Wyniki wyszukiwania

1

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

100%

Dowbor P., Mróz A.

Colloquium Mathematicum

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2007

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tom 107

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nr 2

221-261

EN

Given a module M over an algebra Λ and a complete set 𝓧 of pairwise nonisomorphic indecomposable Λ-modules, the problem of determining the vector $m(M) = (m_X)_{X∈ 𝓧} ∈ ℕ ^{𝓧}$ such that $M ≅ ⨁ _{X∈ 𝓧} X^{m_X}$ 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 $𝔸̃_{p,q}$. In the second case detailed algorithms are given (Algorithms 4.5 and 5.5).

2

On a separation of orbits in the module variety for domestic canonical algebras

100%

Dowbor P., Mróz A.

Colloquium Mathematicum

|

2008

|

tom 111

|

nr 2

283-295

EN

Given a pair M,M' of finite-dimensional modules over a domestic canonical algebra Λ, we give a fully verifiable criterion, in terms of a finite set of simple linear algebra invariants, deciding if M and M' lie in the same orbit in the module variety, or equivalently, if M and M' are isomorphic.

3

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

100%

Dowbor P., Mróz A.

Colloquium Mathematicum

|

2008

|

tom 111

|

nr 2

221-282

EN

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 $dim_{k} Hom_{Λ}(M,X)$, 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. Math. 107 (2007) leads to algorithms with the complexity $𝒪((dim_{k} M)⁴)$. A precise description of algorithms determining the multiplicities $m(M)_{x}$ for postprojective roots x ∈ X is given (Algorithms 6.1, 6.2 and 6.3).

Ograniczanie wyników

3 Colloquium Mathematicum

3 Dowbor P.

3 Mróz A.

2 2008

1 2007

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

On a separation of orbits in the module variety for domestic canonical algebras

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