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Liftings of Jordan and Super Jordan Planes

Part of: Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  12 April 2018

Nicolás Andruskiewitsch ,
Iván Angiono  and
István Heckenberger
Nicolás Andruskiewitsch*
Affiliation:
FaMAF-CIEM (CONICET), Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, Argentina (andrus@famaf.unc.edu.ar; angiono@famaf.unc.edu.ar)
Iván Angiono
Affiliation:
FaMAF-CIEM (CONICET), Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, Argentina (andrus@famaf.unc.edu.ar; angiono@famaf.unc.edu.ar)
István Heckenberger
Affiliation:
Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Hans-Meerwein-Strasse, D-35032 Marburg, Germany (heckenberger@mathematik.uni-marburg.de)
*
*Corresponding author.
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Abstract

We classify pointed Hopf algebras with finite Gelfand–Kirillov dimension whose infinitesimal braiding has dimension 2 but is not of diagonal type, or equivalently is a block. These Hopf algebras are new and turn out to be liftings of either a Jordan or a super Jordan plane over a nilpotent-by-finite group.

Keywords

Hopf algebrasGelfand–Kirillov dimensionJordan plane

MSC classification

Primary: 16T05: Hopf algebras and their applications
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 3 , August 2018 , pp. 661 - 672
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Andruskiewitsch, N. and Angiono, I., On Nichols algebras with generic braiding, In Modules and comodules (ed. Brzezinski, T., Gómez Pardo, J. L., Shestakov, I. and Smith, P. F.), Trends in Mathematics, pp. 47–64 (Birkhäuser, 2008).CrossRefGoogle Scholar
2.Andruskiewitsch, N., Angiono, I., García Iglesias, A., Masuoka, A. and Vay, C., Lifting via cocycle deformation, J. Pure Appl. Alg. 218 (2014), 684–703.CrossRefGoogle Scholar
3.Andruskiewitsch, N., Angiono, I. and Heckenberger, I., On finite GK-dimensional Nichols algebras over abelian groups. arXiv:1606.02521.Google Scholar
4.Andruskiewitsch, N., Angiono, I. and Rossi Bertone, F., The divided powers algebra of a finite-dimensional Nichols algebra of diagonal type, Math. Res. Lett. 24 (2017), 619–643.Google Scholar
5.Andruskiewitsch, N. and Schneider, H.-J., Pointed Hopf algebras, New Directions in Hopf Algebras, MSRI Series, pp. 1–68 (Cambridge University Press, 2002).Google Scholar
6.Andruskiewitsch, N. and Schneider, H.-J., A characterization of quantum groups, J. Reine Angew. Math. 577 (2004), 81–104.Google Scholar
7.Artin, M. and Schelter, W., Graded algebras of global dimension 3, Adv. Math. 66 (1987), 171–216.Google Scholar
8.Brown, K. A., On zero divisors in group rings, Bull. London Math. Soc. 8 (1976), 251–256.Google Scholar
9.Brown, K., Goodearl, K., Lenagan, T. and Zhang, J., Mini-workshop: Infinite dimensional Hopf algebras, Oberwolfach Rep. 11 (2014), 1111–1137.Google Scholar
10.Brown, K. A. and Zhang, J. J., Prime regular Hopf algebras of GK-dimension one, Proc. London Math. Soc. 101 (2010), 260–302.Google Scholar
11.Etingof, P. and Gelaki, S., Quasisymmetric and unipotent tensor categories, Math. Res. Lett. 15 (2008), 857–866.Google Scholar
12.Farkas, D. R. and Snider, R., K 0 and noetherian group rings, J. Algebra 42 (1976), 192–198.Google Scholar
13.Goodearl, K. R. and Zhang, J. J., Noetherian Hopf algebra domains of Gelfand-Kirillov dimension two, J. Algebra 324 (2010), 3131–3168.CrossRefGoogle Scholar
14.Günther, R., Crossed products for pointed Hopf algebras, Comm. Algebra 27 (1999), 4389–4410.Google Scholar
15.Masuoka, A., Abelian and non-abelian second cohomologies of quantized enveloping algebras, J. Algebra 320 (2008), 1–47.Google Scholar
16.Moody, J. A., Torsion-free solvable group rings are Ore domains. Unpublished note (1987).Google Scholar
17.Schneider, H.-J., Normal basis and transitivity of crossed products for Hopf algebras, J. Algebra 152 (1992), 289–312.CrossRefGoogle Scholar
18.Wang, D.-G., Zhang, J. J. and Zhuang, G., Primitive cohomology of Hopf algebras, J. Algebra 464 (2016), 36–96.Google Scholar
19.Zhuang, G., Properties of pointed and connected Hopf algebras of finite Gelfand-Kirillov dimension, J. Lond. Math. Soc. 87 (2013), 877–898.CrossRefGoogle Scholar