研究路径Algebra课程已上线

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

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前置知识

Quivers and path algebras Representation theory of finite groups Homological algebra

36 项必需的前置知识（一直追溯到底）

你将学会

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.

定义

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.

定理

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 a quiver without loops 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.

常见误解

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; direct sums of copies of a simple representation already give infinitely many (decomposable) representations over any field.

可选发现

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

学习路径

Research preparation