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The Construction of Representations of Lie Algebras of Characteristic Zero

Published online by Cambridge University Press:  20 November 2018

B. Noonan
B. Noonan*
Affiliation:
University of Manitoba
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In this paper a procedure is given whereby, from a representation of an ideal contained in the radical, explicit representations of a Lie algebra by matrices can be constructed in an algebraically closed field of characteristic zero. The construction is sufficiently general to permit one arbitrary eigenvalue to be assigned to the representation of each basis element of the radical not in the ideal. The theorem of Ado is proved as an application of the construction. While Ado's theorem has several proofs (1; 3; 5; 6), the present one has a value in its explicitness and in the fact that the degree of the representation can be given.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 304 - 312
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Ado, I. D., The representation of Lie algebras by matrices, Uspehi, Matem. Nauk (N.S.), 2, no. 6(22) (1947). (An improved version of the proof given in Bull. Soc. Phys.-Math. Kazan (1934-35)3-43.)Google Scholar
2. Birkhoff, G., Representability of Lie algebras and Lie groups by matrices, Ann. Math., 88 (1937), 526–532.Google Scholar
3. Cartan, E., Les représentations linéaires des groupes de Lie, J. Math. Pures Appl., 17 (1938), 1–12.Google Scholar
4. Dynkin, E. B., The structure of semi-simple algebras, Uspehi, Matem. Nauk (N.S.), 2, no. 4 (1947), 59–127 (Amer. Math. Soc. translation no. 17).Google Scholar
5. Harish-Chandra, , Faithful representations of Lie algebras, Ann. Math., 50 (1949), 68–76.Google Scholar
6. Hochschild, G., Lie algebras and differentiations in rings of power series, Amer. J. Math., 70 (1950), 58–80.Google Scholar
7. Jacobson, N., Completely reducible Lie algebras of linear transformations, Proc. Amer. Math. Soc, 2 (1951), 105–113.Google Scholar
8. Noonan, B., Representations subduced on an ideal of a Lie algebra, Can. J. Math., 14 (1962), 293–303.Google Scholar
9. Whitehead, J. H. C., Proc. Camb. Phil. Soc, 82 (1936), 229–237.Google Scholar
10. Witt, E., Treue Darstellung Liescher Ringe, J. Reine Angew. Math., 177 (1937), 152–160.Google Scholar
11. Zassenhaus, H., Darstellungstheorie nilpotenter Lie-Ringe bei Charakteristik p > 0, J. Reine Angew. Math., 182 (1940), 150–155.+0,+J.+Reine+Angew.+Math.,+182+(1940),+150–155.>Google Scholar
12. Zassenhaus, H., Ueber Liesche Ringe mit Primzahlcharakteristik, Abh. Math. Sem. Hamburg 13 (1939), 1–100.Google Scholar