Lecture 1 (3/28/11): A quiver Q is a directed graph. Paths in a quiver and the path algebra KQ. Examples
of path algebras.
Lecture 2 (3/30/11): Representations of Q. Correspondence between f.d. left modules over KQ and representations of Q.
Lecture 3 (4/1/11): Review of categories. More formal formulation of result of last time: the category of f.d. left
modules over KQ is equivalent to the category of representations of Q.
Lecture 4 (4/4/11): Simple representations of KQ. Examples on small quivers. If Q has no oriented cycles,
then simples are classified: there is one simple up to isomorphism for each vertex.
Lecture 5 (4/6/11): Indecomposable representations. Examples on small quivers. Statement of Krull-Schmidt theorem: every
finite-dimensional module over an algebra has an essentially unique expression as a direct sum of indecomposables. Projective
modules. KQ is the direct sum of the indecomposable projective modules e_iKQ, and these are all indecomposable projectives if
Q has no oriented cycles. Examples of projectives on a small quiver.
Lecture 6 (4/8/11): Injective modules. General comments on module structures on Hom(M,N) when either M or N is a bimodule.
The duality between left and right R-modules of finite K-dimension via Hom_K( --, K). Classification of f.d. injectives via f.d. projectives on the other side. The corresponding duality for the quiver reps.
Lecture 7 (4/11/11): Some ring theory along the way to proof of Krull-Schmidt. Review of Jacobson radical J(A) and the
structure of a f.d. K-algebra. Local rings. Lifting idempotents. Proof that a f.d. K-algebra A is local if and only if 0,1 are its only idempotents. Application: the endomorphism ring of a f.d. module M is local iff the module is indecomposable.
Lecture 8 (4/13/11): Proof of Krull-Schmidt. Projective resolutions and global dimension.
No lecture (4/15/11)
Lecture 9 (4/18/11): The Ringel resolution of a module over KQ. The path algebra KQ is hereditary.
Lecture 10 (4/20/11): Ext and its basic properties. Dimension vectors and the Ringel form < , >. The formula
dim_K Hom(a, b) - dim_K Ext^1(a,b) = < a , b >, for dimension vectors a,b.
Lecture 11 (4/22/11): Finite representation type. Reflection functors and some properties.
Lecture 12 (4/25/11): More properties of reflection functors. Applying a reflection functor at a vertex i
sends an indecomposable V to another indecomposable unless it is the simple rep S(i), which it kills. Reflection functors act on the dimension vectors of indecomposables via reflection
operations s_i: a --> a - [2(a,e_i)/(e_i,e_i)] e_i, where ( , ) is the symmetrized Ringel form.
Lecture 13 (4/27/11): positive (semi)-definite forms. Classification of which quivers have a Ringel form ( , ) which is positive
definite (underlying graph is Dynkin of type A, D, or E) or positive semi-definite (underlying graph is Euclidean.)
Lecture 14 (4/29/11): Finish the classification of positive (semi)-definite graphs. If ( , ) positive definite, then study roots and the Weyl group. There are finitely many roots and the Weyl group is finite.
Lecture 15 (5/02/11): Proof of first half of Gabriel's theorem (using reflection functors): If Q has underlying graph which is Dynkin, then Q has finite representation type. Introduction to the representation space Rep(Q, alpha) parametrizing reps of a fixed
dimension vector alpha, and its GL(alpha)-action. The oribts correspond to isomorphism classes of reps.
Lecture 16 (5/04/11): More on the representation space. Orbits of an algebraic group action and the orbit-stabilizer theorem. Proof of the second half of Gabriel's theorem using algebraic geometry: if Q has finite representation type, then ( , ) is positive
definite and thus Q is Dynkin.
Lecture 16 (5/06/11): Quivers with relations. Deformed preprojective algebras.
Lecture 17 (5/09/11): Kac's theorem on the dimension vectors of indecomposable modules on path algebras of not necessarily
Dynkin type. Reflection functors for deformed preprojective algebras.
Lecture 18 (5/11/11): Verification of some properties of reflection functors.
Lecture 19 (5/13/11): Relation between representations of Q and representations of a deformed preprojective algebra formed from Q.
Lecture 20 (5/16/11): Proof of part of Kac's theorem using deformed preprojective algebras (method due to Crawley-Boevey and Holland.)
Lecture 21 (5/18/11): idempotent decomposition in finite dimensional algebras. Every finite-dimensional algebra over an algebraically closed field kis Morita equivalent to a basic algebra (one where when we mod out by the Jacobson radical we just get
a product of copies of k)
5/20/11 NO CLASS
5/23/11 NO CLASS
Lecture 22 (5/25/11): Skew group algebras and examples.
Lecture 23 (5/27/11): The skew group algebra of an abelian group acting by linear automorphisms of
a free algebra is isomorphic to the path algebra of the McKay quiver.
5/30/11, NO CLASS (Memorial Day)
Lecture 24 (6/1/11): Generalizations of the theorem of 5/27 to group actions on a presented algebra or to non-abelian groups.
An introduction to the quotient of an affine variety by an finite group action.
Lecture 25 (6/3/11): Survey of the idea of a noncommutative resolution of singularities (without proofs)