Unzerlegbare Darstellungen I

Gabriel, Peter

doi:10.1007/bf01298413

Cited by 745 publications

(615 citation statements)

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“…A quiver is, by definition, a directed graph. A representation of the quiver is given (see [6], and also [2,14]) by assigning to each vertex a vector space and to each arrow a linear mapping of the corresponding vector spaces. For example, the quivers 1 e e 1 * * 4 4 2 1 9 9 e e correspond respectively to the problems of classifying:…”

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confidence: 99%

Classification Problems for Systems of Forms and Linear Mappings

Sergeichuk¹

1988

Math. USSR Izv.

210

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A method is proposed that allow the reduction of many classification problems of linear algebra to the problem of classifying Hermitian forms. Over the complex, real, and rational numbers classifications are obtained for bilinear forms, pairs of quadratic forms, isometric operators, and selfadjoint operators.Many problems of linear algebra can be formulated as problems of classifying the representations of a quiver. A quiver is, by definition, a directed graph. A representation of the quiver is given (see [6], and also [2,14]) by assigning to each vertex a vector space and to each arrow a linear mapping of the corresponding vector spaces. For example, the quivers 1 e e 1 * * 4 4 2 1 9 9 e e correspond respectively to the problems of classifying:• linear operators (whose solution is the Jordan or Frobenius normal form),• pairs of linear mappings from one space to another (the matrix pencil problem, solved by Kronecker), and

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confidence: 99%

Classification Problems for Systems of Forms and Linear Mappings

Sergeichuk¹

1988

Math. USSR Izv.

210

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“…It is finite-dimensional iff the quiver has no oriented cycles. Gabriel [24] has shown that the quiver algebra of a finite quiver has only a finite number of k-finite-dimensional indecomposable modules (up to isomorphism) iff the underlying graph of the quiver is a disjoint union of Dynkin diagrams of type A, D, E.…”

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confidence: 99%

Derived categories and tilting

Keller¹

2007

Handbook of Tilting Theory

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Abstract. We review the basic definitions of derived categories and derived functors. We illustrate them on simple but non trivial examples. Then we explain Happel's theorem which states that each tilting triple yields an equivalence between derived categories. We establish its link with Rickard's theorem which characterizes derived equivalent algebras. We then examine invariants under derived equivalences. Using t-structures we compare two abelian categories having equivalent derived categories. Finally, we briefly sketch a generalization of the tilting setup to differential graded algebras.

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“…An algebra A is of finite representation type if the category ind A admits only a finite number of pairwise nonisomorphic modules. It is well known that a hereditary algebra A is of finite representation type if and only if A is of Dynkin type, that is, the valued quiver Q A of A is a Dynkin quiver of type A n (n 1), B n (n 2), C n (n 3), D n (n 4), E 6 , E 7 , E 8 , F 4 and G 2 (see [8], [9], [10]). A distinguished class of algebras of finite representation type is formed by the tilted algebras of Dynkin type, that is, the algebras of the form End H (T ) for a hereditary algebra H of Dynkin type and a (multiplicity-free) tilting module T in mod H. An algebra A is called selfinjective if A A is an injective module, or equivalently, the projective modules in mod A are injective.…”

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confidence: 99%