Graphical Calculus on Representations of Quantum Lie Algebras
Abstract
We develop graphical calculation methods. Jones-Wenzl projectors for Uq(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 Uq(sl(2,C)) has significant meaning and it is called 3j symbols. Using single clasp expansions for Uq(sl(3,C)), we find some trihedron coefficients of the representation theory of Uq(sl(3,C)). We study representation theory for Uq(sl(4,C)). We conjecture a complete set of relations for Uq(sl(4,C)).
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