Document Zbl 0454.17005

The irreducible representations of the Lie algebra sl(2) and of the Weyl algebra. (English) Zbl 0454.17005

Adv. Math. 39, 69-110 (1981).

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MSC:

17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B20	Simple, semisimple, reductive (super)algebras
17B30	Solvable, nilpotent (super)algebras
16D30	Infinite-dimensional simple rings (except as in 16Kxx)
16P40	Noetherian rings and modules (associative rings and algebras)

Keywords:

Heisenberg algebra; classification of irreducible representation; nonabelian Lie algebra

Citations:

Zbl 0335.16033; Zbl 0143.053; Zbl 0165.04901; Zbl 0202.043; Zbl 0266.16031; Zbl 0249.17005; Zbl 0298.17003; Zbl 0357.17008; Zbl 0215.095; Zbl 0132.296; Zbl 0174.105; Zbl 0414.16017; Zbl 0412.17009; Zbl 0308.17007

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References:

[1]	Amitsur, S., Differential polynomials and division algebras, Ann. of Math., 59, 245-278 (1954) · Zbl 0055.02901
[2]	Arnal, D.; Pinczon, G., Sur certaines représentations de l’algèbre de Lie sl(2), C.R. Acad. Sci. Paris, 272, A, 1369-1372 (1971) · Zbl 0249.17005
[3]	Arnal, D.; Pinczon, G., On algebraically irreducible representations of the Lie algebra sl(2), J. Math. Phys., 15, 350-359 (1974) · Zbl 0298.17003
[4]	Arnal, D.; Pinczon, G., Idéaux à gauche dans les quotients simple de l’algèbre enveloppante de sl(2), Bull. Soc. Math. France, 101, 381-395 (1973) · Zbl 0357.17008
[5]	Bamba, K. S., Sur les idéaux maximaux de l’algèbre de Weyl \(A_1\), C.R. Acad. Sci. Paris, 283, A, 71-74 (1976) · Zbl 0335.16033
[6]	Bamba, K. S., Sur les Modules Simples de L’Algèbre de Weyl \(A_1\), (Thesis (3rd Cycle) (1977), University of Paris)
[7]	Block, R. E., Determination of the differentiably simple rings with a minimal ideal, Ann. of Math., 90, 433-459 (1969) · Zbl 0216.07303
[8]	Block, R. E., Classification of the irreducible representations of \(sl\)(2, C), Bull. Amer. Math. Soc. (N.S.), 1, 247-250 (1979) · Zbl 0412.17009
[9]	Block, R. E., The irreducible representations of the Weyl algebra \(A_1\), (Malliavin, M. P., Séminaire d’Algèbre Paul Dubreil (Proceedings, Paris 1977-1978). Séminaire d’Algèbre Paul Dubreil (Proceedings, Paris 1977-1978), Lecture Notes in Mathematics, No. 740 (1979), Springer-Verlag: Springer-Verlag Berlin/New York), 69-79 · Zbl 0414.16017
[10]	Borho, W.; Gabriel, P.; Rentschler, R., Primideale in Einhüllenden auflösbarer Lie-Algebren, (Lecture Notes in Mathematics, No. 357 (1973), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0293.17005
[11]	Conze, N., Algèbres d’opérateurs différentiels et quotients des algèbres enveloppantes, Bull. Soc. Math. France, 102, 379-415 (1974) · Zbl 0298.17012
[12]	Dixmier, J., Représentations irréductibles des algèbres de Lie nilpotentes, An. Acad. Brasil. Ci., 35, 491-519 (1963) · Zbl 0143.05302
[13]	Dixmier, J., Sur les algèbres de Weyl, Bull. Soc. Math. France, 96, 209-242 (1968) · Zbl 0165.04901
[14]	Dixmier, J., Sur les algèbres de Weyl, II, Bull. Sci. Math., 94, 289-301 (1970) · Zbl 0202.04303
[15]	(Enveloping Algebras (1977), Gauthier-Villars: Gauthier-Villars Amsterdam), English transl. · Zbl 0308.17007
[16]	Dixmier, J., Quotients simples de l’algèbre enveloppante de \(sl_2\), J. Algebra, 24, 551-564 (1973) · Zbl 0252.17004
[17]	Jacobson, N., Pseudo-linear transformations, Ann. of Math., 38, 484-507 (1937) · JFM 63.0087.01
[18]	Jacobson, N., The Theory of Rings (1943), Amer. Math. Soc.,: Amer. Math. Soc., Providence, R.I. · Zbl 0060.07302
[19]	Kaplansky, I., Commutative Rings (1970), Allyn & Bacon: Allyn & Bacon Boston · Zbl 0203.34601
[20]	Kolchin, E. R., Algebraic matrix groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations, Ann. of Math., 49, 1-42 (1948) · Zbl 0037.18701
[21]	Kostant, B., On Whittaker vectors and representation theory, Invent. Math., 48, 101-184 (1978) · Zbl 0405.22013
[22]	Kovacic, J., An Eisenstein criterion for noncommutative polynomials, (Proc. Amer. Math. Soc., 34 (1972)), 25-29 · Zbl 0236.12106
[23]	Lemire, F. W., Existence of weight space decompositions for irreducible representations of simple Lie algebras, Canad. Math. Bull., 14, 113-115 (1971) · Zbl 0215.09502
[24]	McConnell, J. C.; Robson, J. C., Homomorphisms and extensions of modules over certain polynomial rings, J. Algebra, 26, 319-342 (1973) · Zbl 0266.16031
[25]	Miller, W., On Lie Algebras and Some Special Functions of Mathematical Physics, (Amer. Math. Soc. Memoir, 50 (1964), Amer. Math. Soc.,: Amer. Math. Soc., Providence, R.I.) · Zbl 0132.29602
[26]	Miller, W., Lie Theory and Special Functions (1968), Academic Press: Academic Press New York · Zbl 0174.10502
[27]	Nouazé, Y.; Gabriel, P., Idéaux premiers de l’algèbre enveloppante d’une algèbre de Lie nilpotente, J. Algebra, 6, 77-99 (1967) · Zbl 0159.04101
[28]	Ore, O., Theory of non-commutative polynomials, Ann. of Math., 34, 480-508 (1933) · JFM 59.0925.01
[29]	(Analogies Entre la Théorie des Équations Differentielles Linéaire et la Théorie des Équations Algebriques (1936), Academic Press: Academic Press Paris), Reprinted as · JFM 63.1031.03
[30]	Posner, E. C., Differentiably simple rings, (Proc. Amer. Math. Soc., 11 (1960)), 337-343 · Zbl 0103.26802
[31]	Sweedler, M. E., Hopf Algebras (1969), Benjamin: Benjamin New York · Zbl 0194.32901

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