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...

We study bialgebra structures on quiver coalgebras and monoidal structures on the categories of locally nilpotent and locally finite quiver representations. It is shown that the path coalgebra of an arbitrary quiver admits natural bialgebra structures. This endows the category of locally nilpotent and locally finite representations of an arbitrary quiver with natural monoidal structures from bialgebras. We also obtain theorems of Gabriel type for pointed bialgebras and hereditary finite pointed...

Given a tuple $({𝒳}_1,...,{𝒳}_k)$ of irreducible characters of $G{L}_n\left(F_q\right)$ we define a star-shaped quiver $\Gamma$ together with a dimension vector $v$. Assume that $({𝒳}_1,...,{𝒳}_k)$ is generic. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product ${𝒳}_1\otimes \cdots \otimes {𝒳}_k$ as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to $(\Gamma ,v)$. The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied, we...

Given the category $X$ of coherent sheaves over a weighted projective line $X=X(\lambda ,p)$ (of any representation type), the endomorphism ring $\Sigma =\left(𝒯\right)$ of an arbitrary tilting sheaf - which is by definition an almost concealed canonical algebra - is shown to satisfy a rank additivity property (Theorem 3.2). Moreover, this property extends to the representationinfinite quasi-tilted algebras of canonical type (Theorem 4.2). Finally, it is demonstrated that rank additivity does not generalize to the case of tilting complexes...

We study the Zariski closures of orbits of representations of quivers of type $A$, $D$ ou $E$. With the help of Lusztig’s canonical base, we characterize the rationally smooth orbit closures and prove in particular that orbit closures are smooth if and only if they are rationally smooth.

The Dynkin and the extended Dynkin graphs are characterized by representations over the real numbers.

In this article, we survey recent work on the construction and geometry of representations of a quiver in the category of coherent sheaves on a projective algebraic manifold. We will also prove new results in the case of the quiver • ← • → •.

We apply van den Dries's test to the class of algebras (over algebraically closed fields) which are not representation-directed and prove that this class is axiomatizable by a positive quantifier-free formula. It follows that the representation-directed algebras form an open ℤ-scheme.