Classification of the simple modules of the quantum Weyl algebra and the quantum plane

 Access to full text

 Full (PDF)

1. [AP 1] D. Arnal and G. Pinczon, On algebraically irreducible representations of the Lie algebra sl(2), J. Math. Phys. 15 (1974), 350-359. Zbl0298.17003
2. [AP 2] D. Arnal and G. Pinczon, Idéaux à gauche dans les quotients simples de l'algèbre enveloppante de sl(2), Bull. Soc. Math. France 101 (1973), 381-395. Zbl0357.17008
3. [Bam] K. S. Bamba, Sur les idéaux maximaux de l’algèbre de Weyl $A_1$, C. R. Acad. Sci. Paris (A) 283 (1976), 71-74. 
4. [BVO] V. V. Bavula, and F. van Oystaeyen, The simple modules of certain generalized crossed products, Trans. Amer. Math. Soc. (to appear). Zbl0927.16002
5. [Bav 1] V. V. Bavula, The finite-dimensionality of Ext${}^n$’s and Tor${}_n$’s of simple modules over a class of algebras, Funktsional. Anal. i Prilozhen. 25 (1991),no. 3, 80-82. 
6. [Bav 2] V. V. Bavula, The simple D[X,Y;σ,a]-modules, Ukrainian Math. J. 44 (1992), 1628-1644. Zbl0810.16003
7. [Bav 3] V. V. Bavula, Generalized Weyl algebras, kernel and tensor-simple algebras, their simple modules, Representations of algebras. Sixth International Conference, August 19-22, 1992. CMS Conference proceedings (V. Dlab and H. Lenzing Eds.), v. 14 (1993), 83-106. Zbl0806.17023
8. [Bav 4] V. V. Bavula, Generalized Weyl algebras and their representations, Algebra i Analiz, 4 (1992), no. 1, 75-97; English trans. in St.Petersburg Math. J. 4 (1993), no. 1, 71-92. 
9. [Bav 5] V. V. Bavula, Tensor homological minimal algebras, global dimension of the tensor product of algebras and of generalized Weyl algebras, Bull. Sci. Math. 120 (1996), 293-335. Zbl0855.16010
10. [Bav 6] V. V. Bavula, Global dimension of generalized Weyl algebras. Proceedings of the 7th Int. Conf. on Represent. of Algebras, August 22-26, 1994. CMS Conference proceedings (R. Bautista, R. Martinez-Villa and J. A. de la Pena Eds), 18 (1996), 81-107. Zbl0857.16025
11. [Bl 1] R. E. Block, Classification of the irréducible representations of sl(2, C), Bull. Amer. Math. Soc., 1 (1979), 247-250. sl(2) and of the Weyl algebra, Adv. Math. 39 (1981), 69-110. 
12. [Bl 2] R. E. Block, The irreducible representations of the Weyl algebra $A_1$, in ’Séminaire d’Algèbre Paul Dubreil (Proceedings, Paris 1977-1978)’ (M. P. Malliavin, Ed.), Lecture Notes in Mathematics no. 740, pp. 69-79, Springer-Verlag, Berlin/New York, 1979. 
13. [Bl 3] R. E. Block, The irreducible representations of the Lie algebra sl(2) and of the Weyl algebra, Adv. Math. 39 (1981), 69-110. Zbl0454.17005
14. [Di 1] J. Dixmier, Représentations irréductibles des algèbres de Lie nilpotentes, An. Acad. Brasil. Ci. 35 (1963), 491-519. 
15. [Di 2] J. Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968), 209-242. 
16. [EL] A. Van den Essen and A. Levelt, An explicit description of all simple K[[X]][∂]-modules, Contemp. Math., 130 (1992), 121-131. Zbl0812.16037
17. [Jac] N. Jacobson, The Theory of Rings, Amer. Math. Soc., Providence, R. I., 1943. 
18. [Jor 1] D. A. Jordan, Krull and global dimension of certain iterated skew polynomial rings, Contemp. Math., 130 (1992), 201-213. Zbl0779.16011
19. [Jor 2] D. A. Jordan, Primitivity in skew Laurent polynomial rings and related rings, Math. Z., 213 (1993), 353-371. Zbl0797.16037
20. [Hod 1] T. J. Hodges, Noncommutative deformation of type-A Kleinian singularities, J. Algebra 161 (1993), no. 2, 271-290. Zbl0807.16029
21. [Le] F. W. Lemire, Existence of weight space decompositions for irreducible representations of simple Lie algebras, Canad. Math. Bull. 14 (1971), 113-115. Zbl0215.09502
22. [Ma] M.-P. Malliavin, L'algèbre d'Heisenberg quantique, C. R. Acad. Sci. Paris, Sér. 1, 317 (1993), 1099-1102. 
23. [MR] J. C. McConnell and J. C. Robson, Homomorphism and extensions of modules over certain polynomial rings, J. Algebra 26 (1973), 319-342. Zbl0266.16031
24. [Sm] S. P. Smith, A class of algebras similar to the enveloping algebra of sl(2), Trans. Amer. Math. Soc. 322 (1990), 285-314. 
25. [Sm 1] S. P. Smith, Quantum qroups: An introduction and survey for ring theoretists, in Noncommutative Rings (S.Montgomery and L.W.Small, Eds.) pp. 131-178, MSRI publ. 24, Springer-Verlag, Berlin (1992). 
26. [Za] C. Zachos, Elementary paradigms of quantum algebras, Contemporary Math. 134 (1992), 351-377. Zbl0784.17028