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ISRN Algebra
Volume 2012 (2012), Article ID 858959, 13 pages
doi:10.5402/2012/858959
Another Proof of the Faithfulness of the Lawrence-Krammer Representation of the Braid Group
Department of Mathematics, Beirut Arab University, P.O. Box 11-5020, Beirut 11072809, Lebanon
Received 16 March 2012; Accepted 6 May 2012
Academic Editors: P. Koshlukov, H. Li, S. Yang, and Y. Zhou
Copyright © 2012 Mohammad N. Abdulrahim and Mariam Hariri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The Lawrence-Krammer representation of the braid group was proved to be faithful for by Bigelow and Krammer. In our paper, we give a new proof in the case by using matrix computations. First, we prove that the representation of the braid group is unitary relative to a positive definite Hermitian form. Then we show the faithfulness of the representation by specializing the indeterminates q and t to complex numbers on the unit circle rather than specializing them to real numbers as what was done by Krammer.