Graphical Calculus on Representations of Quantum Lie Algebras

By Dongseok Kim. 2003

70 citations recorded by the source corpus.

We develop graphical calculation methods. Jones-Wenzl projectors for U_q(sl(2,C)) are very powerful tools to find not only invariants of links but also invariants of 3-manifolds. We find single clasp expansions of generalized Jones-Wenzl projectors for simple Lie algebras of rank 2. Trihedron coefficients of the representation theory for U_q(sl(2,C)) has significant meaning and it is called 3j symbols. Using single clasp expansions for U_q(sl(3,C)), we find some trihedron coefficients of the representation theory of U_q(sl(3,C)). We study representation theory for U_q(sl(4,C)). We conjecture a complete set of relations for U_q(sl(4,C)).

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1b2a8f5581cc765dd25d41ff95f454eaccb73d12

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Citations in the corpus

Light ladders and clasp conjectures (2015, background): “As mentioned before, Kim [18] has computed some recursion formulas for clasps in rank 2, such as a single and a double clasp expansion in type A2 and B2, and a quadruple clasp expansion in type A2.”
Webs and quantum skew Howe duality (2012, methodology): “(For the simplest Kekulé relation, originally found by Kim [Kim03], we do have such a derivation.)”
Skein theory for the D_2pi planar algebras (2008, background): “Some progress has been made on the representation categories of Uq(sln) [24, 16, 22].”
Buildings, spiders, and geometric Satake (2011, background): “Kim conjectured for n= 4 (see [Kim03]), and Morrison for general n (see [Mor07]), a set of generating relations for the kernel of fsp(SL(n))→ rep(SL(n))min.”
Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, result): “These descriptions have recently been generalized by Cautis, Kamnitzer, and Morrison [18] to general n, building on earlier work of Kim [42] and Morrison [57].”
A diagrammatic category for the representation theory of ( Uqsln ) (2007, methodology): “Similarly, Kim’s proposed spider for Uq(sl4) does not specify orientations on the ‘thick’ edges, that is, those edges labelled by 2.”
Promotion and cyclic sieving via webs (2008, unknown): “f�(q) = q−�(�) [N]q! �(i,j)∈�[hij]q , 20 T. K . PETERSEN, P. PYLYAVSKYY, AND B. RHOADES”