Quivers and path algebras

Se basa en

Requisitos previos obligatorios, hasta el fondo: 29

Aprenderás a

• Compute the path algebra kQ of a small quiver and show that it is finite-dimensional exactly when Q is finite with no oriented cycles.
• Prove that representations of Q correspond to kQ-modules, giving an equivalence of categories.
• Classify the indecomposable representations of a linearly oriented quiver of type Aₙ.

Definición

• A quiver Q is a directed graph, possibly with loops and multiple arrows.
• The path algebra kQ has a basis consisting of all paths in Q, including a trivial path at each vertex, with product given by concatenation when the paths compose and zero otherwise.
• A representation of Q assigns a vector space to each vertex and a linear map to each arrow.

Teorema

• For a finite quiver, the category of representations of Q is equivalent to the category of kQ-modules.
• For a finite quiver Q, kQ is finite-dimensional if and only if Q has no oriented cycles.
• The path algebra of a finite acyclic quiver is hereditary: submodules of projective modules are projective.

Errores frecuentes

• Mixing conventions: texts compose paths left-to-right or right-to-left, which interchanges left and right modules.
• Treating a quiver as an undirected graph; orientation and multiple arrows matter.

Conduce a

Descubrimientos opcionales

• Categories The paths in a quiver form the free category generated by that quiver, and the path algebra is its linearisation.
• ← Graphs

Itinerarios de aprendizaje

• Research preparation

Lecturas complementarias

• Ralf Schiffler, Quiver Representations (2014)
• Ibrahim Assem, Daniel Simson and Andrzej Skowroński, Elements of the Representation Theory of Associative Algebras, Volume 1 (2006)