Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-25T07:45:23.679Z Has data issue: false hasContentIssue false
  • English
  • Français

Representations Subduced on an Ideal of a Lie Algebra

Published online by Cambridge University Press:  20 November 2018

B. Noonan
B. Noonan*
Affiliation:
Summer Research Institute Canadian Mathematical Congress
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 293 - 303
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Clifford, A. H., Representations induced on an invariant subgroup, Ann. Math., 88 (1937), 533–550.Google Scholar
2. Lomont, J. S., Applications of finite groups (Academic Press, 1959).Google Scholar
3. Magnus, W., Ueber Beziehungen zwischen hôheren Kommutatoren, J. Reine Angew. Math., 177 (1937) 105–115.Google Scholar
4. Magnus, W., Ueber Gruppen und zugeordnete Liesche Ringe, J. Reine Angew. Math., 182 (1940) 142–149.Google Scholar
5. van der Waerden, B. L., Gruppen von linearen Transformationen, Ergebnisse der Mathematik und ihrer Grenzgebiete, 4, no. 2 (Berlin, 1937).Google Scholar
6. van der Waerden, B. L., Modem algebra, II (Frederick Ungar Publishing Co., 1949).Google Scholar
7. 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
8. Zassenhaus, H., Ueoer Liesche Ringe mit Primzahlcharakteristik, Abh. Math. Sem. Hamburg, 13 (1939), 1–100.Google Scholar
9. Zassenhaus, H., The theory of groups (Chelsea Publishing Co., 1949).Google Scholar