Sumários
Lecture 33
19 dezembro 2024, 16:30 • Pedro Resende
Category Rep F(Q) of representations of a quiver Q over a field F. Equivalent formulation as a category of functors.
Dimension vector of a finite dimensional representation. Statement of Gabriel's theorem.
Presentation of the F-algebra of a quiver by generators and relations.
Equivalence of Rep
F(Q) and the category of modules over the quiver algebra (for quivers with finitely many vertices).
Several examples of quivers, and of their algebras and representations.
This lecture ended the material for the fourth written test and for the exam.
Bibliography: These
lecture notes ("Lecture 13"); these
other notes are worth reading (they contain much more than we covered).
Lecture 32
17 dezembro 2024, 12:30 • Pedro Resende
Representation theory beyond semisimple rings.
Jacobson's density theorem.
Filtrations of modules.
The radical ideal of a finite dimensional algebra over a field. Rad(A) as the largest nilpotent ideal of A.
The irreducible representations of a finite dimensional algebra A over a field, and proof that A/Rad(A) is semisimple.