स्नातकोत्तर प्रथम वर्षAlgebraपाठ उपलब्ध
Quivers and path algebras
Quivers and their path algebras: paths as a basis, multiplication by concatenation, representations of quivers as modules over the path algebra, and simple, projective and indecomposable representations.
math.algebra.path-algebras
इस पर आधारित
29 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- 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.
परिभाषा
- 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.
प्रमेय
- 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.
आम भ्रांतियाँ
- 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.
आगे ले जाता है
वैकल्पिक खोजें
- Categories The paths in a quiver form the free category generated by that quiver, and the path algebra is its linearisation.
- ← Graphs
अधिगम पथ
आगे पढ़ने के लिए
- Ralf Schiffler, Quiver Representations (2014)
- Ibrahim Assem, Daniel Simson and Andrzej Skowroński, Elements of the Representation Theory of Associative Algebras, Volume 1 (2006)