Caminos de investigaciónAlgebraPlanificado
Quiver representations
Optional research direction: Gabriel's theorem on quivers of finite representation type, Auslander–Reiten theory, Kac's theorem, and links with Lie theory, cluster algebras and geometric representation theory.
math.algebra.quiver-representations
La lección de este tema todavía no se ha escrito. El esquema siguiente forma parte del mapa del plan de estudios y no ha sido revisado.
Se basa en
Requisitos previos obligatorios, hasta el fondo: 36
Aprenderás a
- State Gabriel's theorem and verify it for small Dynkin quivers using reflection functors.
- Compute Auslander–Reiten quivers of path algebras of Dynkin type.
- Read a recent survey on quiver representations and separate established results from open questions.
Definición
- A quiver has finite representation type if it has only finitely many indecomposable representations up to isomorphism.
- The Tits form of Q is q(x) = Σᵢ xᵢ² − Σ_{arrows i→j} xᵢxⱼ.
- An Auslander–Reiten sequence is a non-split short exact sequence 0 → A → B → C → 0 with A and C indecomposable, such that every map into C that is not a split epimorphism factors through B → C.
Teorema
- Gabriel (1972): a connected quiver has finite representation type if and only if its underlying graph is a Dynkin diagram of type A, D or E, and then dimension vectors give a bijection between indecomposables and positive roots.
- Kac: over an algebraically closed field, the dimension vectors of indecomposable representations of any quiver are exactly the positive roots of the associated Kac–Moody root system.
- Auslander–Reiten sequences exist for finite-dimensional algebras, ending in any indecomposable non-projective module.
Errores frecuentes
- Thinking Gabriel's theorem depends on the orientation of the arrows; finite type depends only on the underlying graph.
- Confusing finite representation type with having finitely many representations; there are always infinitely many decomposable ones over an infinite field.
Descubrimientos opcionales
- Categorical representation theory Lusztig used the geometry of spaces of quiver representations to construct canonical bases of quantum groups, an important source of categorification.
- ← Derived categories
Itinerarios de aprendizaje
Lecturas complementarias
- Ralf Schiffler, Quiver Representations (2014)
- Harm Derksen and Jerzy Weyman, An Introduction to Quiver Representations (2017)
- Maurice Auslander, Idun Reiten and Sverre O. Smalø, Representation Theory of Artin Algebras (1995)