Representations of the braid group B_3 and of SL(2,Z)
| dc.creator | Tuba, Imre | |
| dc.creator | Wenzl, Hans | |
| dc.date | 1999-12-02 | |
| dc.date.accessioned | 2026-07-07T05:32:05Z | |
| dc.date.available | 2026-07-07T05:32:05Z | |
| dc.description | We give a complete classification of simple representations of the braid group B_3 with dimension $\leq 5$ over any algebraically closed f ield. In particular, we prove that a simple d-dimensional representation $ρ: B_3 \to GL(V)$ is determined up to isomorphism by the eigenvalues $λ_1, λ_2, ..., λ_d$ of the image of the generators for d=2,3 and a choice of a $δ=\sqrt{\det ρ(σ_1)}$ for d=4 or a choice of $δ=\sqrt[5]{\det ρ(σ_1)}$ for d=5. We also s howed that such representations exist whenever the eigenvalues and $δ$ are not roots of certain polynomials $Q_{ij}^{(d)}$, 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. | |
| dc.description | To appear in the Pacific Journal of Mathematics | |
| dc.identifier | https://arxiv.org/abs/math/9912013 | |
| dc.identifier | http://arxiv.org/abs/math/9912013 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/79529 | |
| dc.subject | Representation Theory | |
| dc.subject | Group Theory | |
| dc.subject | Quantum Algebra | |
| dc.subject | Rings and Algebras | |
| dc.subject | 20F36, 20C07, 81R10 (Primary); 16S34, 15A69 (Secondary) | |
| dc.title | Representations of the braid group B_3 and of SL(2,Z) | |
| dc.type | text |