Graphical Calculus on Representations of Quantum Lie Algebras

Abstract

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