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Finite quantum groups over Abelian groups of prime exponent. (English) Zbl 1007.16028
The classification of the pointed finite-dimensional complex Hopf algebras whose group of group-like elements is Abelian of prime exponent \(p\), \(p>17\), is obtained.
From Dynkin diagrams and fixed \(s\in\mathbb{N}\) and \(p\) a prime number, \(p>2\), the authors construct a general family of Hopf algebras which are pointed, whose group of group-like elements is isomorphic to \(\Gamma(s)=(\mathbb{Z}/(p))^s\) and whose dimension is given by \(p^{s+|\Phi^+|}\), where \(\Phi^+\) is the set of positive roots of the root system associated to the corresponding Cartan matrix. Once this result is stated, they prove that, if \(p>17\) and \(\mathcal A\) is a pointed finite-dimensional Hopf algebra with \(\Gamma(s)\simeq G({\mathcal A})\), where \(G({\mathcal A})\) is the group of group-like elements of \(\mathcal A\), then \(\mathcal A\) is isomorphic to an algebra of the mentioned general family. As a consequence, the complete classification of these pointed Hopf algebras is obtained. As an important step in the proof of this result, the authors prove that this family of Hopf algebras is generated by group-like and skew-primitive elements.

MSC:
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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References:
[1] Andersen H.H. , Jantzen J. , Soergel W. , Representations of quantum groups at a p th root of unity and of semisimple groups in characteristic p : Independence of p , Astérisque , 220 , 1994 . MR 1272539 | Zbl 0802.17009 · Zbl 0802.17009
[2] Andruskiewitsch N. , Dăscălescu S , On quantum groups at - 1, Algebr. Represent. Theory , to appear. Zbl 02176192 · Zbl 1136.16032
[3] Andruskiewitsch N. , Graña M. , Braided Hopf algebras over non-abelian groups , Bol. Acad. Ciencias (Córdoba) 63 ( 1999 ) 45 - 78 , available at www.mate.uncor.edu/andrus/articulos.html . MR 1714540 | Zbl 1007.17010 · Zbl 1007.17010
[4] Andruskiewitsch N. , Schneider H.-J. , Lifting of quantum linear spaces and pointed Hopf algebras of order p 3 , J. Algebra 209 ( 1998 ) 658 - 691 . MR 1659895 | Zbl 0919.16027 · Zbl 0919.16027
[5] Andruskiewitsch N. , Schneider H.-J. , Finite quantum groups and Cartan matrices , Adv. in Math. 154 ( 2000 ) 1 - 45 . MR 1780094 | Zbl 1007.16027 · Zbl 1007.16027
[6] Andruskiewitsch N. , Schneider H.-J. , Lifting of Nichols algebras of type A 2 and pointed Hopf algebras of order p 4 , in: Caenepeel S. (Ed.), Proceedings of the Colloquium in Brussels 1998, Hopf Algebras and Quantum Groups , Marcel Dekker , 2000 , pp. 1 - 18 . MR 1763604 | Zbl 1020.16022 · Zbl 1020.16022
[7] Andruskiewitsch N. , Schneider H.-J. , Pointed Hopf algebras, in: New Directions in Hopf Algebras , MSRI Series Cambridge Univ. Press, to appear. MR 1913436 | Zbl 1011.16025 · Zbl 1011.16025
[8] Beattie M. , Dăscălescu S. , Grünenfelder L. , On the number of types of finite-dimensional Hopf algebras , Inventiones Math. 136 ( 1999 ) 1 - 7 . MR 1681117 | Zbl 0922.16021 · Zbl 0922.16021
[9] Beattie M., Dăscălescu S., Raianu S. , Lifting of Nichols algebras of type B 2 , Preprint, 2001.
[10] Caenepeel S. , Dăscălescu S. , Pointed Hopf algebras of dimension p 3 , J. Algebra 209 ( 1998 ) 622 - 634 . MR 1659887 | Zbl 0917.16016 · Zbl 0917.16016
[11] Caenepeel S. , Dăscălescu S. , Raianu S. , Classifying pointed Hopf algebras of dimension 16 , Comm. Algebra 28 ( 2000 ) 541 - 568 . MR 1736746 | Zbl 0952.16030 · Zbl 0952.16030
[12] de Concini C. , Procesi C. , Quantum groups , in: D-modules, Representation Theory and Quantum Groups , Lecture Notes in Maths. , 1565 , Springer-Verlag , 1993 , pp. 31 - 140 . MR 1288995 | Zbl 0795.17005 · Zbl 0795.17005
[13] Didt D. , Linkable Dynkin diagrams , Preprint, 2001. arXiv | MR 1935506 · Zbl 1062.16042
[14] Doi Y. , Takeuchi M. , Multiplication alteration by two-cocycles. The quantum version , Comm. Algebra 22 ( 14 ) ( 1994 ) 5715 - 5732 . MR 1298746 | Zbl 0821.16038 · Zbl 0821.16038
[15] Drinfeld V.G. , Quasi-Hopf algebras , Leningrad Math. J. 1 ( 1990 ) 1419 - 1457 . MR 1047964
[16] Gelaki S. , On pointed Hopf algebras and Kaplansky’s tenth conjecture , J. Algebra 209 ( 1998 ) 635 - 657 . MR 1659891 | Zbl 0922.16023 · Zbl 0922.16023
[17] Graña M. , Pointed Hopf algebras of dimension 32 , Comm. Algebra 28 ( 2000 ) 2935 - 2976 . MR 1757439 | Zbl 0952.16036 · Zbl 0952.16036
[18] Graña M. , On Pointed Hopf algebras of dimension p 5 , Glasgow Math. J. 42 ( 2000 ) 405 - 419 . MR 1793809 | Zbl 0970.16016 · Zbl 0970.16016
[19] Graña M. , On Nichols algebras of low dimension , Contemp. Math. 267 ( 2000 ) 111 - 134 . MR 1800709 | Zbl 0974.16031 · Zbl 0974.16031
[20] Kac V. , Infinite Dimensional Lie Algebras , Cambridge Univ. Press , 1995 . MR 1104219 | Zbl 0716.17022 · Zbl 0716.17022
[21] Lusztig G. , Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 , 257-296. MR 1013053 | Zbl 0695.16006 · Zbl 0695.16006
[22] Lusztig G. , Quantum groups at roots of 1 , Geom. Dedicata 35 ( 1990 ) 89 - 114 . MR 1066560 | Zbl 0714.17013 · Zbl 0714.17013
[23] Lusztig G. , Introduction to Quantum Groups , Birkhäuser , 1993 . MR 1227098 | Zbl 0788.17010 · Zbl 0788.17010
[24] Majid S. , Crossed products by braided groups and bosonization , J. Algebra 163 ( 1994 ) 165 - 190 . MR 1257312 | Zbl 0807.16036 · Zbl 0807.16036
[25] Masuoka A. , Defending the negated Kaplansky conjecture , Proc. Amer. Math. Soc. 129 ( 2001 ) 3185 - 3192 . MR 1844991 | Zbl 0985.16026 · Zbl 0985.16026
[26] Montgomery S. , Hopf Algebras and Their Actions on Rings , CBMS Lecture Notes , 82 , American Mathematical Society , 1993 . MR 1243637 | Zbl 0793.16029 · Zbl 0793.16029
[27] Müller E. , Some topics on Frobenius-Lusztig kernels, I , J. Algebra 206 ( 1998 ) 624 - 658 . MR 1637096 | Zbl 0948.17010 · Zbl 0948.17010
[28] Musson I. , Finite quantum groups and pointed Hopf algebras , Preprint, 1999. MR 1903966
[29] Nichols W.D. , Bialgebras of type one , Comm. Algebra 6 ( 1978 ) 1521 - 1552 . MR 506406 | Zbl 0408.16007 · Zbl 0408.16007
[30] Nichols W.D. , Zoeller M.B. , A Hopf algebra freeness theorem , Amer. J. Math. 111 ( 1989 ) 381 - 385 . MR 987762 | Zbl 0672.16006 · Zbl 0672.16006
[31] Radford D. , Hopf algebras with projection , J. Algebra 92 ( 1985 ) 322 - 347 . MR 778452 | Zbl 0549.16003 · Zbl 0549.16003
[32] Ringel C. , PBW-bases of quantum groups , J. Reine Angew. Math. 470 ( 1996 ) 51 - 88 . MR 1370206 | Zbl 0840.17010 · Zbl 0840.17010
[33] Rosso M. , Groupes quantiques et algèbres de battage quantiques , C.R.A.S. (Paris) 320 ( 1995 ) 145 - 148 . MR 1320345 | Zbl 0929.17005 · Zbl 0929.17005
[34] Rosso M. , Quantum groups and quantum shuffles , Inventiones Math. 133 ( 1998 ) 399 - 416 . MR 1632802 | Zbl 0912.17005 · Zbl 0912.17005
[35] Schauenburg P. , A characterization of the Borel-like subalgebras of quantum enveloping algebras , Comm. Algebra 24 ( 1996 ) 2811 - 2823 . MR 1396857 | Zbl 0856.17017 · Zbl 0856.17017
[36] Stefan D. , van Oystaeyen F. , Hochschild cohomology and coradical filtration of pointed Hopf algebras , J. Algebra 210 ( 1998 ) 535 - 556 . MR 1662284 | Zbl 0918.16030 · Zbl 0918.16030
[37] Woronowicz S.L. , Differential calculus on compact matrix pseudogroups (quantum groups) , Comm. Math. Phys. 122 ( 1989 ) 125 - 170 . Article | MR 994499 | Zbl 0751.58042 · Zbl 0751.58042
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