Proceedings of the 44th Symposium on Ring Theory and ...

◦ side
Γ = (I, Ω) cycle (acyclic)
C[Γ] S(i) i ∈ I simple C[Γ]-module, P (i)
projective cover I = {1, 2, · · · , n}
Definition 5. n C Γ = (c i,j ) i,j∈I
c i,j := dim C[Γ] (P (i), P (j))
C Γ path algebra C[Γ] Cartan matrix
(i, j ∈ I).
(i) Cartan matrix C Γ
(ii) Cartan matrix C Γ c i,j
(iii) Cartan matrix C Γ
Lie <strong>the</strong>ory
Cartan matrix
(i) (ii) (iii)
Ringel [13] Z I s(i) := dimS(i),
p(i) := dimP (i)
p(i) = s(i) t C Γ (4.1.1)
s(i) i 1 0
(4.1.1) p(i) t C Γ i
C[Γ]-mod C[Γ]-modules abelian categoryK(C[Γ]) := K(C[Γ]-mod)
Gro<strong>the</strong>ndieck
Z-module
K(C[Γ]) ∋ [V ] ↦→ dimV ∈ Z I
Φ Γ : K(C[Γ]) ∼ → Z I
[V ] C[Γ]-module V K(C[Γ])
C[Γ] hereditary S(i) projective resolution
K(C[Γ]) [S(i)] [P (j)] (j ∈ I) Φ Γ
Z I n P ′
E n =
⎛
⎜
⎝
s(1)
s(2)
.
s(n)
⎞ ⎛
⎟
⎠ = ⎜
⎝
p(1)
p(2)
.
p(n)
⎞
⎟
⎠ P ′
(4.1.1) P ′ = t C −1
Γ
C Γ Z I bilinear form
〈〈x, y〉〉 := x (t C −1
Γ
(E n n )
) t
y (x, y ∈ Z I )
–172–
(iii)