Mathematics – Rings and Algebras
Scientific paper
2010-09-24
Mathematics
Rings and Algebras
32 pages, no figures, minor improvements of the presentation of the material
Scientific paper
In this paper we show that each non-zero ideal of a twisted generalized Weyl algebra (TGWA) $A$ intersects the centralizer of the distinguished subalgebra $R$ in $A$ non-trivially. We also provide a necessary and sufficient condition for the centralizer of $R$ in $A$ to be commutative, and give examples of TGWAs associated to symmetric Cartan matrices satisfying this condition. By imposing a certain finiteness condition on $R$ (weaker than Noetherianity) we are able to make an Ore localization which turns out to be useful when investigating simplicity of the TGWA. Under this mild assumption we obtain necessary and sufficient conditions for the simplicity of TGWAs. We describe how this is related to maximal commutativity of $R$ in $A$ and the (non-) existence of non-trivial $\Z^n$-invariant ideals of $R$. Our result is a generalization of the rank one case, obtained by D. A. Jordan in 1993. We illustrate our theorems by considering some special classes of TGWAs and providing concrete examples.
Hartwig Jonas T.
Öinert Johan
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