×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.