Representations of the braid group B_3 and of SL(2,Z)

We give a complete classification of simple representations of the braid group B_3 with dimension ≤ 5 over any algebraically closed f ield. In particular, we prove that a simple d-dimensional representation ρ: B₃ → GL(V) is determined up to isomorphism by the eigenvalues λ₁, λ₂, ..., λd of the image of the generators for d=2,3 and a choice of a δ=√ ρ(σ₁) for d=4 or a choice of δ=√[5] ρ(σ₁) for d=5. We also s howed that such representations exist whenever the eigenvalues and $δ$ are not roots of certain polynomials Qᵢⱼ⁽ᵈ⁾, which are explicitly given. In this case, we construct the matrices via which the generators act on V. As an application of our techniques, we also obtain nontrivial q-versions of some of Deligne's formulas for dimensions of representations of exceptional Lie groups.