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Quantum subgroups of a simple quantum group at roots of one

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  01 March 2009

Nicolás Andruskiewitsch  and
Gastón Andrés García
Nicolás Andruskiewitsch
Affiliation:
FaMAF-CIEM, Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, República Argentina (email: [email protected])
Gastón Andrés García
Affiliation:
FaMAF-CIEM, Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, República Argentina (email: [email protected])
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Abstract

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Let G be a connected, simply connected, simple complex algebraic group and let ϵ be a primitive ℓth root of one, ℓ odd and 3∤ℓ if G is of type G2. We determine all Hopf algebra quotients of the quantized coordinate algebra 𝒪ϵ(G).

Keywords

quantum groupsquantized enveloping algebrasquantized coordinate algebras

MSC classification

Secondary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 2 , March 2009 , pp. 476 - 500
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Andruskiewitsch, N., Notes on extensions of Hopf algebras, Canad. J. Math. 48 (1996), 3–42.CrossRefGoogle Scholar
[2]Andruskiewitsch, N. and Devoto, J., Extensions of Hopf algebras, Algebra i Analiz 7 (1995), 22–61.Google Scholar
[3]Andruskiewitsch, N. and García, G. A., Extensions of finite quantum groups by finite groups, Transform. Groups 14 (2009), 1–27.CrossRefGoogle Scholar
[4]Andruskiewitsch, N. and Schneider, H.-J., Pointed Hopf algebras, in New Directions in Hopf algebras, Mathematical Sciences Research Institute Publications, vol. 43 (Cambridge University Press, Cambridge, 2002), 1–68.Google Scholar
[5]Andruskiewitsch, N. and Schneider, H.-J., On the classification of finite-dimensional pointed Hopf algebras, Ann. of Math. (2), math.QA/0502157, to appear.Google Scholar
[6]Brown, K. A. and Goodearl, K. R., Lectures on Algebraic Quantum Groups, Advanced Courses in Mathematics—CRM Barcelona (Birkhäuser, Basel, 2002).CrossRefGoogle Scholar
[7]Chin, W. and Musson, I., The coradical filtration for quantized universal enveloping algebras, J. London Math. Soc. 53 (1996), 50–67.CrossRefGoogle Scholar
[8]De Concini, C. and Lyubashenko, V., Quantum function algebra at roots of 1, Adv. Math. 108 (1994), 205–262.CrossRefGoogle Scholar
[9]Dynkin, E. B., Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc., Transl., II. Ser. 6 (1957), 111–243.CrossRefGoogle Scholar
[10]Etingof, P. and Ostrik, V., Finite tensor categories, Mosc. Math. J. 4 (2004), 627–654, 782–783.CrossRefGoogle Scholar
[11]Ferrer Santos, W. and Rittatore, A., Actions and Invariants of Algebraic Groups, Pure and Applied Mathematics, vol. 269 (Chapman & Hall/CRC, Boca Raton, FL, 2005).CrossRefGoogle Scholar
[12]García, G. A., ‘Álgebras de Hopf y grupos cuánticos’. Tesis de doctorado, Universidad de Córdoba (2007). Available at http://www.mate.uncor.edu/∼ggarcia/.Google Scholar
[13]Graña, M., On Nichols algebras of low dimension, Contemp. Math. 267 (2000), 111–134.CrossRefGoogle Scholar
[14]Graña, M., Finite dimensional Nichols algebras of non-diagonal group type, zoo of examples, available at http://mate.dm.uba.ar/∼matiasg/zoo.html.Google Scholar
[15]Heckenberger, I., Classification of arithmetic root systems, Adv. Math. 220 (2009), 59–124.CrossRefGoogle Scholar
[16]Hochschild, G., Basic theory of algebraic groups and Lie algebras, Graduate Texts in Mathematics, vol. 75 (Springer, New York, 1981).CrossRefGoogle Scholar
[17]Jantzen, J. C., Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6 (American Mathematical Society, Providence, RI, 1996).Google Scholar
[18]Kirillov Jr, A. and Ostrik, V., On a q-analogue of the McKay correspondence and the ADE classification of conformal field theories, Adv. Math. 171 (2002), 183–227.CrossRefGoogle Scholar
[19]Kreimer, H. F. and Takeuchi, M., Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J. 30 (1981), 675–692.CrossRefGoogle Scholar
[20]Letzter, G., Coideal subalgebras and quantum symmetric pairs, in New directions in Hopf algebras, Mathematical Sciences Research Institute Publications, vol. 43 (Cambridge University Press, Cambridge, 2002), 117–165.Google Scholar
[21]Lusztig, G., Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990), 257–296.Google Scholar
[22]Lusztig, G., Quantum groups at roots of 1, Geom. Dedicata 35 (1990), 89–113.CrossRefGoogle Scholar
[23]Masuoka, A., Hopf algebra extensions and cohomology, in New Directions in Hopf Algebras, Mathematical Sciences Research Institute Publications, vol. 43, eds S. Montgomery and H.-J. Schneider. (Cambridge University Press, Cambridge, 2002), 167–209.Google Scholar
[24]Montgomery, S., Hopf Algebras and their Actions on Rings, CBMS Regional Conference Series in Mathematics, vol. 82 (American Mathematical Society, Providence, RI, 1993).CrossRefGoogle Scholar
[25]Müller, E., Some topics on Frobenius–Lusztig kernels I, II, J. Algebra 206 (1998), 624–658, 659–681CrossRefGoogle Scholar
[26]Müller, E., Finite subgroups of the quantum general linear group, Proc. London Math. Soc. (3) 81 (2000), 190–210.CrossRefGoogle Scholar
[27]Milinski, A. and Schneider, H.-J., Pointed Indecomposable Hopf Algebras over Coxeter Groups, Contemp. Math. 267 (2000), 215–236.CrossRefGoogle Scholar
[28]Ocneanu, A., The classification of subgroups of quantum SU(N), Contemp. Math. 294 (2002), 33–159.Google Scholar
[29]Podleś, P., Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups, Comm. Math. Phys. 170 (1995), 1–20.CrossRefGoogle Scholar
[30]Schneider, H.-J., Normal basis and transitivity of crossed products for Hopf algebras, J. Algebra 152 (1992), 289–312.CrossRefGoogle Scholar
[31]Schneider, H.-J., Some remarks on exact sequences of quantum groups, Commun. Algebra 21 (1993), 3337–3357.CrossRefGoogle Scholar
[32]Ştefan, D., Hopf algebras of low dimension, J. Algebra 211 (1999), 343–361.CrossRefGoogle Scholar
[33]Sweedler, M., Hopf algebras (Benjamin, New York, 1969).Google Scholar