Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T01:59:37.866Z Has data issue: false hasContentIssue false
  • English
  • Français

Induced Representations of Rings

Published online by Cambridge University Press:  20 November 2018

Marc A. Rieffel
Marc A. Rieffel*
Affiliation:
The University of California, Berkeley, California
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.

At the beginning of the chapter on induced representations in the treatise of Curtis and Reiner [8] on representation theory, they write “Most of the results have not yet found suitable generalization to rings with minimum condition or finite dimensional algebras, …”. The purpose of this paper is to indicate how some of the more basic theorems concerning induced representations can, in fact, be generalized to rings and algebras. In most cases we can do this by bringing together known results, so that in this sense this paper does not contain substantially new results.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 2 , 01 April 1975 , pp. 261 - 270
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Adams, W. W. and Rieffel, M. A., Adjoint functors and derived functors with an application to the cohomology of semigroups, J. Algebra 7 (1967), 25–34.Google Scholar
2. Bass, H., Algebraic K-theory (W. A. Benjamin, New York, 1968).Google Scholar
3. Blattner, R. J., Positive definite measures, Proc. Amer. Math. Soc. 14 (1963), 423–428.Google Scholar
4. Bourbaki, N., Algèbre, Ch. 3, Act. Sci. Ind. no. 10U (Hermann, Paris, 1958).Google Scholar
5. Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, Princeton, 1956).Google Scholar
6. Clifford, A. H., Representations induced in an invariant subgroup, Ann. of Math. 38 (1937), 533–550.Google Scholar
7. Cohn, P. M., Morita equivalence and duality, Queen Mary College Mathematical Notes, London, n.d.Google Scholar
8. Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (Interscience Publ., New York, 1962).Google Scholar
9. Dress, A., An intertwining number theorem for integral representations and applications, Math. Z. 116 (1970), 153–165.Google Scholar
10. Frobenius, G., Lilber Relationen zwischen den Characteren einer Gruppe und denen ihrer Untergruppen, Sitzber. Preuss. Akad. Wiss. (1898), 501–515.Google Scholar
11. Higman, D. G., Induced and produced modules, Can. J. Math. 7 (1955), 490–508.Google Scholar
12. Huppert, B., Endliche Gruppen, I (Springer-Verlag, Berlin, 1967).Google Scholar
13. Kasch, F., Grundlagen einer Théorie der Frobeniuserweiterungen, Math. Ann. 127 (1954), 453–474.Google Scholar
14. Lang, S., Rapport sur la cohomologie des groupes (W. A. Benjamin Inc., New York, 1966).Google Scholar
15. Levitzki, J., ùber vollstàndigreduzible Ringe und Unterringe, Math. Z. 33 (1931), 663–691.Google Scholar
16. Mackey, G. W., On induced reparesentations of groups, Amer. J. Math. 73 (1951), 576–592.Google Scholar
17. MacLane, S., Categories for the working mathematician (Springer-Verlag, New York, 1971).Google Scholar
18. Morita, K., Duality for modules and its applications to the theory of rings with minimum conditions, Tokyo Kyoiku Daigoku, Sect. A, 6 (1958), 83–142.Google Scholar
19. Morita, K., Adjoint pairs of functors and Frobenius extensions, Tokyo Kyoiku Daigaku, Sect. A, 9 (1965), 40–71.Google Scholar
20. Rieffel, M. A., Burnside's theorem for representations of Hopf Algebras, J. Algebra 6 (1967), 123–130.Google Scholar
21. Rieffel, M. A., Induced Banach representations of Banach algebras and locally compact groups, J. Functional Analysis 1 (1967), 443–491.Google Scholar
22. Rieffel, M. A., Unitary representations induced from compact subgroups, Studia Math. Jf.2 (1971), 145–175.Google Scholar
23. Rieffel, M. A., Induced representations of C*-algebras, Bull. Amer. Math. Soc. 78 (1972), 606–609.Google Scholar
24. Takesaki, M., Covariant representations of C*-algebras and their locally compact automorphism groups, Acta Math. 119 (1967), 273–303.Google Scholar
25. Weiss, E., Cohomology of groups (Academic Press, New York, 1969).Google Scholar
26. Weyl, H., Theory of groups and quantum mechanics (Princeton University Press, Princeton, 1931).Google Scholar