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Representation Theory

In: The Concise Handbook of Algebra

Author

Listed:
  • Donald S. Passman
  • I. Martin Isaacs
  • Albert Fässler
  • Peter Fleischmann
  • Joachim Hilgert
  • Kenichi Kanatani
  • Günter F. Pilz

Abstract

If R is a ring and G is a multiplicative group, then we let R[G] denote the group ring of G over R. Thus R[G] is a free R-module with basis {x | x ∈ G} and with multiplication defined distributively using the group multiplication in G. When R = K is a field, then K[G] is the group algebra of G over K, and our main concern here is with group algebras of infinite groups. Nevertheless, we are frequently forced to deal with more general objects such as twisted group rings, skew group rings, crossed products and group-graded rings. In the following, we discuss a selection of fairly well-developed topics which relate the group structure of G to the ring-theoretic structure of K[G]. Basic references for this material include the books by Karpilovsky (1987, 1989), Passi (1979), Passman (1971, 1977, 1989), and Sehgal (1978, 1993), and the papers by Passman (1984, 1998), Roseblade (1978), and Zalesskiĭ (1995).

Suggested Citation

  • Donald S. Passman & I. Martin Isaacs & Albert Fässler & Peter Fleischmann & Joachim Hilgert & Kenichi Kanatani & Günter F. Pilz, 2002. "Representation Theory," Springer Books, in: The Concise Handbook of Algebra, chapter 0, pages 383-415, Springer.
    • Handle: RePEc:spr:sprchp:978-94-017-3267-3_5
    DOI: 10.1007/978-94-017-3267-3_5
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