On a class of unitary representations of the braid groups B3 and B4

Abstract

Abstract We describe a class of irreducible non-equivalent unitary representations of the braid group B 3 in every dimension n ≥ 6 which depends continuously on n 2 / 6 + 1 real parameters. We show that the upper bound on the number of the parameters of which the class of irreducible non-equivalent unitary representations of B 3 depends smoothly is equal to n 2 / 4 + 2 . The proof is achieved by a construction of such a class. We also prove that the tensor product of the Burau unitarisable representation of B 4 and the irreducible unitary representation of B 4 that coincide on commuting standard generators always forms irreducible unitary representations for the braid group B 4 . This gives a new class of unitary representations for the braid group B 4 in 3n dimensions.