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单词 PropertiesOfTheOrdinaryQuiver
释义

properties of the ordinary quiver


Let k be a field and A be a finite-dimensional algebra over k. Denote by QA the ordinary quiver (http://planetmath.org/OrdinaryQuiverOfAnAlgebra) of A.

Theorem. The following statements hold:

  1. 1.

    If A is basic and connected, then QA is a connected quiver.

  2. 2.

    If Q is a finite quiver and I is an admissible ideal (http://planetmath.org/AdmissibleIdealsBoundQuiverAndItsAlgebra) in kQ and A=kQ/I, then QA and Q are isomorphicPlanetmathPlanetmathPlanetmath.

  3. 3.

    If A is basic and connected, then A is isomorphic to kQA/I for some (not necessarily unique) admissible ideal (http://planetmath.org/AdmissibleIdealsBoundQuiverAndItsAlgebra) I.

For proofs please see [1, Chapter II.3].

References

  • 1 I. Assem, D. Simson, A. Skowronski, Elements of the Representation Theory of Associative Algebras, vol 1., Cambridge University Press 2006, 2007
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