THE FINITE LENGTH MODULES FOR THIN -GRADED RINGS

Abstract

A -graded ring R = is called thin, provided R ₀ = K is a skew field and each K–K-bimodule R_i has left and right dimension at most 1. Such rings occur, for example, as fiber products of skew polynomial rings, as numerical skew monoid rings or as factor rings or localizations of generalized Weyl algebras. We show that in case R is (graded) affine with no homogeneous units in nonzero degree, the representation type (finite, “twisted tame infinite,” “twisted wild”) can be read off from the combinatorial structure, that is, from the vanishing of products of homogeneous elements of certain degrees. Those rings R which are of twisted wild representation type admit by our definition a representation embedding from the category of finite dimensional modules over some twisted wild algebra T(K, _σ K ⊕_τ K) into the category mod R of finite length R-modules. The rings which we call here “twisted tame” come close to twisted versions of string algebras, for which we obtain the classification of the indecomposable finite length modules as a version of the Butler–Ringel theorem. Finally, for rings which are either not affine or which have homogeneous units in nonzero degree, the representation type does not depend on the combinatorial structure alone, as examples show.

Keywords:

• Z-graded rings
• Generalized Weyl algebras
• Dedekind like rings
• String algebras
• Butler-Ringel theorem
• Tame and wild representation type

ACKNOWLEDGMENTS

The author thanks F. Van Oystaeyen for encouragement and helpful discussions. He is grateful for the opportunity to present his results in the semiprime case to the International Conference on the Representation Theory of Algebras in September 1998 in Bielefeld. During the time of this research, the author was first employed at the Charles University in Prague and has been later supported by the European TMR-project “Algebraic Lie Representations.”