Basic course description

Math 201a will be a course on the theory of finite dimensional algebras over a field and their representations, concentrating on path algebras of quivers with relations. See the syllabus at right for a more detailed description of the course topics and the course requirements.

TA and Professor Contact Information

Professor Rogalski: Office 5131 AP&M, e-mail

Lecture summaries

Jan 4: Introduction to the course. Algebras over a field. Quivers. The path algebra of a quiver. Representations of a quiver.

Jan 6: (no class)

Jan 8: (no class)

Jan 11: Right modules over the path algebra KQ are the same as representations of Q over the field K.

Jan 13: Morphisms of modules are the same as morphisms of representations. Simple and indecomposable modules/reps. The example of one vertex and one loop: in this case KQ is the same as K[x] and modules over it are classified in Math 200.

Jan 15: Simple and indecomposable modules over the quiver with two vertices and one arrow. One vertex quivers with multiple arrows have path algebras isomorphic to free associative algebras, and have simple modules of arbitrarily large dimension. Some indecomposables over the Kronecker quiver (two vertices and two arrows from 1 to 2).

Jan 18: MLK Day (no class)

Jan 20: Artinian and Noetherian modules and algebras. Review of Wedderburn's theorem and the Jacobson radical. Artinian local rings.

Jan 22: Relation betwen indecomposable modules and local rings. Proof of Krull-Schmidt theorem.

Jan 25: Matrix form of Homs between direct sums of modules. Projective modules. Indecomposable projective modules over an artinian algebra are just the e_i A for primitive idempotents e_i.

Jan 27: Simple modules of an artinian algebra. For a path algebra of an acyclic quiver, there is one simple for each vertex. Review of category theory. Equivalences of categories. The category of representations of Q over K is equivalent to the category of right KQ-modules.

Jan 29: Injective modules. Duality between f.d left modules of a K-algebra and f.d. right modules given by M -> M^* = Hom _K(M,K). Indecomposable right injectives over a finite-dimensional K-algebra are just the duals of indecomposable left projectives.

Feb 1: Reflection functors. Definition of the reflection functor C^+_i at a sink i of a quiver Q, and the functor C^-_i at a source i. Examples.

Feb 3: C^+_i and C^-_i are almost inverses: C^+_i sends every indecomposable to another indecomposable, except for the simple S(i) supported at i (which it kills). For indecomposables other than S(i), C^-_i C^+_i(V) = V. The symmetrized bilinear form ( , ) associated to a quiver Q. Reflections s_i on Z^n. When C^+_i(V) is indecomposable again its dimension vector is given by applying s_i to the dimension vector of V.

Feb 5: Classification of those quivers Q for which ( , ) is positive definite (the Dynkin quivers) or positive semidefinite (the Euclidean or extended Dynkin quivers)

Feb 8: Roots are the a s.t (a,a) = 2 in the bilinear form. The Weyl group W is generated by the reflections s_i and it preserves the form. After numbering the quiver Q properly, c = s_n ... s_1 sends Q to itself and C = C^+_n ... C^+_1 is a well defined functor from reps on Q. If a is nonnegative then c^k(a) is not nonnegative for some k.

Feb 10: Proof that if Q is Dynkin then it has finite rep type, every dim vector of an indecomposable is a root, and every positive root is the dimension vector of a unique indecomposable up to isomorphism. Sketch of proof of the converse that only Dynkin type quivers have finite rep type. Introduction to the geometry of reps.

Feb 12: The representation space rep(Q, a) of reps of dimension vector a of a quiver Q. The action of GL(a); orbits are the same as equivalence classes under isomorphism of reps. Review of affine geometry. The action of GL(a) on rep(Q, a) is a regular action of an algebraic group on a variety. Examples. Orbits O_x under the action are irreducible, locally closed, and the complement of O_x in its closure is a union of orbits of strictly smaller dimension.

Feb 15: Presidents' Day (no class)

Feb 17: Using the representation space to prove the converse of Gabriel's theorem: A path algebra of finite rep type must be the path algebra of a Dynkin quiver. Review of projective resolutions.

Feb 19: Proof that every module over a path algebra has a canonical projective resolution of length at most one, so that the path algebra has global dimension one.

Feb 22: Connection between the dimension of Ext^1(M, N) for modules over the path algebra and the numerics of the dimension vectors in the Ringel form.

Feb 24: Basic algebras. Every finite-dimensional algebra is Morita equivalent to a basic algebra. In turn, every basic finite-dimensional algebra A is isomorphic to a path algebra with relations KQ/J.

Feb 26: The skew group algebra. Skew group algebras generalize semidirect products. Review of representation theory. Idempotent decomposition in the group algebra. Idea of the McKay quiver.

Feb 29: Proof that if an Abelian group G acts on the free algebra S = C, the skew group algebra S*G is isomorphic to the path algebra CQ, where Q is the associated McKay quiver. If G acts instead on a free algebra with relations S/J, then we get CQ/J' instead for an ideal J' determined in an easy way from J.

Mar 2: Definition of the McKay quiver of an arbitrary finite group action of G on S = C[x_1, .., x_n]. Example of S_3. The associated action on affine space spec A^n. The quotient variety A^n/G is defined as spec S^G, where S^G is the ring of invariants. Example of Z/2Z action on C[x, y]. Desingularization and blowing up. The idea of noncommutative desingularization.

Mar 4: Introduction to theory of the Auslander-Reiten quiver. Irreducible morphisms.

Mar 7: The radical of a category. Irreducible morphisms X to Y are those that are in rad(X, Y) and not in rad^2(X, Y). Definition of the AR-quiver: vertices are indecomposables, arrows correspond to a basis of irreducible morphisms. Easy examples of the AR-quivers of kQ where Q is a type A quiver.

Mar 9: Almost split sequences. Relation to left minimal almost split and right minimal almost split morphisms and to irreducible morphisms. The translation functor tau.

Mar 11: Application of AR-quivers to prove Brauer and Thrall conjecture: If a finite-dimensional algebra A is not representation finite then it has indecomposables of arbitrary large degree.