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On the representation theory of braid groups. (English. French summary) Zbl 1303.20047
This 67 pages long paper develops a technique to construct representations in a more systematic way and to study them. They are representations of certain groups related with the Artin braid groups \(B_n\) (the \(n\)-th braid group of the disk). As main application the author recovers several well known representations of \(B_n\) as well representations which factor through certain quotients of \(B_n\), among others the quotient of \(B_n\) by the commutator subgroup \([P_n,P_n]\) where \(P_n\) denotes the pure braid group on \(n\) strings (but not all). He also finds new representations. Many of the details are too specific and technical facts about representations to be stated here.
The main stream of the paper uses the fundamental article by V. G. Drinfel’d [Algebra Anal. 2, No. 4, 149-181 (1990; Zbl 0718.16034); translation in Leningr. Math. J. 2, No. 4, 829-860 (1991)] to obtain the set \(\text{Assoc}_\lambda(\mathbb K)\) of all formal series \(\Phi\) in two non-commuting variables \(A\) and \(B\), which satisfy certain relations. For any \(\Phi\in\text{Assoc}_\lambda(\mathbb K)\) by Drinfeld there exists a homomorphism from \(\mathbb KB_n\) to the completion \(\widehat{\mathcal B}_n\) of \(\mathcal B_n\). Here \(B_n\) denotes Artin’s braid group, \(\mathcal B_n\) the semi-direct product \(\mathbb Q\mathcal C_n\ltimes\cup\mathcal T_n\) and \(\mathcal T_n\) the Lie algebra of infinitesimal pure braids on \(n\) strands.
Once we have a representation \(\rho\colon\mathcal B_n\to M_N(\mathbb K)\) of \(\mathcal B_n\) over \(\mathbb K\) we may extend \(\rho\) to a representation \(\rho''\colon\widehat{\mathcal B}_n\to\mathbb K[[h]]\), where \(\mathbb K[[h]]\) is the completion of the polynomial ring \(\mathbb K[h]\) in one variable \(h\). After composing \(\rho''\) with the homomorphism associated with \(\Phi\) and taking the restriction to \(B_n\), we obtain a representation of \(B_n\) denoted by \(\widehat\Phi(\rho)\). The paper extensively studies this correspondence \(\rho\to\widehat\Phi(\rho)\). More specifically which properties of \(\rho\) are preserved under this correspondence like linearly independence, Tannakian, generically Tannakian, etc., whatever they mean. This general approach is used to recover several of the main classical representations. Notoriously the Iwahori-Hecke and Birman-Wenzl-Murakami algebras, Yang-Baxter representation, etc. The following sentence from the paper describes in a precise way the goal of the article: “We show how the representations of the braid groups which appeared in the last decades can be dealt with in our terms, and how our approach – in particular our uniform approach to unitary representations – may simplify or improve either the constructions or the proof of their main properties.”
Some considerations about convergence of the elements \(\Phi\) are made and they appear in the appendix A. The paper is well organized and requires a good background in representations from the reader.

MSC:
20F36 Braid groups; Artin groups
20C15 Ordinary representations and characters
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References:
[1] Artin, E., Theorie der Zöpfe, Abhandlungen Hamburg, 4, 47-72, (1925) · JFM 51.0450.01
[2] Artin, M., On the solutions of analytic equations, Invent. Math., 5, 277-291, (1968) · Zbl 0172.05301
[3] Bellaïche, J., A propos d’un lemme de ribet, Rend. Semin. Mat. Univ. Padova, 109, 45-62, (2003) · Zbl 1048.20032
[4] Bellaïche, J.; Graftieaux, P., Représentations sur un anneau de valuation discrète complet, Math. Ann., 334, 465-488, (2006) · Zbl 1178.20040
[5] Birman, J., Braids, links, and mapping class groups, Annals of Mathematics Studies 82, (1975), Princeton University Press and University of Tokyo Press, Princeton, Tokyo · Zbl 0305.57013
[6] Broué, M.; Malle, G., Zyklotomische heckealgebren, Astérisque, 212, 119-189, (1993) · Zbl 0835.20064
[7] Broué, M.; Malle, G.; Rouquier, R., Complex reflection groups, braid groups, Hecke algebras, J. Reine Angew. Math., 500, 127-190, (1998) · Zbl 0921.20046
[8] Chevalley, C., Théorie des groupes de Lie, vol. 2-3, (1961), Hermann, Paris · Zbl 0186.33104
[9] Cohen, F. R.; Prassidis, S., On injective homomorphisms for pure braid groups, and associated Lie algebras, J. Algebra, 298, 363-370, (2006) · Zbl 1134.20039
[10] Collins, D. J., Relations among the squares of the generators of the braid group, Invent. Math., 117, 525-529, (1994) · Zbl 0822.20040
[11] Crawley-Boevey, W., On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero, Duke Math. J., 118, 339-352, (2003) · Zbl 1046.15013
[12] Drinfeld, V. G., On quasitriangular quasi-Hopf algebras and a group closely connected with \({\rm{{G}}al}(\overline{\mathbb{Q}}/{\mathbb{Q}}),\) Leningrad Math. J., 2, 829-860, (1991) · Zbl 0728.16021
[13] Eisenbud, D., Commutative algebra with a view toward algebraic geometry, (1995), Springer-Verlag, Heidelberg · Zbl 0819.13001
[14] Formanek, E., Braid group representations of low degree, Proc. London Math. Soc., 73, 279-322, (1996) · Zbl 0855.20038
[15] Kohno, T., Série de Poincaré-Koszul associée aux groupes de tresses pures, Inven. Math., 82, 57-76, (195) · Zbl 0574.55009
[16] Kohno, T., Linear representations of braid groups and classical Yang-Baxter equations, Contemp. Math., 78, 339-363, (1988) · Zbl 0661.20026
[17] Le, T. Q.T.; Murakami, J., The universal Vassiliev-kontsevitch invariant for framed oriented links, Compositio Math., 102, 41-64, (1996) · Zbl 0851.57007
[18] Long, D. D., Constructing representations of the braid groups, Comm. in analysis and geometry, 2, 217-238, (1994) · Zbl 0845.20028
[19] Marin, I., On KZ-systems which are irreducible under the action of the symmetric group, C. R. Acad. Sci. Paris Sér. I, 333, 517-522, (2001) · Zbl 1058.32008
[20] Marin, I., Représentations linéaires des tresses infinitésimales, (2001)
[21] Marin, I., Caractérisations de la représentation de Burau, Expo. Math., 21, 263-278, (2003) · Zbl 1064.20042
[22] Marin, I., Infinitesimal Hecke algebras, Comptes Rendus Mathématiques Série I, 337, 297-302, (2003) · Zbl 1056.20004
[23] Marin, I., On the representation theory of braid groups, Preprint Université d’Evry, (2003)
[24] Marin, I., Quotients infinitésimaux du groupe de tresses, Ann. Inst. Fourier (Grenoble), 53, 1323-1364, (2003) · Zbl 1063.20042
[25] Marin, I., Irréductibilité générique des produits tensoriels de monodromies, Bull. Soc. Math. Fr., 132, 201-232, (2004) · Zbl 1073.20003
[26] Marin, I., Caractères de rigidité du groupe de Grothendieck-Teichmüller, Compositio Math., 142, 657-678, (2006) · Zbl 1133.14027
[27] Marin, I., Monodromie algébrique des groupes d’artin diédraux, J. Algebra, 303, 97-132, (2006) · Zbl 1151.20003
[28] Marin, I., L’algèbre de Lie des transpositions, J. Algebra, 310, 742-774, (2007) · Zbl 1171.20010
[29] Marin, I., Sur LES représentations de krammer génériques, Ann. Inst. Fourier (Grenoble), 57, 1883-1925, (2007) · Zbl 1183.20036
[30] Marin, I., Braids inside the Birman-wenzl-murakami algebra, Algebraic Geometric Topology, 10, 1865-1886, (2010) · Zbl 1235.20037
[31] Marin, I., The cubic Hecke algebra on at most 5 strands, J. Pure Applied Algebra, 216, 2754-2782, (2012) · Zbl 1266.20006
[32] Ribet, K., A modular construction of unramified extensions of \({\mathbb{Q}}(μ _p),\) Invent. Math., 34, 151-162, (1976) · Zbl 0338.12003
[33] Shephard, G. C.; Todd, J. A., Finite unitary reflection groups, Canad. J. Math., 6, 274-304, (1954) · Zbl 0055.14305
[34] Squier, C. C., The Burau representation is unitary, Proc. Am. Math. Soc., 90, 199-202, (1984) · Zbl 0542.20022
[35] Sysoeva, I., On irreducible representations of braid groups, (1999)
[36] Sysoeva, I., Dimension \(n\) representations of the braid groups on \(n\) strings, J. Algebra, 243, 518-538, (2001) · Zbl 1008.20029
[37] Tits, J., Normalisateurs de tores I : groupes de Coxeter étendus, J. Algebra, 4, 96-116, (1966) · Zbl 0145.24703
[38] Vassiliev, V. A., Complements of discriminants of smooth maps : topology and applications, Translation of mathematical monographs, 98, (1992) · Zbl 0762.55001
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