×

Homological representations of the Hecke algebra. (English) Zbl 0716.20022

Es werden Darstellungen von Hecke Algebren mit Hilfe von Monodromie- Darstellungen in einem Vektorraumbündel mit einem natürlichen flachen Zusammenhang konstruiert. Dabei sind die Fasern Homologie-Vektorräume von Konfigurationsräumen von Punkten in der komplexen Ebene, wie man sie in der Zopftheorie betrachtet. Der Ansatz führt auf eine topologische Beschreibung des Jones Polynoms, die in einem Preprint vorliegt. Die durchgeführte Konstruktion hängt eng zusammen mit A. Tsuchiya, Y. Kanie [Adv. Stud. Pure Math. 16, 297-372 (1988; Zbl 0661.17021), Erratum ibid. 19, 675-682 (1989; Zbl 0699.17019)]. Als Spezialfall ergibt sich die Burau-Darstellung der Zopfgruppe bzw. das Alexanderpolynom einer Verkettung.
Reviewer: G.Burde

MSC:

20G05 Representation theory for linear algebraic groups
17B65 Infinite-dimensional Lie (super)algebras
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
20F36 Braid groups; Artin groups
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)

References:

[1] [A] Atiyah, M.F.: The geometry and physics of knots. Lezioni Lincee. Cambridge: University Press (1990)
[2] [ABG] Atiyah, M.F., Bott, R., Gårding, L.: Lacunas for hyperbolic differential operators with constant coefficients. Acta Mathematica124, 109–189 1970) · Zbl 0191.11203 · doi:10.1007/BF02394570
[3] [Al1] Alexander, J.: Topological invariants of knots and links. Trans. AMS30, 275–306 (1928) · JFM 54.0603.03 · doi:10.1090/S0002-9947-1928-1501429-1
[4] [Al2] Alexander, J.W.: A lemma on a system of knotted curves. Proc. Nat. Acad. Sci. USA9, 93–95 (1923) · doi:10.1073/pnas.9.3.93
[5] [FYHLMO] Freyd, P., Yetter, D., Hoste, J., Lickorish, W., Millet, K., Ocneanu, A.: A new polynomial invariant of knots and links. Bull. AMS12, 239–246 (1985) · Zbl 0572.57002 · doi:10.1090/S0273-0979-1985-15361-3
[6] [G] Grothendieck, A.: On the de Rham cohomology of algebraic varieties. Publ. IHES29, 351–359 (1966) · Zbl 0145.17602
[7] [H1] Hitchin, N.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc.55, 59–126 (1987) · Zbl 0634.53045 · doi:10.1112/plms/s3-55.1.59
[8] [H2] Hitchin, N.: Reduction to the Abelian case. Oxford Seminar on Jones-Witten Theory 75–92 (1988)
[9] [J] Jones, V.F.R.: Hecke algebra representations of braid groups and link polynomials. Ann. Math.126, 335–388 (1987) · Zbl 0631.57005 · doi:10.2307/1971403
[10] [Ko] Kohno, T.: Hecke algebra representations of braid groups and classical Yang-Baxter equations. Adv. Stud. Pure Maths.16, 255–269 (1988)
[11] [L1] Lawrence, R.J.: Homology representations of braid groups. D. Phil. Thesis, Oxford (June 1989)
[12] [L2] Lawrence, R.J.: A functional appraoch to the one-variable Jones polynomial. Harvard University preprint (1990)
[13] [M] Markov, A.: Über die freie Äquivalenz geschlossener Zöpfe. Recueil Math. Moscow1, 73–78 (1935)
[14] [R] Rolfsen, D.: Knots and links. Publish or Perish Press 1976 · Zbl 0339.55004
[15] [S1] Segal, G.B.: Two-dimensional conformal field theories and modular functors. Proc. IXth Int. Congr. on Mathematical Physics 22–37 (1989)
[16] [S2] Segal, G.B.: The Abelian Theory. Oxford Seminar on Jones-Witten Theory 17–35 (1988)
[17] [S3] Segal, G.B.: Fusion rules and the Verlinde algebra. Oxford Seminar on Jones-Witten Theory 51–74 (1988)
[18] [TK] Tsuchiya, A., Kanie, Y.: Vertex operators in conformal field theory onP 1 and monodromy representations of braid groups. Adv. Stud. Pure Math.16, 297–372 (1988), Erratum ibid Tsuchiya, A., Kanie, Y.: Vertex operators in conformed field theory onP 1 and monodromy representations of braid groups. Adv. Stud. Pure Math.19, 675–682 (1990) · Zbl 0661.17021
[19] [Tu] Turaev, V.G.: The Yang-Baxter equation and invariants of links. Invent. Math.92, 527–553 (1988) · Zbl 0648.57003 · doi:10.1007/BF01393746
[20] [We] Wenzl, H.: Hecke algebra representations of typeA n and subfactors. Invent. Math.92, 349–383 (1988) · Zbl 0663.46055 · doi:10.1007/BF01404457
[21] [Wi] Witten, E.: Some geometrical applications of quantum field theory. IXth Int. Congr. on Mathematical Physics 77–116 (1989) · Zbl 0743.57012
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.