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Braids inside the Birman-Wenzl-Murakami algebra. (English) Zbl 1235.20037
Summary: We determine the Zariski closure of the representations of the braid groups that factor through the Birman-Wenzl-Murakami algebra, for generic values of the parameters $$\alpha,s$$. For $$\alpha,s$$ of modulus 1 and close to 1, we prove that these representations are unitarizable, thus deducing the topological closure of the image when in addition $$\alpha,s$$ are algebraically independent.

MSC:
 20F36 Braid groups; Artin groups 20C08 Hecke algebras and their representations 20C15 Ordinary representations and characters 57M07 Topological methods in group theory
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References:
 [1] M Artin, On the solutions of analytic equations, Invent. Math. 5 (1968) 277 · Zbl 0172.05301 [2] S J Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001) 471 · Zbl 0988.20021 [3] J S Birman, H Wenzl, Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. 313 (1989) 249 · Zbl 0684.57004 [4] V G Drinfel’d, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $$\mathrm{Gal}(\overline\mathbfQ/\mathbfQ)$$, Algebra i Analiz 2 (1990) 149 · Zbl 0718.16034 [5] D Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Math. 150, Springer (1995) · Zbl 0819.13001 [6] 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 · Zbl 1045.20027 [7] R Goodman, N R Wallach, Representations and invariants of the classical groups, Encycl. of Math. and its Appl. 68, Cambridge Univ. Press (1998) · Zbl 0901.22001 [8] D Krammer, The braid group $$B_4$$ is linear, Invent. Math. 142 (2000) 451 · Zbl 0988.20023 [9] D Krammer, Braid groups are linear, Ann. of Math. $$(2)$$ 155 (2002) 131 · Zbl 1020.20025 [10] I Marin, Infinitesimal Hecke algebras, C. R. Math. Acad. Sci. Paris 337 (2003) 297 · Zbl 1056.20004 [11] I Marin, Quotients infinitésimaux du groupe de tresses, Ann. Inst. Fourier (Grenoble) 53 (2003) 1323 · Zbl 1063.20042 [12] I Marin, Caractères de rigidité du groupe de Grothendieck-Teichmüller, Compos. Math. 142 (2006) 657 · Zbl 1133.14027 [13] I Marin, L’algèbre de Lie des transpositions, J. Algebra 310 (2007) 742 · Zbl 1171.20010 [14] I Marin, Sur les représentations de Krammer génériques, Ann. Inst. Fourier (Grenoble) 57 (2007) 1883 · Zbl 1183.20036 [15] M Nazarov, Young’s orthogonal form for Brauer’s centralizer algebra, J. Algebra 182 (1996) 664 · Zbl 0868.20012 [16] H Wenzl, Hecke algebras of type $$A_n$$ and subfactors, Invent. Math. 92 (1988) 349 · Zbl 0663.46055 [17] H Wenzl, On the structure of Brauer’s centralizer algebras, Ann. of Math. $$(2)$$ 128 (1988) 173 · Zbl 0656.20040 [18] H Wenzl, Quantum groups and subfactors of type $$B$$, $$C$$, and $$D$$, Comm. Math. Phys. 133 (1990) 383 · Zbl 0744.17021
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