The braid group surjects onto $G_2$ tensor space

Details The braid group surjects onto $G_2$ tensor space The braid group surjects onto $G_2$ tensor space

2009-07-01

arxiv.org/abs/0907.0256v1

Pacific Journal of Mathematics 249-1 (2011), 189--198

Mathematics

Quantum Algebra

9 pages

Scientific paper

10.2140/pjm.2011.249.189

Let V be the 7-dimensional irreducible representation of the quantum group U_q(g_2). For each n, there is a map from the braid group B_n to the endomorphism algebra of the n-th tensor power of V, given by R-matrices. We can extend this linearly to a map on the braid group algebra. Lehrer and Zhang (MR2271576) prove this map is surjective, as a special case of a more general result. Using Kuperberg's spider for G_2 from arXiv:math.QA/9201302, we give an elementary diagrammatic proof of this result.

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