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Partial flag varieties and preprojective algebras

Christof Geiß, Bernard Leclerc, Jan Schröer (2008)

Annales de l’institut Fourier

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

Path coalgebras of profinite bound quivers, cotensor coalgebras of bound species and locally nilpotent representations

Daniel Simson (2007)

Colloquium Mathematicae

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 is the path coalgebra of a profinite bound quiver (Q,), and the category C-Comod...

Properties of G-atoms and full Galois covering reduction to stabilizers

Piotr Dowbor (2000)

Colloquium Mathematicae

Given a group G of k-linear automorphisms of a locally bounded k-category R it is proved that the endomorphism algebra of a G-atom B is a local semiprimary ring (Theorem 2.9); consequently, the injective -module is indecomposable (Corollary 3.1) and the socle of the tensor product functor 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 and )is full (resp. strictly full) is studied...

Quiver bialgebras and monoidal categories

Hua-Lin Huang, Blas Torrecillas (2013)

Colloquium Mathematicae

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...

Quiver varieties and the character ring of general linear groups over finite fields

Emmanuel Letellier (2013)

Journal of the European Mathematical Society

Given a tuple of irreducible characters of we define a star-shaped quiver together with a dimension vector . Assume that is generic. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to . The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied, we...

Rank additivity for quasi-tilted algebras of canonical type

Thomas Hübner (1998)

Colloquium Mathematicae

Given the category of coherent sheaves over a weighted projective line (of any representation type), the endomorphism ring 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...

Rational smoothness of varieties of representations for quivers of Dynkin type

Philippe Caldero, Ralf Schiffler (2004)

Annales de l’institut Fourier

We study the Zariski closures of orbits of representations of quivers of type , ou . 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.

Real representations of quivers

Lidia Hügeli, Sverre Smalø (1999)

Colloquium Mathematicae

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

Recent results on quiver sheaves

Andreas Laudin, Alexander Schmitt (2012)

Open Mathematics

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 • ← • → •.

Representation-directed algebras form an open scheme

Stanislaw Kasjan (2002)

Colloquium Mathematicae

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.

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