Vol. 249, No. 1, 2011

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 297: 1
Vol. 296: 1  2
Vol. 295: 1  2
Vol. 294: 1  2
Vol. 293: 1  2
Vol. 292: 1  2
Vol. 291: 1  2
Vol. 290: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
Other MSP Journals
The braid group surjects onto G2 tensor space

Scott Morrison

Vol. 249 (2011), No. 1, 189–198
Abstract

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.

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

Mathematica notebook

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