Unitary braid representations with finite image

Larsen, Michael¹ ; Rowell, Eric²

¹Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
²Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA

Algebraic and Geometric Topology, Tome 8 (2008) no. 4, pp. 2063-2079

Résumé

We characterize unitary representations of braid groups Bn of degree linear in n and finite images of such representations of degree exponential in n.

Détail

DOI : 10.2140/agt.2008.8.2063

Keywords: braid group, unitary representations

Bibliographie

[1] B Bakalov, A Kirillov Jr., Lectures on tensor categories and modular functors, University Lecture Series 21, Amer. Math. Soc. (2001)

[2] J S Birman, B Wajnryb, Markov classes in certain finite quotients of Artin's braid group, Israel J. Math. 56 (1986) 160

[3] A Borel, J Tits, Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I, Invent. Math. 12 (1971) 95

[4] P Etingof, D Nikshych, V Ostrik, On fusion categories, Ann. of Math. $(2)$ 162 (2005) 581

[5] P Etingof, E Rowell, S Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math. 234 (2008) 33

[6] E Formanek, Braid group representations of low degree, Proc. London Math. Soc. $(3)$ 73 (1996) 279

[7] E Formanek, W Lee, I Sysoeva, M Vazirani, The irreducible complex representations of the braid group on $n$ strings of degree $\le n$, J. Algebra Appl. 2 (2003) 317

[8] M H Freedman, M J Larsen, Z Wang, The two-eigenvalue problem and density of Jones representation of braid groups, Comm. Math. Phys. 228 (2002) 177

[9] D M Goldschmidt, V F R Jones, Metaplectic link invariants, Geom. Dedicata 31 (1989) 165

[10] V F R Jones, Braid groups, Hecke algebras and type $\mathrm{II}_1$ factors, from: "Geometric methods in operator algebras (Kyoto, 1983)", Pitman Res. Notes Math. Ser. 123, Longman Sci. Tech. (1986) 242

[11] V F R Jones, On a certain value of the Kauffman polynomial, Comm. Math. Phys. 125 (1989) 459

[12] V Landazuri, G M Seitz, On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32 (1974) 418

[13] E C Rowell, From quantum groups to unitary modular tensor categories, from: "Representations of algebraic groups, quantum groups, and Lie algebras", Contemp. Math. 413, Amer. Math. Soc. (2006) 215

[14] G M Seitz, The maximal subgroups of classical algebraic groups, Mem. Amer. Math. Soc. 67 (1987)

[15] G M Seitz, Maximal subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 90 (1991)

[16] R Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. 80, Amer. Math. Soc. (1968) 108

[17] I Sysoeva, Dimension $n$ representations of the braid group on $n$ strings, J. Algebra 243 (2001) 518

[18] V G Turaev, Quantum invariants of knots and $3$–manifolds, de Gruyter Studies in Math. 18, Walter de Gruyter Co. (1994)

[19] B Wajnryb, A braidlike presentation of $\mathrm{Sp}(n,p)$, Israel J. Math. 76 (1991) 265