On the Representation Theory of an Algebra of Braids and Ties

We consider the algebra ${\mathcal{E}}_n(u)$ introduced by F. Aicardi and J. Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor space representation for ${\mathcal{E}}_n(u)$ and show that this is faithful. We use it to give a basis for ${\mathcal{E}}_n(u)$ and to classify its irreducible representations.