International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 806502, 10 pages
Research Article

Krammer's Representation of the Pure Braid Group, P3

Department of Mathematics, Beirut Arab University, P.O. Box 11-5020, Beirut 1107 2809, Lebanon

Received 24 December 2009; Accepted 25 May 2010

Academic Editor: Howard Bell

Copyright © 2010 Mohammad N. Abdulrahim and Madline Al-Tahan. 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.


We consider Krammer's representation of the pure braid group on three strings: P3GL(3,Z[t±1,q±1]), where t and q are indeterminates. As it was done in the case of the braid group, B3, we specialize the indeterminates t and q to nonzero complex numbers. Then we present our main theorem that gives us a necessary and sufficient condition that guarantees the irreducibility of the complex specialization of Krammer's representation of the pure braid group, P3.