Another Proof of the Faithfulness of the Lawrence-Krammer Representation of the Braid Group B3. (25th June 2012)

• Record Type: Journal Article
• Authors: Abdulrahim, Mohammad N.; Hariri, Mariam
• Other Names: Koshlukov P. Academic Editor.; Li H. Academic Editor.; Yang S. Academic Editor.; Zhou Y. Academic Editor.
• Is Part Of: ISRN algebra. Volume 2012(2012)
• Publication Date: 2012-06-25
• Subjects: Algebra -- Periodicals; Algebra; Periodicals; Electronic journals; 512
• DOI: 10.5402/2012/858959
• Languages: English
• ISSNs: 2090-6285

Abstract

Abstract : The Lawrence-Krammer representation of the braid group B n was proved to be faithful for n ≥ 3 by Bigelow and Krammer. In our paper, we give a new proof in the case n = 3 by using matrix computations. First, we prove that the representation of the braid group B 3 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.