Azumaya rings and Maschke's Theorem

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.

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On the Structure of Finite Groupoids and Their Representations

Symmetry ◽

10.3390/sym11030414 ◽

2019 ◽

Vol 11 (3) ◽

pp. 414 ◽

Cited By ~ 7

Author(s):

Alberto Ibort ◽

Miguel Rodríguez

Keyword(s):

Irreducible Representation ◽

Regular Representation ◽

Irreducible Representations ◽

Quantum Mechanical ◽

Linear Representations ◽

Finite Dimensional ◽

Completely Reducible ◽

Quantum Mechanical Systems ◽

Maschke’S Theorem ◽

Totally Disconnected

In this paper, both the structure and the theory of representations of finite groupoids are discussed. A finite connected groupoid turns out to be an extension of the groupoids of pairs of its set of units by its canonical totally disconnected isotropy subgroupoid. An extension of Maschke’s theorem for groups is proved showing that the algebra of a finite groupoid is semisimple and all finite-dimensional linear representations of finite groupoids are completely reducible. The theory of characters for finite-dimensional representations of finite groupoids is developed and it is shown that irreducible representations of the groupoid are in one-to-one correspondence with irreducible representation of its isotropy groups, with an extension of Burnside’s theorem describing the decomposition of the regular representation of a finite groupoid. Some simple examples illustrating these results are exhibited with emphasis on the groupoids interpretation of Schwinger’s description of quantum mechanical systems.

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