A finite dimensional algebra of the diagram of a knot

Cibils, Claude

doi:10.1090/S0002-9939-2014-12293-3

A finite dimensional algebra of the diagram of a knot
HTML articles powered by AMS MathViewer

by Claude Cibils PDF

Proc. Amer. Math. Soc. 142 (2014), 3741-3746 Request permission

Abstract:

To a regular projection of a knot we associate a finite dimensional non-commutative associative algebra which is self-injective and special biserial.

References

Ibrahim Assem, Juan Carlos Bustamante, and Patrick Le Meur, Special biserial algebras with no outer derivations, Colloq. Math. 125 (2011), no. 1, 83–98. MR 2860583, DOI 10.4064/cm125-1-6
K. Bongartz and P. Gabriel, Covering spaces in representation-theory, Invent. Math. 65 (1981/82), no. 3, 331–378. MR 643558, DOI 10.1007/BF01396624
Ragnar-Olaf Buchweitz, Edward L. Green, Dag Madsen, and Øyvind Solberg, Finite Hochschild cohomology without finite global dimension, Math. Res. Lett. 12 (2005), no. 5-6, 805–816. MR 2189240, DOI 10.4310/MRL.2005.v12.n6.a2
Juan Carlos Bustamante, On the fundamental group of a Schurian algebra, Comm. Algebra 30 (2002), no. 11, 5307–5329. MR 1945091, DOI 10.1081/AGB-120015654
Juan Carlos Bustamante and Diane Castonguay, Fundamental groups and presentations of algebras, J. Algebra Appl. 5 (2006), no. 5, 549–562. MR 2269406, DOI 10.1142/S021949880600179X
Claude Cibils and Eduardo N. Marcos, Skew category, Galois covering and smash product of a $k$-category, Proc. Amer. Math. Soc. 134 (2006), no. 1, 39–50. MR 2170541, DOI 10.1090/S0002-9939-05-07955-4
Claude Cibils, María Julia Redondo, and Andrea Solotar, Connected gradings and the fundamental group, Algebra Number Theory 4 (2010), no. 5, 625–648. MR 2679101, DOI 10.2140/ant.2010.4.625
Claude Cibils, María Julia Redondo, and Andrea Solotar, Fundamental group of Schurian categories and the Hurewicz isomorphism, Doc. Math. 16 (2011), 581–595. MR 2837545
Claude Cibils, María Julia Redondo, and Andrea Solotar, The intrinsic fundamental group of a linear category, Algebr. Represent. Theory 15 (2012), no. 4, 735–753. MR 2944440, DOI 10.1007/s10468-010-9263-1
Claude Cibils, Maria Julia Redondo, and Andrea Solotar, Full and convex linear subcategories are incompressible, Proc. Amer. Math. Soc. 141 (2013), no. 6, 1939–1946. MR 3034421, DOI 10.1090/S0002-9939-2013-11470-X
C. Cibils, M. J. Redondo, and A. Solotar, On universal gradings, versal gradings and Schurian generated categories. arXiv:1210.4098v1. To appear in J. Noncommut. Geom.
Piotr Dowbor and Andrzej Skowroński, Galois coverings of representation-infinite algebras, Comment. Math. Helv. 62 (1987), no. 2, 311–337. MR 896100, DOI 10.1007/BF02564450
Claus Michael Ringel, The minimal representation-infinite algebras which are special biserial, Representations of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, pp. 501–560. MR 2931904, DOI 10.4171/101-1/11
Karin Erdmann and Andrzej Skowroński, On Auslander-Reiten components of blocks and self-injective biserial algebras, Trans. Amer. Math. Soc. 330 (1992), no. 1, 165–189. MR 1144759, DOI 10.1090/S0002-9947-1992-1144759-7
I. M. Gel′fand and V. A. Ponomarev, Indecomposable representations of the Lorentz group, Uspehi Mat. Nauk 23 (1968), no. 2 (140), 3–60 (Russian). MR 0229751
Yang Han, Hochschild (co)homology dimension, J. London Math. Soc. (2) 73 (2006), no. 3, 657–668. MR 2241972, DOI 10.1112/S002461070602299X
Patrick Le Meur, The universal cover of an algebra without double bypass, J. Algebra 312 (2007), no. 1, 330–353. MR 2320460, DOI 10.1016/j.jalgebra.2006.10.035
Patrick Le Meur, The fundamental group of a triangular algebra without double bypasses, C. R. Math. Acad. Sci. Paris 341 (2005), no. 4, 211–216 (English, with English and French summaries). MR 2164674, DOI 10.1016/j.crma.2005.07.004
P. Le Meur, Revêtements galoisiens et groupe fondamental d’algèbres de dimension finie, Ph.D. thesis, Université Montpellier 2, 2006. http://tel.archives-ouvertes.fr/ tel-00011753
R. Martínez-Villa and J. A. de la Peña, The universal cover of a quiver with relations, J. Pure Appl. Algebra 30 (1983), no. 3, 277–292. MR 724038, DOI 10.1016/0022-4049(83)90062-2
Zygmunt Pogorzały, Algebras stably equivalent to self-injective special biserial algebras, Comm. Algebra 22 (1994), no. 4, 1127–1160. MR 1261252, DOI 10.1080/00927879408824898
Nicole Snashall and Rachel Taillefer, The Hochschild cohomology ring of a class of special biserial algebras, J. Algebra Appl. 9 (2010), no. 1, 73–122. MR 2642814, DOI 10.1142/S0219498810003781
Burkhard Wald and Josef Waschbüsch, Tame biserial algebras, J. Algebra 95 (1985), no. 2, 480–500. MR 801283, DOI 10.1016/0021-8693(85)90119-X