1999/12/02 by Imre Tuba, Tuba, Imre, Hans Wenzl +1 · 4 citations
Mathematics · #15A69 (Secondary) #16S34 #20C07 #20F36 #81R10 (Primary) #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.GR #math.QA #math.RA #math.RT #msc:15A69 #msc:16S34 #msc:20C07 #msc:20F36 #msc:81R10
paper · pdf · doi:10.48550/arxiv.math/9912013
To appear in the Pacific Journal of Mathematics
arxiv created 1999/12/02 · openalex publication_date 1999/12/02 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a complete classification of simple representations of the braid group B3 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=√(det ρ(σ1)) for d=4 or a choice of δ=√[5]det 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.