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The structure of the Lie field connected with a split semisimple Lie algebra. (English. Russian original) Zbl 0244.17007

Funct. Anal. Appl. 3, 6-21 (1969); translation from Funkts. Anal. Prilozh. 3, No. 1, 7-26 (1969).

MSC:

17B20 Simple, semisimple, reductive (super)algebras
17B35 Universal enveloping (super)algebras
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References:

[1] I. M. Gel’fand and A. A. Kirillov, ”On fields connected with enveloping Lie algebras,” Dokl. Akad. Nauk SSSR,167, No. 3, 503-506 (1966).
[2] I. M. Gel’fand and M. I. Graev, ”The Fourier transformation of rapidly decreasing functions on complex semisimple groups,” Dokl. Akad. Nauk SSSR,131, No. 3, 496-499 (1960).
[3] B. Konstant, ”Lie group representations on polynomial rings,” Amer. J. Math.,85, No. 3, 327-404 (1963). · Zbl 0124.26802
[4] I. M. Gel’fand and A. A. Kirillov, ”On the structure of the quotient field of the enveloping algebra of a semisimple Lie algebra,” Dokl. Akad. Nauk SSSR,180, No. 4, 775-777 (1961).
[5] A. Borel and J. Tits, ”Groupes reductifs,” Publs. Math. Inst. des Hautes-Etudes Sci.,27, 55-151 (1965). · Zbl 0145.17402
[6] J. Tits, ”Classification of algebraic semisimple groups,” Symposium,Boulder, Colorado (1965). · Zbl 0238.20052
[7] The Theory of Lie Algebras. The Topology of Lie Groups [Russian translation], IL, Moscow (1962).
[8] A. Grothendieck, ”Eléments de géometrie algébrique. IV (Troisiéme partie),” Publs. Math. Inst. des Hautes-Etudes Sci.,28, 1-248 (1966).
[9] N. Jacobson, Lie Algebras, Interscience, New York (1962).
[10] Harish-Chandra, ”On some applications of the universal enveloping algebra of a semisimple Lie algebra,” Trans. Amer. Math. Soc.,70, 28-96 (1951). · Zbl 0042.12701
[11] C. Chevalley, ”Invariants of finite groups generated by reflections,” Amer. J. Math.,77, No. 4, 778-782 (1955). · Zbl 0065.26103
[12] I. M. Gel’fand and V. A. Ponomarev, ”The category of Harish-Chandra modules over the Lie algebra of the Lorentz group,” Dokl. Akad. Nauk SSSR,176, No. 2, 243-246 (1967). [For more detail see: Uspekhi Mat. Nauk,23, No. 2, 3-60 (1968)].
[13] I. M. Gel’fand and M. I. Graev, ”The geometry of homogeneous spaces, the representation of groups in homogeneous spaces, and related problems of integral geometry,” Trudy Mosk. Matem. Obshch.,8, 321-390 (1959).
[14] I. M. Gel’fand, M. I. Graev, and I. I. Pyatetski-Shapiro, Generalized Functions. Vol. 6. The Theory of Representations and Automorphic Functions [in Russian], ”Nauka,” Moscow (1966).
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