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

Euler form 7 (i),(ii),(iii)
• Euler form 〈〈·, ·〉〉
• Euler form 〈〈·, ·〉〉 Z
Euler form A
A 8 V projective dimension
A-module, W injective dimension A-module
〈〈dimV, dimW 〉〉 = ∑ l≥0
A = C[Γ]
dim C Ext l A(V, W )
〈〈dimV, dimW 〉〉 = dim C Hom C[Γ] (V, W ) − dim C Ext C[Γ] (V, W )
V, W simple module
〈〈s(i), s(j)〉〉 = δ i,j − dim C Ext C[Γ] (S(i), S(j))
{
1 (i = j),
=
−(Ω i j arrow ) (i ≠ j)
“”
Euler form
(x, y) alg := 1 2 x ( C −1
Γ
+ ) t C −1 t
Γ
y (x, y ∈ Z I )
Z I symmetric bilinear form Γ = (I, Ω) arrow
quiver Γ := (I, Ω)
C Γ = t C Γ
symmetric bilinear form (·, ·) alg
1 (
)
t C −1
2
+ t C −1
Γ Γ
symmetric bilinear form
{ 1 (i = j),
(s(i), s(j)) alg =
−(H = Ω ∪ Ω i j arrow )/2 (i ≠ j)
◦ Lie <strong>the</strong>ory side
quiver Γ = (I, Ω) cycle loop Γ
arrow Γ Dyn Γ Dyn quiver Γ
(underlying) Dynkin diagram 9
7 〈·, ·〉 〈·, ·〉
8 A global dimension
9 “Dynkin” Lie <strong>the</strong>ory C[Γ]
“Dynkin case” “non-Dynkin case” Lie <strong>the</strong>ory C[Γ]
Γ Dyn “Dynkin diagram”
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