Let $\Lambda$ be a preprojective algebra of type $A,D,E$, and let $G$ be the corresponding semisimple simply connected complex algebraic group. We study rigid modules in subcategories $\mathrm{Sub}Q$ for $Q$ an injective $\Lambda$-module, and we introduce a mutation operation between complete rigid modules in $\mathrm{Sub}Q$. This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to $G$.

We prove that the study of the category C-Comod of left comodules over a K-coalgebra C reduces to the study of K-linear representations of a quiver with relations if K is an algebraically closed field, and to the study of K-linear representations of a K-species with relations if K is a perfect field. Given a field K and a quiver Q = (Q₀,Q₁), we show that any subcoalgebra C of the path K-coalgebra K◻Q containing $K^{◻}Q₀\oplus K^{◻}Q₁$ is the path coalgebra $K^{◻}(Q,)$ of a profinite bound quiver (Q,), and the category C-Comod...

Given a group G of k-linear automorphisms of a locally bounded k-category R it is proved that the endomorphism algebra $En{d}_R\left(B\right)$ of a G-atom B is a local semiprimary ring (Theorem 2.9); consequently, the injective $En{d}_R\left(B\right)$-module ${\left(En{d}_R\left(B\right)\right)}^{*}$ is indecomposable (Corollary 3.1) and the socle of the tensor product functor $-{\otimes }_R{B}^{*}$ is simple (Theorem 4.4). The problem when the Galois covering reduction to stabilizers with respect to a set U of periodic G-atoms (defined by the functors ${\Phi }^U:{\coprod }_{B\in U}modk{G}_B\to mod(R/G)$ and ${\Psi }^U:mod(R/G)\to {\prod }_{B\in U}modk{G}_B$)is full (resp. strictly full) is studied...