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Abstract

Given a group $G$ of $k$-linear automorphisms of a locally bounded $k$-category $R$, the problem of existence and construction of non-orbicular indecomposable $R/G$-modules is studied. For a suitable finite sequence $B$ of $G$-atoms with a common stabilizer $H$, a representation embedding ${\mit \Phi }^{B} : \mathop {I_n\hbox {\rm-spr}}\nolimits (H)\to \mathop {\hbox {mod}}(R/G)$, which yields large families of non-orbicular indecomposable $R/G$-modules, is constructed (Theorem 3.1). It is proved that if a $G$-atom $B$ with infinite cyclic stabilizer admits a non-trivial left Kan extension $\widetilde {\! B}$ with the same stabilizer, then usually the subcategory of non-orbicular indecomposables in $\mathop {\hbox {mod}}_{\{ \widetilde {B},B\} }(R/G)$ is wild (Theorem 4.1, also 4.5). The analogous problem for the case of different stabilizers is discussed in Theorem 5.5. It is also shown that if $R$ is tame then $\widetilde {B}\simeq B$ for any infinite $G$-atom $B$ with $\mathop {\hbox {End}}_R(B)/J(\mathop {\hbox {End}}_R(B)) \simeq k$ (Theorem 7.1). For this purpose the techniques of neighbourhoods (Theorem 7.2) and extension embeddings for matrix rings (Theorem 6.3) are developed.