The Simple Modules of Certain Generalized Crossed Products

Abstract We consider a Dedekind domainDand aZ-graded ring R = ;⊕ i ∈ Z R i withR0 = Dand eachRi = Dvibeing a freeD-module of rank 1. The structure ofRis described by an automorphism ofDand a generalized 2-cocyclec : Z × Z → Dnot necessarily taking its values in the units ofD. The aim of this paper is to classify the simpleR-modules, say in the case wherec(i, j) ≠ 0 for alli, j ∈ Z. We also deal with this problem in theN-graded case. As a consequence we obtain a description of the simple modules of some classical algebras and of generalized Weyl algebras.

[1]  V. Bavula.  Tensor homological minimal algebras, global dimension of the tensor product of algebras and of generalized Weyl algebras , 1996 .

[2]  T. Hodges.  Noncommutative Deformations of Type-A Kleinian Singularities , 1993 .

[3]  D. Jordan.  Primitivity in skew Laurent polynomial rings and related rings , 1993 .

[4]  M. Malliavin.  L'algèbre d'Heisenberg quantique , 1993 .

[5]  A. Rosenberg.  The unitary irreducible representations of the quantum heisenberg algebra , 1992 .

[6]  C. K. Zachos.  PARADIGMS OF QUANTUM ALGEBRAS , 1991 .

[7]  S. P. Smith.  A Class of Algebras Similar to the Enveloping Algebra of sl(2) , 1990 .

[8]  S. P. Smith.  A class of algebras similar to the enveloping algebra of (2) , 1990 .

[9]  R. Block.  The irreducible representations of the Lie algebra sl(2) and of the Weyl algebra , 1981 .

[10]  R. Block.  Classification of the irreducible representations of $\mathfrak{s}\mathfrak{l}\left( {2,{\text{C}}} \right)$ , 1979 .

[11]  R. Block.  The irreducible representations of the Weyl algebra A1 , 1979 .

[12]  M. Perroud.  On the irreducible representations of the Lie algebra chain G2⊇A2 , 1976 .

[13]  D. Arnal,et al.  On algebraically irreducible representations of the Lie algebra sl(2) , 1974 .

[14]  J. C. McConnell,et al.  Homomorphisms and extensions of modules over certain differential polynomial rings , 1973 .

[15]  D. Arnal,et al.  Idéaux à gauche dans les quotients simples de l'algèbre enveloppante de $sl(2)$ , 1973 .

[16]  F. Lemire.  Existence of Weight Space Decompositions for Irreducible Representations of Simple Lie Algebras , 1971, Canadian Mathematical Bulletin.

[17]  J. Dixmier.  Sur les algèbres de Weyl , 1968 .

[18]  Nathan Jacobson,et al.  Theory of rings , 1943 .