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

Proposition 17 ([10]). ( B; wt, ε i , ϕ i , ẽ i , ˜f i
)
root datum
(
Q I , {s(i)}, (·, ·) )
crystal
Pro<strong>of</strong>. (C1)(C4) (C1)(C3)
(C4) ϕ i (Λ) = −∞ Λ ∈ B OK
□
4.4. .
B = ⊔ d∈Z
IrrΛ(d) crystal structure preprojective
I
≥0
algebra
Qcrystal structure
preprojective algebra
crystal
Theorem 18 (Kashiwara-S [10]). crystal B B(∞) 20 .
B(∞) “ crystal basis” crystal
“ crystal basis”
Theorem 18
Acrystal preprojective
algebra nilpotent variety
Γ Dynkin case A 5
4.5. Ano<strong>the</strong>r crystal structure on B.
4.3 crystal structure “ top ”
top
socle
OK top socle B crystal structure
B crystal structure 4.3
wt 4.3 ε i 4.3
∗ ε ∗ i
20 crystal
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