Irreducibility of the tensor product of Albeverio's representations of the Braid groups $B_3$ and $B_4$
الموضوعات :A. Taha 1 , M. N. Abdulrahim 2
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
الکلمات المفتاحية: Braid group, irreducible,
ملخص المقالة :
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.
[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
