Vol. 249, No. 1, 2011

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The braid group surjects onto G2 tensor space

Scott Morrison

Vol. 249 (2011), No. 1, 189–198

Let V be the 7-dimensional irreducible representation of the quantum group Uq(g2). For each n, there is a map from the braid group n to the endomorphism algebra of the n-th tensor power of V , given by matrices. Extending linearly to the braid group algebra, we get a map

𝒜 ℬn → EndUq(g2)(V⊗n ).

Lehrer and Zhang have proved that map is surjective, as a special case of a more general result.

Using Kuperberg’s spider for G2, we give an elementary diagrammatic proof of this result.

braid group, spider, G2, tensor category, representation theory
Mathematical Subject Classification 2000
Primary: 17B37
Secondary: 20G42, 17B10, 18D10
Supplementary material

Mathematica notebook

Received: 22 September 2009
Revised: 4 January 2010
Accepted: 4 January 2010
Published: 3 January 2011
Scott Morrison
Microsoft Station Q
CNSI Building
University of California
Santa Barbara, CA 93106-6105