A twofold generalization of Gabriel's theorem

2019/04/04 Thu 17:00 - 17:00
Bernard Leclerc

(joint work with C. Geiss and J. Schröer)

By a celebrated theorem of Gabriel, isomorphism classes of indecomposable representations of an A-D-E quiver are in one-to-one correspondence with positive roots of the corresponding root system.

Let C be a Cartan matrix (of type A, B, C, ..., F4, G2), and let D be a symmetrizer for C (i.e. a diagonal matrix with positive integer entries such that DC is symmetric). Fix an orientation of the Dynkin diagram corresponding to C, and an arbitrary field F. To this datum, in joint work with Geiss and Schröer we have introduced an F-algebra H and studied its representation theory. When C is of A-D-E type and the symmetrizer is equal to k times the identity matrix, H is isomorphic to the path algebra over the truncated polynomial ring F[t]/(t^k) of the quiver corresponding to C and the fixed orientation. I will present a twofold generalization of Gabriel's theorem in this situation. Namely there are two bijections:

(1) between isoclasses of indecomposable rigid locally free modules in rep(H) and positive roots of C;

(2) between isoclasses of bricks in rep(H) and positive roots of the tranpose of C.

I will also sketch how this generalizes to symmetrizable generalized Cartan matrices C.

The seminar will be hold at RIMS 006