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The multiplicity problem for indecomposable decompositions of modules over domestic canonical algebras

Piotr Dowbor, Andrzej Mróz (2008)

Colloquium Mathematicae

Given a module M over a domestic canonical algebra Λ and a classifying set X for the indecomposable Λ-modules, the problem of determining the vector m ( M ) = ( m x ) x ∈ X ∈ ℕ X such that M ≅ ⨁ x ∈ X X x m x is studied. A precise formula for d i m k H o m Λ ( M , X ) , for any postprojective indecomposable module X, is computed in Theorem 2.3, and interrelations between various structures on the set of all postprojective roots are described in Theorem 2.4. It is proved in Theorem 2.2 that a general method of finding vectors m(M) presented by the authors in Colloq....

The multiplicity problem for indecomposable decompositions of modules over a finite-dimensional algebra. Algorithms and a computer algebra approach

Piotr Dowbor, Andrzej Mróz (2007)

Colloquium Mathematicae

Given a module M over an algebra Λ and a complete set of pairwise nonisomorphic indecomposable Λ-modules, the problem of determining the vector m ( M ) = ( m X ) X ∈ ∈ ℕ such that M ≅ ⨁ X ∈ X m X is studied. A general method of finding the vectors m(M) is presented (Corollary 2.1, Theorem 2.2 and Corollary 2.3). It is discussed and applied in practice for two classes of algebras: string algebras of finite representation type and hereditary algebras of type ̃ p , q . In the second case detailed algorithms are given (Algorithms 4.5 and 5.5).

The nil radical of an Archimedean partially ordered ring with positive squares

Boris Lavrič (1994)

Commentationes Mathematicae Universitatis Carolinae

Let R be an Archimedean partially ordered ring in which the square of every element is positive, and N ( R ) the set of all nilpotent elements of R . It is shown that N ( R ) is the unique nil radical of R , and that N ( R ) is locally nilpotent and even nilpotent with exponent at most 3 when R is 2-torsion-free. R is without non-zero nilpotents if and only if it is 2-torsion-free and has zero annihilator. The results are applied on partially ordered rings in which every element a is expressed as a = a 1 - a 2 with positive a 1 ,...

The number of complete exceptional sequences for a Dynkin algebra

Mustafa Obaid, Khalid Nauman, Wafa S. M. Al-Shammakh, Wafaa Fakieh, Claus Michael Ringel (2013)

Colloquium Mathematicae

The Dynkin algebras are the hereditary artin algebras of finite representation type. The paper determines the number of complete exceptional sequences for any Dynkin algebra. Since the complete exceptional sequences for a Dynkin algebra of Dynkin type Δ correspond bijectively to the maximal chains in the lattice of non-crossing partitions of type Δ, the calculations presented here may also be considered as a categorification of the corresponding result for non-crossing partitions.

The periodicity conjecture for blocks of group algebras

Karin Erdmann, Andrzej Skowroński (2015)

Colloquium Mathematicae

We describe the representation-infinite blocks B of the group algebras KG of finite groups G over algebraically closed fields K for which all simple modules are periodic with respect to the action of the syzygy operators. In particular, we prove that all such blocks B are periodic algebras of period 4. This confirms the periodicity conjecture for blocks of group algebras.

Currently displaying 3061 – 3080 of 3997