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Infinitesimal quotients of the braid group. (Quotients infinitésimaux du groupe de tresses.) (French) Zbl 1063.20042
Summary: We define and study infinitesimal analogues of the main quotients of the group algebra of Artin’s groups, namely the Temperley-Lieb, Hecke and Birman-Wenzl-Murakami algebras, in terms of KZ-systems. These analogues are Hopf algebras which correspond to reductive groups: we give then a general framework for the study of representations deduced from the classical representations of $$B_n$$ through tensor constructions. We use this to analyse representations related to the Burau representation, and we fully decompose the infinitesimal Temperley-Lieb algebra. As a by-product, we obtain several irreducibility properties.

##### MSC:
 20F36 Braid groups; Artin groups 20C08 Hecke algebras and their representations 20C15 Ordinary representations and characters
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##### References:
 [1] A new approach to the word and conjugacy problems in the braid groups, Adv. Math, 139, 2, 322-353, (1998) · Zbl 0937.20016 [2] Groupes et algèbres de Lie, ch. 1, (1960), Hermann · Zbl 0199.35203 [3] On algebras which are connected with the semisimple continuous groups, Annals of Math (2), 38, 857-872, (1937) · JFM 63.0873.02 [4] Zyclotomische heckealgebren, Astérisque, 212, 7-92, (1993) [5] Hecke algebras and characters of parabolic type of finite groups with $$(B, N)$$-pairs, Inst. Hautes Études Sci. Publ. Math., 40, 81-116, (1971) · Zbl 0254.20004 [6] Le groupe fondamental de la droite projective moins trois points, Galois groups over $$\mathbb{Q}, 16, 79-287, (1989),$$ Springer-Verlag · Zbl 0742.14022 [7] Applications of Murphy’s elements, (1989) [8] Quasi-Hopf algebras, Leningrad Math. J, 1, 6, 1419-1457, (1990) · Zbl 0718.16033 [9] On quasitriangular quasi-Hopf algebras and a group closely connected with $$\text{Gal}(\overline{\mathbb{Q}/\mathbb{Q}),$$ Leningrad Math. J, 2, 4, 829-860, (1991) · Zbl 0728.16021 [10] Coxeter Graphs and Towers of Algebras, 14, (1989), Springer-Verlag · Zbl 0698.46050 [11] Nouvelle classification des (super-)algèbres de Lie simples par le biais de leurs invariants tensoriels, (1994) [12] Representation theory, a first course, 129, (1991), Springer-Verlag · Zbl 0744.22001 [13] On a generalization of Hilbert’s 21st problem, Ann. Sci. E.N.S. 4e sér., t. 19, 609-627, (1986) · Zbl 0616.14004 [14] On the decomposition of the tensor algebra of the classical Lie algebras, Adv. in Math, 56, 3, 238-282, (1985) · Zbl 0577.17003 [15] Polynomial invariants of knots via von Neumann algebras, Bull. AMS, 12, 103-111, (1985) · Zbl 0564.57006 [16] Symmetric polynomials and the center of the symmetric group ring, Rep. Math. Phys, 5, 107-112, (1974) · Zbl 0288.20014 [17] Quantum groups, 155, (1995), Springer-Verlag · Zbl 0808.17003 [18] Monodromy representations of braid groups and Yang-Baxter equations, Ann. Inst. Fourier (Grenoble), 37, 4, 139-160, (1987) · Zbl 0634.58040 [19] Linear representations of braid groups and classical Yang-Baxter equations, Braids (Santa Cruz, CA, 1986), 78, 339-363, (1988), Amer. Math. Soc., Providence, RI · Zbl 0661.20026 [20] Série de Poincaré-Koszul associée au groupe de tresses pures, Invent. Math, 82, 57-75, (1985) · Zbl 0574.55009 [21] Mémoires sur la théorie des systèmes des équations différentielles linéaires, (1953), Chelsea · Zbl 0011.34903 [22] Une caractérisation tensorielle des représentations standard, Expositiones Mathematicae, 18, 243-254, (2000) · Zbl 0977.17006 [23] Représentations linéaires des tresses infinitésimales, (2001) [24] On KZ-systems which are irreducible under the action of the symmetric group, Comptes Rendus Acad. Sci, Série I, 333, 517-522, (2001) · Zbl 1058.32008 [25] Normalisateurs de tores I. groupes de Coxeter étendus, J. Alg, 4, 96-116, (1966) · Zbl 0145.24703 [26] On the structure of Brauer’s centralizer algebras, Ann. of Math (2), 128, 173-193, (1988) · Zbl 0656.20040
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