Representations of the Braid Group B3 and of Sl(2,z)

We give a complete classification of all simple representations of B3 for dimension d ≤ 5 over an algebraically closed field K of any characteristic. To describe our result in detail, recall that B3 is given by generators σ1 and σ2 which satisfy the relation σ1σ2σ1 = σ2σ1σ2. Moreover, it is well-known that the center of B3 is generated by ζ = (σ1σ2). It is easy to see that ζ acts on a simple d-dimensional B3module via the scalar δ which satisfies the equation δd = det(A)6, where A is the linear endomorphism via which σ1 acts on V . Our main result states that the eigenvalues of A and the scalar δ completely determine a simple representation of B3 for dimension ≤ 5, up to equivalence; for d ≤ 3 it is uniquely determined by the eigenvalues of A. Moreover, such simple representations exist if and only if the eigenvalues do not belong to the zero set of certain polynomials in the eigenvalues and δ which we list explicitly (see Proposition 2.8, Section 2.10 and 2.11, Remark 4). One of the motivations for this paper was studying braided tensor categories by analysing braid representations. The categories under consideration have a Grothendieck semiring isomorphic to the one of a semisimple Lie group. It turns out that in this context it suffices to classify representations of B3 up to dimension 5. Indeed, as an application, we obtain uniform formulas for the categorical dimensions of objects in the second tensor power of the adjoint representation in braided tensor categories closely related to the conjectured series of exceptional Lie algebras, as proposed by Deligne and Vogel. Our approach in this paper is quite elementary: We first show that assuming a certain triangular form of the matrices A and B of the generators of B3, the braid relation reduces to checking the values of certain coefficients of the matrix BA. We then show that for dimension d ≤ 5 one can