A torsion class is a collection of modules for an algebra that is closed under quotients and extensions. It turns out that many combinatorial objects can be realized as torsion classes for certain algebras, and the containment order on torsion classes recovers interesting partial orders on those objects. We will give a brief introduction to several examples, including the Tamari lattice, the weak order, and the oriented exchange graph of a cluster algebra. We will also outline the schedule for the first half of the semester.

We will define quivers and their representations and use examples to gain intuition. We will then define algebraic objects associated to quivers (the path algebra and the \(K\)-group) and use these to state the standard representation of a quiver and explore properties of the category of quiver representations. This talk will cover sections 1.1–1.4 of Quiver Representations and Quiver Varieties by Alexander Kirillov Jr.

We review notions from last lecture, including projective and simple modules over the path algebra \(kQ\) of a quiver. From the projective resolution of any module over \(kQ\), we compute dimensions of Hom and Ext functors. We use this to motivate the definition of the Euler form, classify when the symmetrized Euler form is positive (semi)definite, and draw connections to Lie theory. We finally state the definition and classification of finite type quivers. This talk will cover sections 1.4–1.7 and 3.1 of Quiver Representations and Quiver Varieties by Alexander Kirillov Jr.