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

Abstract

We describe a class of irreducible non-equivalent unitary representations of the braid group B3 in every dimension n≥6 which depends continuously on n2/6+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 B3 depends smoothly is equal to n2/4+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 B4 and the irreducible unitary representation of B4 that coincide on commuting standard generators always forms irreducible unitary representations for the braid group B4. This gives a new class of unitary representations for the braid group B4 in 3n dimensions.