Geodesic
Classification of $U_q(\mathfrak{sl}_2)$-module algebra structures on the quantum plane
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 6 (2010) no. 4, pp. 406-430 Cet article a éte moissonné depuis la source Math-Net.Ru

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A complete list of $U_q(\mathfrak{sl}_2)$-module algebra structures on the quantum plane is produced and the (uncountable family of) isomorphism classes of these structures are described. The composition series of representations in question are computed. The classical limits of the $U_q(\mathfrak{sl}_2)$-module algebra structures are discussed. 
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S. Duplij; S. D. Sinel'shchikov. Classification of $U_q(\mathfrak{sl}_2)$-module algebra structures on the quantum plane. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 6 (2010) no. 4, pp. 406-430. http://geodesic.mathdoc.fr/item/JMAG_2010_6_4_a4/

[1] E. Abe, Hopf Algebras, Cambridge Univ. Press, Cambridge, 1980 | MR | Zbl

[2] J. Alev, M. Chamarie, “Dérivations et Automorphismes de Quelques Algèbres Quantiques”, Comm. Algebra, 20 (1992), 1787–1802 | DOI | MR | Zbl

[3] V. G. Drinfeld, “On Almost Cocommutative Hopf Algebras”, Leningrad Math. J., 1 (1989), 321–342 | MR

[4] S. Duplij, F. Li, “Regular Solutions of Quantum Yang-Baxter Equation from Weak Hopf Algebras”, Czech. J. Phys., 51 (2001), 1306–1311 | DOI | MR | Zbl

[5] S. Duplij, S. Sinel'shchikov, “Quantum Enveloping Algebras with von Neumann Regular Cartan-like Generators and the Pierce Decomposition”, Commun. Math. Phys., 287 (2009), 769–785 | DOI | MR | Zbl

[6] A. Gonález-López, N. Kamran, P. Olver, “Quasi-exactly Solvable Lie Algebras of Differential Operators in two Complex Variables”, J. Phys. A: Math. Gen., 24 (1991), 3995–4078 | DOI | MR

[7] J. C. Jantzen, Lectures on Quantum Groups, Amer. Math. Soc., Providence, RI, 2005 | MR

[8] C. Kassel, Quantum Groups, Springer-Verlag, New York, 1995 | MR

[9] L. A. Lambe, D. E. Radford, Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: An Algebraic Approach, Kluwer, Dordrecht, 1997 | MR

[10] F. Li, S. Duplij, “Weak Hopf Algebras and Singular Solutions of Quantum Yang–Baxter Equation”, Commun. Math. Phys., 225 (2002), 191–217 | DOI | MR | Zbl

[11] Yu. I. Manin, Topics in Noncommutative Differential Geometry, Princeton Univ. Press, Princeton, 1991 | MR | Zbl

[12] S. Montgomery, S. P. Smith, “Skew Derivations and $U_q(\mathfrak{sl}_2)$”, Israel J. Math., 72 (1990), 158–166 | DOI | MR | Zbl

[13] M. E. Sweedler, Hopf Algebras, Benjamin, New York, 1969 | MR | Zbl