Lidia Angeleri, Francesca Mantese Universit`a di Verona, 2010/11 ...

Introduzione alla teoria delle rappresentazioni di grafi 1
TAME AND WILD ALGEBRAS
Definition. For each module A ∈ Λ mod denote by dimA = (m1, . . . , mn) ∈ Z n the
dimension vector of A given by the Jordan-Hölder multiplicities, that is, mi is the number
of composition factors of A that are isomorphic to the simple module Si for aech 1 ≤ i ≤ n.
We set
ei = (0, . . . , 1, 0, . . . , 0) = dimSi
pi = dim Λei = dimPi
qi = dim D(eiΛ) = dimIi
Remark. (1) For every exact sequence 0 → A ′ → A → A ′′ → 0 we have
(2) If dimA = (m1, . . . , mn), then l(A) =
dimA = dimA ′ + dimA ′′
n
mi.
i=1
(3) Consider the Grothendieck group K0(Λ) defined as the group generated by the isomorphism
classes [A] of Λ mod with the relations [A ′ ] + [A ′′ ] = [A] whenever 0 → A ′ →
A → A ′′ → 0 is exact in Λ mod. Note that K0(Λ) is a free abelian group with basis
[S1], . . . , [Sn], see [1, I,1.7]. The assignment [A] ↦→ dimA defines an isomorphism between
K0(Λ) and Z n .
Theorem [Gabriel 1972] Let Λ be a finite dimensional hereditary algebra over an
algebraically closed field k, and let Q be the Gabriel-quiver of Λ. The following statements
are equivalent.
(a) Λ is of finite representations type.
(b) Q is of Dynkin type, that is, its underlying graph belongs to the following list.
An :• • • . . . • •
•
Dn : ❅•
•
• . . . •
•
•
E6 :• • • • •
•
E7 :• • • • • •
•
E8 :• • • • • • •
Moreover, if (a) - (b) are satisfied, the finite dimensional indecomposable modules are
uniquely determined by their dimension vector.