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

( B Ω ; wt, ε i , ϕ i , ẽ i , ˜f i
)
27 ∗ crystal
structure
Q Ω R Ω Ω 0
R Ω 0
Ω
A ([3],[17])
5.5. Open problems.
Problem 1D E Z N ≥0 crystal structure
B = ⊔ d IrrΛ(d) Z N ≥0 N = |∆+ |
Ψ : B ∼ → Z N ≥0
Γ = (I, Ω) Dynkin quiverA
D E B crystal structure Ψ Ω Z N ≥0
A
Z N ≥0 crystal structure
Problem 2Tame case
B ∼ = B(∞) loop quiver Γ Dynkin case
tame Γ extended Dynkin Lie <strong>the</strong>ory
side “affine case”
B(∞)
Λ(d)
Problem 3Rigid crystals
variety <strong>of</strong> nilpotent representations Λ(d) G(d)-
dimension vector d nilpotent P (Γ)-module
Λ(d) IrrΛ(d)
27 ẽ i
(
ẽ i a Ω =
R Ω Ω 0
◦ ẽ i ◦ R Ω0
Ω
)
(a Ω ) (a Ω ∈ B Ω )
ẽ i B Ω0 ẽ i
5.2 explicit formula RΩ Ω 0
R Ω 0
Ω
–191–