The Liu-Schulz example

Claus Michael Ringel

Abstract. Liu Shiping and Rainer Schulz have constructed a symmetric
algebra \Lambda = \Lambda(q) of dimension 8 and an indecomposable
\Lambda-module M such that all the syzygy modules \Omega^t M are
4-dimensional and pairwise non-ismorphic. In this way, they have
exhibited an Auslander-Reiten component which contains infinitely
many isomorphism classes of modules of dimension 4. The algebra
\Lambda(q) depends on a non-zero parameter q \in k, and the interesting
behaviour occurs when q is not a root of unity. On the other hand,
the case q = 1 yields the exterior algebra \Lambda(1) on the 
3-dimensional vector space k^3, thus we may consider \Lambda(q) as a 
sort of quantization of \Lambda(1). The purpose of this note is to
present a more detailed study of properties of this algebra \Lambda
= \Lambda(q) and some of the \Lambda-modules.