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1998 | 42 | 1 | 297-306
Tytuł artykułu

Skein algebra of a group

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We define for each group G the skein algebra of G. We show how it is related to the Kauffman bracket skein modules. We prove that skein algebras of abelian groups are isomorphic to symmetric subalgebras of corresponding group rings. Moreover, we show that, for any abelian group G, homomorphisms from the skein algebra of G to C correspond exactly to traces of SL(2,C)-representations of G. We also solve, for abelian groups, the conjecture of Bullock on SL(2,C) character varieties of groups - we show that skein algebras are isomorphic to the coordinate rings of the corresponding character varieties.
Słowa kluczowe
Rocznik
Tom
42
Numer
1
Strony
297-306
Opis fizyczny
Daty
wydano
1998
Twórcy
  • Department of Mathematics, The George Washington University, 2201 G Str., room 428, Funger Hall, Washington, D.C. 20052, U.S.A.
  • Department of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa, Poland
Bibliografia
  • [B-1] D. Bullock, Estimating a skein module with $SL_2(C)$ characters, Proc. Amer. Math. Soc. 125 (1997), 1835-1839.
  • [B-2] D. Bullock, Estimating the states of the Kauffman bracket skein module, this volume.
  • [B-P] D. Bullock, J. H. Przytycki, Kauffman bracket quantization of symmetric algebra and so(3), preprint 1996.
  • [C-S] M. Culler and P. B. Shalen, Varieties of group representations and splittings of 3-manifolds, Ann. of Math. 117 (1983), 109-146.
  • [F-K] R. Fricke, F. Klein, Vorlesungen über die Theorie der automorphen Functionen, Vol. 1, pp. 365-370. Leipzig: B.G. Teubner 1897. Reprint: New York, Johnson reprint Corporation (Academic Press) 1965.
  • [Ho] R. D. Horowitz, Characters of free groups represented in two-dimensional special linear group, Comm. Pure and Appl. Math. 25, 1972, 635-649.
  • [H-P-1] J. Hoste, J. H. Przytycki, A survey of skein modules of 3-manifolds; in Knots 90, Proceedings of the International Conference on Knot Theory and Related Topics, Osaka (Japan), August 15-19, 1990, Editor A. Kawauchi, Walter de Gruyter 1992, 363-379.
  • [H-P-2] J. Hoste, J. H. Przytycki, The (2,∞)-skein module of lens spaces; a generalization of the Jones polynomial, Journal of Knot Theory and Its Ramifications, 2(3), 1993, 321-333.
  • [H-P-3] J. Hoste, J. H. Przytycki, The Kauffman bracket skein module of $S^1 × S^2$, Math. Z., 220(1), 1995, 63-73.
  • [Hu] T. W. Hungerford, Algebra, Graduate Texts in Mathematics, Springer-Verlag 1974.
  • [Jo] T. Jorgensen, Closed geodesics on Riemann surfaces, Proc. Amer. Math. Soc., 72(1), 1978, 140-142.
  • [L-M] M. Lustig, W. Metzler, Integral representations of $Aut F^n$ and presentation classes of groups, Contemporary Mathematics, Vol 44, 1985, 51-67 (in Combinatorial Methods in Topology and Algebraic Geometry, Ed. J.R.Harper, R.Mandelbaum, Proceedings of a conference in honor of Arthur M.Stone, Rochester 1982).
  • [Ma-1] W. Magnus, Rings of Fricke characters and automorphism groups of free groups, Math. Z., 170, 1980, 91-103.
  • [Ma-2] W. Magnus, The use of 2 by 2 matrices in combinatorial group theory. A survey, Resultate der Mathematik, Vol. 4, 1981, 171-722.
  • [Pr] J. H. Przytycki, Skein modules of 3-manifolds, Bull. Polish Acad. Science, 39(1-2), 1991, 91-100.
  • [P-S-1] J. H. Przytycki, A. S. Sikora, On Skein Algebras And $Sl_2(C)$-Character Varieties, Topology, submitted.
  • [P-S-2] J. H. Przytycki, A. S. Sikora, in preparation.
  • [V] H. Vogt, Sur les invariants fondamentaux des équations différentielles linéaires du second ordre. Ann. Sci. Ecole Norm. Sup. (3) 6, Suppl. 3-72 (1889) (Thèse, Paris).
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-bcpv42i1p297bwm
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