Document Type: Research Paper

Authors

1 Department of Mathematics and Computer Science, ‎Beirut Arab University‎, ‎PO‎. ‎Box 11-5020‎, ‎Beirut‎, ‎Lebanon

2 Department of Mathematics and Computer Science‎, ‎Beirut Arab University‎, ‎PO‎. ‎Box 11-5020‎, ‎Beirut‎, ‎Lebanon

Abstract

‎We consider Albeverio's linear representations of the braid groups $B_3$ and $B_4$‎. ‎We specialize the indeterminates used in defining these representations to non zero complex numbers‎. ‎We then consider the tensor products of the representations of $B_3$ and the tensor products of those of $B_4$‎. ‎We then determine necessary and sufficient conditions that guarantee the irreducibility of the tensor products of the representations of $B_3$‎. ‎As for the tensor products of the representations of $B_4$‎, ‎we only find sufficient conditions for the irreducibility of the tensor product‎.
 

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[1] S. Albeverio, S. Rabanovich, On a class of unitary representations of the braid groups B3and B4, Bull. Sci. Math. 153 (2019), 35-56.

[2] S. J. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001), 471-486.

[3] J. S. Birman, Braids, Links, and Mapping Class Groups, Annals of Mathematical Studies, Princeton University Press, Newjersy, 1975.

[4] V. L. Hansen, Braids and Coverings, London Mathematical Society Student Texts, 1989.

[5] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Annals of Math. 126 (1987), 335-366.

[6] D. Krammer, Braid groups are linear, Ann. of Math. 155 (2002), 131-156.

[7] R. Lawrence, Homological representations of the Hecke algebra, Comm. Math. Physics, 135 (1990), 141-191.

[8] C. I. Levaillant, D. B. Wales, Parameters for which the Lawrence-Krammer representation is reducible, J. Algebra. 323 (2010), 1966-1982