# LisMath Seminar

### Quiver Representations

A quiver is a directed graph where multiple arrows between two vertices and loops are allowed. A representation of a quiver $Q$, over a field $K$, is an assignment of a finite dimensional $K$-vector space $V_i$ to each vertex $i$ of $Q$ and a linear map $f_a:V_i\rightarrow V_j$ to each arrow $a:i\rightarrow j$. Given a quiver $Q$, the set of all representations of $Q$ forms a category, denoted by $\mathrm{Rep}(Q)$. A connected quiver is said to be of finite type if it has only finitely many isomorphism classes of indecomposable representations.

Quiver representations have remarkable connections to other algebraic topics, such as Lie algebras or quantum groups, and provide important examples of moduli spaces in algebraic geometry [3].

The main goal of this work would consist, first, of good comprehension of the category $\mathrm{Rep}(Q)$. Then, the student would cover the basics on quiver representations to be able to prove Gabriel's theorem [1], following a modern approach, as in [2]:

A connected quiver is of finite type if and only if its underlying graph is one of the ADE Dynkin diagrams $A_n$, $D_n$, for $n \in \mathbb N$, $E_6$, $E_7$ or $E_8$. Moreover, the indecomposable representations of a given quiver of finite type are in one-to-one correspondence with the positive roots of the root system of the Dynkin diagram.

These basic concepts involve topics such as the Jacobson radical, Dynkin diagrams or homological algebra of quiver representations.

Bibliography:

[1] P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Mathematica 6, pp. 71–103 (1972).

[2] H. Derksen and J. Weyzman, An Introduction to Quiver Representations, Graduate Studies in Mathematics 184, American Mathematical Society (2017).

[3] A. Soibelman, Lecture Notes on Quiver Representations and Moduli Problems in Algebraic Geometry, arXiv:1909.03509 (2019)