Spin networks and SL(2,C)-Character varieties

Details Spin networks and SL(2,C)-Character varieties Spin networks and SL(2,C)-Character varieties

2005-11-10

arxiv.org/abs/math/0511271v3

Handbook of Teichmuller theory. Volume II. European Mathematical Society (EMS). IRMA Lectures in Mathematics and Theoretical P

Mathematics

Quantum Algebra

Updated historical treatment of the subject. Figures drawn with PGF/TikZ; Handbook of Teichmuller Theory II, A. Papadopoulos (

Scientific paper

Denote the free group on 2 letters by F_2 and the SL(2,C)-representation variety of F_2 by R=Hom(F_2,SL(2,C)). The group SL(2,C) acts on R by conjugation. We construct an isomorphism between the coordinate ring C[SL(2,C)] and the ring of matrix coefficients, providing an additive basis of C[R]^SL(2,C) in terms of spin networks. Using a graphical calculus, we determine the symmetries and multiplicative structure of this basis. This gives a canonical description of the regular functions on the SL(2,C)-character variety of F_2 and a new proof of a classical result of Fricke, Klein, and Vogt.

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