its applications

Cited by in corpus (20)

• Quantum Picturalism
• Skein theory for the D_{2n} planar algebras
• Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m)
• Categories for the practising physicist
• Light ladders and clasp conjectures
• sl(N)-Web categories
• A Categorification of Quantum sl_3 Projectors and the sl_3 Reshetikhin-Turaev Invariant of Tangles
• $U_q({\mathfrak{sl}}_n)$ web models and ${\mathbb{Z}}_n$ spin interfaces
• Skein Modules from Skew Howe Duality and Affine Extensions
• A web basis of invariant polynomials from noncrossing partitions
• A simple bijection between standard (n,n,n) tableaux and irreducible webs for sl_3
• Bases in diagrammatic quantum protocols
• Braided Subfactors, Spectral Measures, Planar algebras and Calabi-Yau algebras associated to SU(3) modular invariants
• On webs in quantum type $C$
• An explicit bijection between semistandard tableaux and non-elliptic sl_3 webs
• A comparison between $S{L}_n$ spider categories
• The Robinson-Schensted Correspondence and $A_2$-web Bases
• Recoupling theory for quantum spinors
• A classification of transitive links and periodic links
• The transition matrix between the Specht and ${\mathfrak{sl}}_3$ web bases is unitriangular with respect to shadow containment