Multi-parameter Burau representations

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Mohammad N. Abdulrahim
Madline Al- Tahan
Samer S. Habre

Abstract

We consider the multi-parameter representation of Artin's braid group introduced by D. D. Long and J. P. Tian, namely $\alpha: B_{n}\rightarrow GL_{m}(C)$, where $m=n!n$ . First, we show that there exists a complex specialization of the multi-parameter representation that does not arise from any Hecke algebra. Second, we find conditions under which the images of the generators of the braid group on three strings under the multi-parameter representation are unitary relative to a nonsingular hermitian matrix.

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How to Cite
Abdulrahim, M. N., Tahan, M. A.-., & Habre, S. S. (2013). Multi-parameter Burau representations. Tamkang Journal of Mathematics, 44(1), 91–98. https://doi.org/10.5556/j.tkjm.44.2013.1081
Section
Papers
Author Biographies

Mohammad N. Abdulrahim

Department ofMathematics, Beirut Arab University, P.O. Box 11-5020, Beirut, Lebanon. E-mail: mna@bau.edu.lb

Madline Al- Tahan

Department ofMathematics, Beirut Arab University, P.O. Box 11-5020, Beirut, Lebanon.

Samer S. Habre

Department of Computer Science and Mathematics, Lebanese American University, P.O. Box 13-5053 Chouran, Beirut, Lebanon.

References

S. Bigelow and J. P. Tian, Generalized Long- Moody representations of braid groups, Communications in Contemporary Math, 10(2008), (suppl.1), 1093--1102.

J. S. Birman, Braids, Links and Mapping Class Groups. Annals of Mathematical Studies. Princeton University Press, 82, New Jersey, 1975.

D.D. Long, On the linear representation of braid groups. Transaction of the American Mathematical Society, 311(1989), 535--560

V. Shpilrain, Representing Braids by Automorphisms, International Journal of Algebra and Computation, 11(2001), 773--777.

C. C. Squier, The Burau representation is unitary, Proceedings of the American Mathematical Society, 90(1984),199--202.

J. P. Tian, Spin representations of Artin's braid group, Preprint, 2007.

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