Quivers in Representation Theory (18.735, Spring

Quivers are directed graphs. The term is the term used in representation theory, which goes along with the following notion: a representation of a quiver is an assignment of vector spaces to vertices and linear maps between the vector spaces to the arrows. Quivers appear in many areas of mathematics: (1) Algebraic geometry (Hilbert schemes, moduli spaces (represent these as varieties of quiver representations); derived categories Db(Coh X), where X is a projective variety or a scheme of finite type (give an equivalence with the derived category of dg representations of certain quivers with relations)) (2) Representation theory (quiver algebras, groups, local systems/vector bundles with flat connection on curves) (3) Lie theory (Kac-Moody Lie algebras are related to the combinatorics of representations of the associated quiver; quantum enveloping algebras of Kac-Moody algebras can be constructed from the category of quiver representations) (4) Noncommutative geometry (computable examples, Calabi-Yau 3-algebras in terms of quivers with potentials) (5) Physics (Calabi-Yau 3-folds X and their branes: one can represent Db(X) as a derived category of certain quiver representations; or in the affine case X = SpecA, one can sometimes present A using a quiver and a potential)