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A characterization of representation-finite algebras

Andrzej Skowroński, M. Wenderlich (1991)

Fundamenta Mathematicae

Let A be a finite-dimensional, basic, connected algebra over an algebraically closed field. Denote by Γ(A) the Auslander-Reiten quiver of A. We show that A is representation-finite if and only if Γ(A) has at most finitely many vertices lying on oriented cycles and finitely many orbits with respect to the action of the Auslander-Reiten translation.

A density theorem for algebra representations on the space (s)

W. Żelazko (1998)

Studia Mathematica

We show that an arbitrary irreducible representation T of a real or complex algebra on the F-space (s), or, more generally, on an arbitrary infinite (topological) product of the field of scalars, is totally irreducible, provided its commutant is trivial. This provides an affirmative solution to a problem of Fell and Doran for representations on these spaces.

A general form of non-Frobenius self-injective algebras

Andrzej Skowroński, Kunio Yamagata (2006)

Colloquium Mathematicae

Applying the classical work of Nakayama [Ann. of Math. 40 (1939)], we exhibit a general form of non-Frobenius self-injective finite-dimensional algebras over a field.

An explicit construction for the Happel functor

M. Barot, O. Mendoza (2006)

Colloquium Mathematicae

An easy explicit construction is given for a full and faithful functor from the bounded derived category of modules over an associative algebra A to the stable category of the repetitive algebra of A. This construction simplifies the one given by Happel.

Applications of spinor class fields: embeddings of orders and quaternionic lattices

Luis Arenas-Carmona (2003)

Annales de l'Institut Fourier

We extend the theory of spinor class fields and relative spinor class fields to study representation problems in several classical linear algebraic groups over number fields. We apply this theory to study the set of isomorphism classes of maximal orders of central simple algebras containing a given maximal Abelian suborder. We also study isometric embeddings of one skew-Hermitian Quaternionic lattice into another.

CB-degenerations and rigid degenerations of algebras

Adam Hajduk (2006)

Colloquium Mathematicae

The main aim of this note is to prove that if k is an algebraically closed field and a k-algebra A₀ is a CB-degeneration of a finite-dimensional k-algebra A₁, then there exists a factor algebra Ā₀ of A₀ of the same dimension as A₁ such that Ā₀ is a CB-degeneration of A₁. As a consequence, Ā₀ is a rigid degeneration of A₁, provided A₀ is basic.

Cohen-Macaulay modules over two-dimensional graph orders

Klaus Roggenkamp (1999)

Colloquium Mathematicae

For a split graph order ℒ over a complete local regular domain 𝒪 of dimension 2 the indecomposable Cohen-Macaulay modules decompose - up to irreducible projectives - into a union of the indecomposable Cohen-Macaulay modules over graph orders of type •—• . There, the Cohen-Macaulay modules filtered by irreducible Cohen-Macaulay modules are in bijection to the homomorphisms ϕ : 𝒪 L ( μ ) 𝒪 L ( ν ) under the bi-action of the groups ( G l ( μ , 𝒪 L ) , G l ( ν , 𝒪 L ) ) , where 𝒪 L = 𝒪 / π for a prime π. This problem strongly depends on the nature of 𝒪 L . If 𝒪 L is regular,...

Differentiation and splitting for lattices over orders

Wolfgang Rump (2001)

Colloquium Mathematicae

We extend our module-theoretic approach to Zavadskiĭ’s differentiation techniques in representation theory. Let R be a complete discrete valuation domain with quotient field K, and Λ an R-order in a finite-dimensional K-algebra. For a hereditary monomorphism u: P ↪ I of Λ-lattices we have an equivalence of quotient categories ̃ u : Λ - l a t / [ ] δ u Λ - l a t / [ B ] which generalizes Zavadskiĭ’s algorithms for posets and tiled orders, and Simson’s reduction algorithm for vector space categories. In this article we replace u by a more...

Directing components for quasitilted algebras

Flávio Coelho (1999)

Colloquium Mathematicae

We show here that a directing component of the Auslander-Reiten quiver of a quasitilted algebra is either postprojective or preinjective or a connecting component.

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