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Schubert varieties and representations of Dynkin quivers

Grzegorz Bobiński, Grzegorz Zwara (2002)

Colloquium Mathematicae

We show that the types of singularities of Schubert varieties in the flag varieties Flagₙ, n ∈ ℕ, are equivalent to the types of singularities of orbit closures for the representations of Dynkin quivers of type 𝔸. Similarly, we prove that the types of singularities of Schubert varieties in products of Grassmannians Grass(n,a) × Grass(n,b), a, b, n ∈ ℕ, a, b ≤ n, are equivalent to the types of singularities of orbit closures for the representations of Dynkin quivers of type 𝔻. We also show that...

Selfinjective algebras of euclidean type with almost regular nonperiodic Auslander-Reiten components

Grzegorz Bobiński, Andrzej Skowroński (2001)

Colloquium Mathematicae

We give a complete description of finite-dimensional selfinjective algebras of Euclidean tilted type over an algebraically closed field whose all nonperiodic Auslander-Reiten components are almost regular. In particular, we describe the tame selfinjective finite-dimensional algebras whose all nonperiodic Auslander-Reiten components are almost regular and generalized standard.

Selfinjective algebras of strictly canonical type

Marta Kwiecień, Andrzej Skowroński (2009)

Colloquium Mathematicae

We develop the representation theory of selfinjective algebras of strictly canonical type and prove that their Auslander-Reiten quivers admit quasi-tubes maximally saturated by simple and projective modules.

Selfinjective algebras of tubular type

Jerzy Białkowski, Andrzej Skowroński (2002)

Colloquium Mathematicae

We classify all tame self/injective algebras having simply connected Galois coverings and the stable Auslander-Reiten quivers consisting of stable tubes. Moreover, the classification of nondomestic polynomial growth standard self/injective algebras is completed.

Selfinjective algebras of wild canonical type

Helmut Lenzing, Andrzej Skowroński (2003)

Colloquium Mathematicae

We develop the representation theory of selfinjective algebras which admit Galois coverings by the repetitive algebras of algebras whose derived category of bounded complexes of finite-dimensional modules is equivalent to the derived category of coherent sheaves on a weighted projective line with virtual genus greater than one.

Simply connected right multipeak algebras and the separation property

Stanisław Kasjan (1999)

Colloquium Mathematicae

Let R=k(Q,I) be a finite-dimensional algebra over a field k determined by a bound quiver (Q,I). We show that if R is a simply connected right multipeak algebra which is chord-free and ˜ -free in the sense defined below then R has the separation property and there exists a preprojective component of the Auslander-Reiten quiver of the category prin(R) of prinjective R-modules. As a consequence we get in 4.6 a criterion for finite representation type of prin(R) in terms of the prinjective Tits quadratic...

Sincere posets of finite prinjective type with three maximal elements and their sincere prinjective representations

Justyna Kosakowska (2002)

Colloquium Mathematicae

Assume that K is an arbitrary field. Let (I,⪯) be a poset of finite prinjective type and let KI be the incidence K-algebra of I. A classification of all sincere posets of finite prinjective type with three maximal elements is given in Theorem 2.1. A complete list of such posets consisting of 90 diagrams is presented in Tables 2.2. Moreover, given any sincere poset I of finite prinjective type with three maximal elements, a complete set of pairwise non-isomorphic sincere indecomposable prinjective...

Singularity categories of skewed-gentle algebras

Xinhong Chen, Ming Lu (2015)

Colloquium Mathematicae

Let K be an algebraically closed field. Let (Q,Sp,I) be a skewed-gentle triple, and let ( Q s g , I s g ) and ( Q g , I g ) be the corresponding skewed-gentle pair and the associated gentle pair, respectively. We prove that the skewed-gentle algebra K Q s g / ⟨ I s g ⟩ is singularity equivalent to KQ/⟨I⟩. Moreover, we use (Q,Sp,I) to describe the singularity category of K Q g / ⟨ I g ⟩ . As a corollary, we find that g l d i m K Q s g / ⟨ I s g ⟩ < ∞ if and only if g l d i m K Q / ⟨ I ⟩ < ∞ if and only if g l d i m K Q g / ⟨ I g ⟩ < ∞ .

Slice modules over minimal 2-fundamental algebras

Zygmunt Pogorzały, Karolina Szmyt (2007)

Open Mathematics

We consider a class of algebras whose Auslander-Reiten quivers have starting components that are not generalized standard. For these components we introduce a generalization of a slice and show that only in finitely many cases (up to isomorphism) a slice module is a tilting module.

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