Research pathwaysAlgebraPlanned
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
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 36
You will learn to
- 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.
Definition
- 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.
Theorem
- 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.
Common misconceptions
- 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.
Optional discoveries
- 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
Learning pathways
Further reading
- 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)