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Abstract:

Over fields of arbitrary characteristic we classify all braid-indecomposable tuples of at least two absolutely simple Yetter-Drinfeld modules over non-abelian groups such that the group is generated by the support of the tuple and the Nichols algebra of the tuple is finite-dimensional. Such tuples are classified in terms of analogs of Dynkin diagrams which encode much information about the Yetter-Drinfeld modules. We also compute the dimensions of these finite-dimensional Nichols algebras. Our proof uses essentially theWeyl groupoid of a tuple of simple Yetter-Drinfeld modules and our previous result on pairs. © European Mathematical Society 2017.

Registro:

Documento: Artículo
Título:A classification of Nichols algebras of semisimple Yetter-Drinfeld modules over non-abelian groups
Autor:Heckenberger, I.; Vendramin, L.
Filiación:Philipps-Universität Marburg, FB Mathematik und Informatik, Hans-Meerwein-Straße, Marburg, 35032, Germany
Departamento de Matemática, FCEN, Universidad de Buenos Aires, Pabellón 1, Ciudad Universitaria (1428), Buenos Aires, Argentina
Palabras clave:Hopf algebra; Nichols algebra; Weyl groupoid
Año:2017
Volumen:19
Número:2
Página de inicio:299
Página de fin:356
DOI: http://dx.doi.org/10.4171/JEMS/667
Título revista:Journal of the European Mathematical Society
Título revista abreviado:J. Eur. Math. Soc.
ISSN:14359855
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_14359855_v19_n2_p299_Heckenberger

Referencias:

  • Andruskiewitsch, N., On Finite-dimensional Hopf Algebras, , Accepted for publication in Proceedings of the International Congress of Mathematicians. arXiv: 1403.7838
  • Andruskiewitsch, N., About finite dimensional Hopf algebras (2002) Quantum Symmetries in Theoretical Physics and Mathematics (Bariloche, 2000), Contemp. Math, 294, pp. 1-57. , Amer. Math. Soc., Providence, RI, Zbl 1135.16306 MR 1907185
  • Andruskiewitsch, N., Angiono, I.E., On Nichols algebras with generic braiding (2008) Modules and Comodules, Trends Math, pp. 47-64. , Birkhäuser, Basel, Zbl 1227.16022 MR 2742620
  • Andruskiewitsch, N., Cuadra, J., On the structure of (co-Frobenius) Hopf algebras (2013) J. Noncommut. Geom, 7, pp. 83-104. , Zbl 1279.16024 MR 3032811
  • Andruskiewitsch, N., Fantino, F., Graña, M., Vendramin, L., Finite-dimensional pointed Hopf algebras with alternating groups are trivial (2011) Ann. Mat. Pura Appl, 190 (4), pp. 225-245. , Zbl 1234.16019 MR 2786171
  • Andruskiewitsch, N., Fantino, F., Graña, M., Vendramin, L., Pointed Hopf algebras over the sporadic simple groups (2011) J. Algebra, 325, pp. 305-320. , Zbl 1217.16026 MR 2745542
  • Andruskiewitsch, N., Heckenberger, I., Schneider, H.-J., The Nichols algebra of a semisimple Yetter-Drinfeld module (2010) Amer. J. Math, 132, pp. 1493-1547. , Zbl 1214.16024 MR 2766176
  • Andruskiewitsch, N., Schneider, H.-J., Lifting of quantum linear spaces and pointed Hopf algebras of order p3 (1998) J. Algebra, 209, pp. 658-691. , Zbl 0919.16027 MR 1659895
  • Andruskiewitsch, N., Schneider, H.-J., Finite quantum groups and Cartan matrices (2000) Adv. Math, 154, pp. 1-45. , Zbl 1007.16027 MR 1780094
  • Andruskiewitsch, N., Schneider, H.-J., Pointed Hopf algebras (2002) New Directions in Hopf Algebras, Math. Sci. Res. Inst. Publ, 43, pp. 1-68. , Cambridge Univ. Press, Cambridge, Zbl 1011.16025 MR 1913436
  • Andruskiewitsch, N., Schneider, H.-J., A characterization of quantum groups (2004) J. Reine Angew. Math, 577, pp. 81-104. , Zbl 1084.16027 MR 2108213
  • Andruskiewitsch, N., Schneider, H.-J., On the classification of finite-dimensional pointed Hopf algebras (2010) Ann. of Math, 171 (2), pp. 375-417. , Zbl 1208.16028 MR 2630042
  • Angiono, I., On Nichols algebras of diagonal type (2013) J. Reine Angew. Math, 683, pp. 189-251. , Zbl 1331.16023 MR 3181554
  • Angiono, I.E., A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems (2015) J. Eur. Math. Soc, 17, pp. 2643-2671. , Zbl 1343.16022 MR 3420518
  • Bazlov, Y., Nichols-Woronowicz algebra model for Schubert calculus on Coxeter groups (2006) J. Algebra, 297, pp. 372-399. , Zbl 1101.16027 MR 2209265
  • Cuntz, M., Heckenberger, I., Weyl groupoids with at most three objects (2009) J. Pure Appl. Algebra, 213, pp. 1112-1128. , Zbl 1169.20020 MR 2498801
  • Cuntz, M., Heckenberger, I., Finite Weyl groupoids of rank three (2012) Trans. Amer. Math. Soc, 364, pp. 1369-1393. , Zbl 1246.20037 MR 2869179
  • Cuntz, M., Heckenberger, I., Finite Weyl groupoids (2015) J. Reine Angew. Math, 702, pp. 77-108. , Zbl 06435243 MR 3341467
  • Heckenberger, I., Weyl equivalence for rank 2 Nichols algebras of diagonal type (2005) Ann. Univ. Ferrara Sez. VII (N.S.), 51, pp. 281-289. , Zbl 1121.16032 MR 2294771
  • Heckenberger, I., The Weyl groupoid of a Nichols algebra of diagonal type (2006) Invent. Math, 164, pp. 175-188. , Zbl 1174.17011 MR 2207786
  • Heckenberger, I., Classification of arithmetic root systems of rank 3 (2007) Proc. XVIth Latin American Algebra Colloquium (Spanish), Bibl. Rev. Mat. Iberoamericana, pp. 227-252. , Rev. Mat. Iberoamericana, Madrid Zbl 1197.17003 MR 2500361
  • Heckenberger, I., Examples of finite-dimensional rank 2 Nichols algebras of diagonal type (2007) Compos. Math, 143, pp. 165-190. , Zbl 1169.16027 MR 2295200
  • Heckenberger, I., Rank 2 Nichols algebras with finite arithmetic root system (2008) Algebr. Represent. Theory, 11, pp. 115-132. , Zbl 1175.17003 MR 2379892
  • Heckenberger, I., Classification of arithmetic root systems (2009) Adv. Math, 220, pp. 59-124. , Zbl 1176.17011 MR 2462836
  • Heckenberger, I., Schneider, H.-J., Nichols algebras over groups with finite root system of rank two i (2010) J. Algebra, 324, pp. 3090-3114. , Zbl 1219.16032 MR 2732989
  • Heckenberger, I., Schneider, H.-J., Root systems and Weyl groupoids for Nichols algebras (2010) Proc. London Math. Soc, 101 (3), pp. 623-654. , Zbl 1210.17014 MR 2734956
  • Heckenberger, I., Schneider, H.-J., Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid (2013) Israel J. Math, 197, pp. 139-187. , Zbl 1301.16033 MR 3096611
  • Heckenberger, I., Vendramin, L., The classification of Nichols algebras with finite root system of rank two (2013) J. Eur. Math. Soc. to Appear, , arXiv: 1311.2881
  • Heckenberger, I., Vendramin, L., Nichols algebras over groups with finite root system of rank two II (2014) J. Group Theory, 17, pp. 1009-1034. , Zbl 1305.16026 MR 3276225
  • Heckenberger, I., Vendramin, L., Nichols algebras over groups with finite root system of rank two III (2015) J. Algebra, 422, pp. 223-256. , Zbl 1306.16028 MR 3272075
  • Heckenberger, I., Wang, J., Rank 2 Nichols algebras of diagonal type over fields of positive characteristic (2015) SIGMA Symmetry Integrability Geom. Methods Appl, 11, 24p. , paper 011 Zbl 1318.16031 MR 3313687
  • Kac, V.G., (1985) Infinite-Dimensional Lie Algebras, , 2nd ed. Cambridge Univ. Press Cambridge,Zbl 0574.17010 MR 823672
  • Kharchenko, V.K., A quantum analogue of the Poincaré-Birkhoff-Witt theorem (1999) Algebra Logika 38, 476, pp. 507-509. , (in Russian) Zbl 0936.16034 MR 1763385
  • Lentner, S., New large-rank Nichols algebras over nonabelian groups with commutator subgroup Z2 (2014) J. Algebra, 419, pp. 1-33. , Zbl 1306.16029 MR 3253277
  • Majid, S., Noncommutative differentials and Yang-Mills on permutation groups Sn (2005) Hopf Algebras in Noncommutative Geometry and Physics, Lecture Notes in Pure Appl. Math, 239, pp. 189-213. , Dekker, New York Zbl 1076.58004 MR 2106930
  • Nichols, W.D., Bialgebras of type one (1978) Comm. Algebra, 6, pp. 1521-1552. , Zbl 0408.16007 MR 0506406
  • Rosso, M., Quantum groups and quantum shuffles (1998) Invent. Math, 133, pp. 399-416. , Zbl 0912.17005 MR 1632802
  • Semikhatov, A.M., Fusion in the entwined category of Yetter-Drinfeld modules of a rank-1 Nichols algebra (2012) Theoret. Math. Phys, 173, pp. 1329-1358
  • (2012) Russian Version: Teoret. Mat. Fiz, 173, pp. 3-37. , Zbl 1280.81130 MR 3171534
  • Semikhatov, A.M., Tipunin, I.Y., The Nichols algebra of screenings (2012) Comm. Contemp. Math, 14, 66p. , Zbl 1264.81285 MR 2965674
  • Semikhatov, A.M., Tipunin, I.Y., Logarithmic bs'.2/ CFT models from Nichols algebras (2013) I. J. Phys. A, 46, 53p. , Zbl 1283.81113 MR 3146017
  • Woronowicz, S.L., Compact matrix pseudogroups (1987) Comm. Math. Phys, 111, pp. 613-665. , Zbl 0627.58034 MR 901157
  • Woronowicz, S.L., Differential calculus on compact matrix pseudogroups (quantum groups) (1989) Comm. Math. Phys, 122, pp. 125-170. , Zbl 0751.58042 MR 994499

Citas:

---------- APA ----------
Heckenberger, I. & Vendramin, L. (2017) . A classification of Nichols algebras of semisimple Yetter-Drinfeld modules over non-abelian groups. Journal of the European Mathematical Society, 19(2), 299-356.
http://dx.doi.org/10.4171/JEMS/667
---------- CHICAGO ----------
Heckenberger, I., Vendramin, L. "A classification of Nichols algebras of semisimple Yetter-Drinfeld modules over non-abelian groups" . Journal of the European Mathematical Society 19, no. 2 (2017) : 299-356.
http://dx.doi.org/10.4171/JEMS/667
---------- MLA ----------
Heckenberger, I., Vendramin, L. "A classification of Nichols algebras of semisimple Yetter-Drinfeld modules over non-abelian groups" . Journal of the European Mathematical Society, vol. 19, no. 2, 2017, pp. 299-356.
http://dx.doi.org/10.4171/JEMS/667
---------- VANCOUVER ----------
Heckenberger, I., Vendramin, L. A classification of Nichols algebras of semisimple Yetter-Drinfeld modules over non-abelian groups. J. Eur. Math. Soc. 2017;19(2):299-356.
http://dx.doi.org/10.4171/JEMS/667