Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-31T14:54:42.719Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized Casimir Operators

Published online by Cambridge University Press:  20 November 2018

A. J. Douglas
A. J. Douglas*
Affiliation:
The University, Sheffield, England
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.

Throughout this paper, S will be a ring (not necessarily commutative) with an identity element ls ≠ 0s. We shall use R to denote a second ring, and ϕ: S→ R will be a fixed ring homomorphism for which ϕ1S = 1R.

In (7), Higman generalized the Casimir operator of classical theory and used his generalization to characterize relatively projective and injective modules. As a special case, he obtained a theorem which contains results of Eckmann (3) and of Higman himself (5), and which also includes Gaschütz's generalization (4) of Maschke's theorem. (For a discussion of some of the developments of Maschke's idea of averaging over a finite group, we refer the reader to (2, Chapter IX).) In the present paper, we define the Casimir operator of a family of S-homomorphisms of one R-module into another, and we again use this operator to characterize relatively projective and injective modules.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1496 - 1505
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Cartan, H. and Eilenberg, S., Homologuai algebra (Princeton Univ. Press, Princeton, N.J., 1956).Google Scholar
2. Curtis, C. W. and I. Reiner, Representation theory of finite groups and associative algebras (Interscience, New York, 1962).Google Scholar
3. Eckmann, B., On complexes with operators, Proc. Nat. Acad. Sci. U.S.A. 89 (1953), 35–42.Google Scholar
4. Gaschiïtz, W., Uber den Fundamentalsatz von Maschke zur Darstellungstheorie der endlichen Gruppen, Math. Z. 56 (1952), 376–387.Google Scholar
5. Higman, D. G., On modules with a group of operators, Duke Math. J. 21 (1954), 369–376.Google Scholar
6. Higman, D. G., Indecomposable representations of characteristic p, Duke Math. J. 21 (1954), 377–382.Google Scholar
7. Higman, D. G., Induced and produced modules. Can. J. Math. 7 (1955), 490–508 Google Scholar
8. Higman, D. G., Relative cohomology, Can. J. Math. 9 (1957), 19–34.Google Scholar
9. Popescu, N., Modules à différentielle généralisée, Rev. Roumaine Math. Pures Appl. 9 (1964), 549–559.Google Scholar