Low - Dimensional Unitary Representations Of

We characterize all simple unitarizable representations of the braid group B 3 on complex vector spaces of dimension d ≤ 5. In particular, we prove that if σ 1 and σ 2 denote the two generating twists of B 3 , then a simple representation ρ : B 3 → GL(V) (for dim V ≤ 5) is unitarizable if and only if the eigenvalues λ 1 , λ 2 ,. .. , λ d of ρ(σ 1) are distinct, satisfy |λ i | = 1 and µ (d) 1i > 0 for 2 ≤ i ≤ d, where the µ (d) 1i are functions of the eigenvalues, explicitly described in this paper. Unitary braid representations have been constructed in several ways using the representation theory of Kac-Moody algebras and quantum groups, see e.g. [1], [2], and [4]. Such representations easily lead to representations of PSL(2, Z) = B 3 /Z, where Z is the center of B 3 , and PSL(2, Z) = SL(2, Z)/{±1}, where {±1} is the center of SL(2, Z). We give a complete classification of simple unitary representations of B 3 of dimension d ≤ 5 in this paper. In particular, the unitarizability of a braid representation depends only on the the eigenvalues λ 1 , λ 2 ,. .. , λ d of the images the two generating twists of B 3. The condition for unitarizability is a set of linear inequalities in the logarithms of these eigenvalues. In other words, the representation is unitarizable if and only if the (ln λ 1 , ln λ 2 ,. .. , ln λ d) is a point inside a polyhedron in (R/2π) d , where we give the equations of the hyperplanes that bound this polyhedron. This classification shows that the approaches mentioned previously do not produce all possible unitary braid representations. We obtain representations that seem to be new for d ≥ 3. As any unitary representation of B n restricts to a unitary representation of B 3 in an obvious way, these results may also be useful in classifying such representation of B n. I would like to thank my advisor, Hans Wenzl for the ideas he contributed to this paper. Let B 3 be generated by σ 1 and σ 2 with the relation σ 1 σ 2 σ 1 = σ 2 σ 1 σ 2. It is well-known that the center …