ADE (Ryo Fujita) 1 Introduction Lie g U(g) q 2 q q Hopf Drinfeld- U q (g) C S 1 g U(g) q U q (g) U(Lg) q U q (Lg) Lg := g C[t ±1 ] Lie U q (Lg) 2 R ADE Lie g U q (Lg) ADE Dynkin Dynkin Q Dynkin Q Hernandez-Leclerc [6] U q (Lg) C Q Q 2 2.1 Dynkin Lie An (n Z 1 ) Lie Dynkin Dynkin 3 2 3 D n (n Z 4 ) E n (n = 6, 7, 8) 1 2 3 n 1 n 1 2 1 3 4 5 2 n 3 n 2 4 A n (n Z 1 ), B n (n Z 2 ), C n (n Z 2 ), D n (n Z 4 ) E n (n = 6, 7, 8), F 4, G 2 ADE Dynkin Dynkin n 1 n n E-mail:rfujita@math.kyoto-u.ac.jp
Dynkin Lie g X = A, D, E X n Dynkin I = {1, 2,..., n} 2 i, j I i j Cartan A = (a ij ) i,j I 2 i = j ; a ij = 1 i j ; 0 X n Lie g {e i, f i, h i } i I [h i, h j ] = 0, [h i, e j ] = a ij e j, [h i, f j ] = a ij f j, [e i, f j ] = δ ij h i, ad(e i ) 1 a ij (e j ) = ad(f i ) 1 a ij (f j ) = 0 (i j), ad(x)(y) := [x, y] Lie g U(g) Lie [x, y] xy yx C Lie g C U(g) Cartan h := i I Ch i g {h i } i I {ϖ i } i I h P := i I Zϖ i h U(g) M M = M λ, M λ := {v M h v = λ(h)v (h h)}. λ P i I α i := j I a ijϖ j P P Q := i I Zα i Q + := i I Z 0α g = α Q g α R := {α Q \ {0} g α 0} R + := R Q + R = R + ( R + ) n + (resp. n ) {e i } i I (resp. {f i } i I ) g Lie n ± = α ±R + g α, h = g 0 g = n h n + α R dim g α dim n ± = R +. = 1 U(g) U(g)-mod fd Weyl Irr U(g)-mod fd P + := i I Z 0α i P 1 : 1 λ P + V (λ) v e i v = 0, h i v = λ(h i )v, f λ(h i)+1 i v = 0 ( i I) U(g) X g (X) = X 1 + 1 X : U(g) U(g) U(g) 2 U(g) V, W C V W U(g) U(g)-mod fd v w w v U(g) V W = W V 2.2 C Lie g Laurent C[t ±1 ] Lg := g C[t ±1 ] Lie [X f(t), Y g(t)] = [X, Y ] f(t)g(t), (X, Y g, f(t), g(t) C[t ±1 ])
Lg C Lie C U(Lg) ADE U q (Lg) ADE Lie g Lg U(Lg) q 1 q C 2.1. X = A, D, E X n U q (Lg) {e i,r, f i,r i I, r Z} {K ±1 i i I} {h i,m i I, m Z 0 } C : K i K 1 i = K 1 i K i = 1, [K i, K j ] = [K i, h j,l ] = [h i,m, h j,l ] = 0, K i e j,r K 1 i = q aij e j,r, K i f j,r K 1 i = q aij f j,r, (w q ±a ij z)ψi ε (z)x ± j (w) = (q±a ij w z)x ± j (w)ψε i (z), [x + i (z), x i (w)] = δ ( ( ij z ) ( w ) ) q q 1 δ ψ + i w (w) δ ψ i z (z), (w q ±aij z)x ± i (z)x± j (w) = (q±aij w z)x ± j (w)x± i (z), { x ± i (z 1)x ± j (z 2)x ± j (w) (q + q 1 )x ± i (z 1)x ± j (w)x± i (z 2) + x ± j (w)x± i (z 1)x ± i (z 2) } + {z 1 z 2 } = 0 (i j ). ε {+, } δ(z), ψ ± i (z), x± i (z) ( U q(lg)[[z, z 1 ]]) : δ(z) := r= x + i z r, ψ ± i (z) := (z) := K±1 i exp r= ( ±(q q 1 ) e i,r z r, x i (z) := r= ) h i,±m z ±m, m=1 f i,r z r. 2 {z 1 z 2 } 1 z 1 z 2 e i,r, f i,r, h i,m Lg e i t r, f i t r, h i t m K i q hi 1 U q (Lg) U q (Lg)-mod fd U q (Lg) M M = M λ, M λ = {v M K i v = q λ(hi) v (i I)} λ P 1 1 U q (Lg) C C Lie g C : U q (Lg) U q (Lg) U q (Lg) *1. i I (K i ) = K i K i, (e i,0 ) = e i,0 K 1 i + 1 e i,0, (f i,0 ) = f i,0 1 + K i f i,0, *1 U q(lg) U q(ĝ) Lie Hopf 0 U q (Lg) U q (ĝ) 0
1 C U q (Lg)-mod fd 2 3 V, W C V W W V V W W V U q (Lg) Grothendiek K(C) 2.3 C 2.2 (Chari-Pressley [1]). L C 1 Cv L 1 I π = (π i (u)) i I (1 + uc[u]) I 3 *2 (1) e i,r v = 0 (i I, r Z); (2) K i v = q λ(hi) v (i I); [ (3) L[[z ±1 ]] ψ ± i (z) v = πi (q qλ(h i) 2 ] z) π i (z) v z ±1 =0 (i I). λ := i I (deg π i)ϖ i P + [ ] z ±1 =0 z ±1 = 0 L = L(π) Irr C L(π) π (1 + uc[u]) I 1 : 1 I π (1 + uc[u]) I Drinfeld (1), (2), (3) v U q (Lg) l v l l 2.2 C l U(g)-mod fd 2.4 / Weyl Chari-Pressley [2] l Weyl U q (Lg) M(π) = U q(lg)v (1),(2),(3) /(3) /r π M(λ) = U q(lg)v (1),(2) L(π) W (π) W(λ) K v 2.2 (1), (2), (3) l M(π) L(π) C M(π) C Weyl (local Weyl module) M(π) W (π) C Weyl W (π) v (1), (2), (3) f λ(hi)+1 i,r v = 0 (i I, r Z) (2.1) v 2.2 (1), (2) M(λ) *3 *2 (2) (3) *3 M(λ) Weyl W(λ) Drinfeld λ P +
M(λ) *4 W(λ) Weyl (global Weyl module) Weyl W(λ) (1), (2) (2.1) Chari-Pressley [2] [9] R λ := End Uq (Lg)(W(λ)) = i I C[z ±1 i,1,..., z±1 i,λ(h i ) ]S λ(h i ) W(λ) R λ R λ Specm(R λ ) = (C ) λ := i I (C ) λ(hi) /S λ(hi ) ( (C ) λ [(c i,1,..., c i,λ(hi ))] i I λ(hi ) ) j=1 (1 c i,ju) (1 + uc[u]) I λ Drinfeld λ i I Drinfeld π R λ r π Weyl W (π) Weyl W(λ) r π Weyl W(λ) (C ) λ Weyl W (π) π R π := lim k R λ /r k π Weyl W(λ) π Ŵ (π) := W(λ) R λ Rπ Weyl 2.5 L(λ) M(λ) [9] π {0} M 0 (λ) λ P + (quiver variety) 2 C M(λ) M 0 (λ) M(λ) ν Q + M(ν, λ) M 0 (λ) ν Q + M 0 (ν, λ) M(λ) M 0 (λ) G(λ) := i I GL λ(h i )(C) C G(λ) π : M(λ) M 0 (λ) M 0 (λ) 0 π L(λ) (central fiber) A 1 I = {1} ϖ = ϖ 1, α = α 1 λ = lϖ P +, ν = kα Q + M(ν, λ) Grassmann T Gr(k, l) M 0 (λ) = {x End(C l ) x 2 = 0} Gr(k, l) T Gr(k, l) 0 k l 0 k l π {0} {x End(C l ) x 2 = 0} T Gr(k, l) = {(x, V ) End(C l ) Gr(k, l) x(c l ) V, x(v ) = 0} π 1 L(λ) Grassmann Gr(k, l) G(λ) = GL(C l ) C T Gr(k, l) GL(C l ) g (x, V ) = (gxg 1, gv ) C G X K X G Coh G (X) Grothendiek K(Coh G (X)) K G (X) X 1 pt Coh G (pt) G Rep(G) K G (pt) G R(G) V Rep(G) G X G [V ] R(G) [F] K G (X) *4 e i,r, f i,r U q (ĝ) 0
[V ] [F] := [V OX F] K K G (X) R(G) M(λ) M 0 (λ) G G(λ) A := R(C ) = Z[v ±1 ], R(GL k (C)) = Z[z 1 ±1,..., z±1 k ]S k R(G(λ)) = i I A[z±1 i,1,..., z±1 i,λ(h i) ]S λ(h i) A v A U q (Lg) q C C A R λ R λ = R(G(λ)) A C G(λ) X K G(λ) (X) := K G(λ) (X) A C R λ M(λ) M 0 (λ) Steinberg Z(λ) := M(λ) M0(λ) M(λ) K K G(λ) (Z(λ)) (convolution) K K G(λ) (Z(λ)) R λ K K G(λ) (L(λ)) K G(λ) (Z(λ)) 2.3 ( [9]). λ P + C Φ λ : U q (Lg) K G(λ) (Z(λ)) U q (Lg) Φ λ (KG(λ) (L(λ))) Weyl W(λ) R λ 2.4. Φ λ 4.2 3 Dynkin Hernandez-Leclerc C Q Dynkin C Q C 3.1 Gabriel (quiver) I Ω Q = (I, Ω) a Ω a I a I I Ω Q C i I C V i a Ω f a Hom(V a, V a ) ((V i ) i I, (f a ) a Ω ) Q = (I, Ω) p = (a 1, a 2,..., a l ) a k = a k+1 (1 k < l) a 1 a l (path) l p a Ω 1 0 i i 0 ϵ i Q C CQ = p Cp 2 0 *5 C CQ V V i := ϵ i V a f a : V a V a Q ((V i ), (f a )) Q CQ CQ V CQ-mod fd dimv := (dim V i ) i I (Z 0 ) I d = (d i ) i I (Z 0 ) I E d := a Ω Hom(C d a, C d a ) G d := i I GL(C di ) *5 (a 1,..., a l ) (a l+1,..., a l+m ) = δ al,a l+1 (a 1,..., a l+m ), ϵ i ϵ j = δ ij ϵ i, ϵ i (a 1,..., a l ) = δ i,a1 (a 1,... a l ), (a 1,..., a l ) ϵ i = δ al,i(a 1,... a l ).
G d (g i ) i I : (f a ) a Ω (g a f a g 1 a ) a Ω E d dimv = d V CQ-mod fd G d E d G d E d /G d d 1 : 1 3.1 (Gabriel ). (1) Q d Z I 0 E d/g d < Q Dynkin Q ADE Dynkin (2) Q Dynkin (Z 0 ) I d i I d iα i Q + (Z 0 ) I Q + CQ-mod fd R + Q + 1 : 1 Dynkin Q 3.1 (2) α R + dimm α = α M α CQ-mod fd M CQ-mod fd Krull-Schmidt dim [ ] 3.2. M m α α R+ α (m α ) Dynkin Q β β Kostant KP(β) := {(m α ) (Z 0 ) R + α m αα = β} 1 : 1 β = i I d iα i E d G d KP(β) 3.2 Auslander-Reiten Krull-Schmidt C A *6 Auslander-Reiten(AR) Γ(A) Γ(A) A Γ(A) X Y X Y *7 Dynkin Q D b (CQ-mod fd ) AR D b (CQ-mod fd ) {M α [k] α R +, k Z} M α [k] stalk H i (M α [k]) := { M α i = k ; 0 Dynkin Q = (I, Ω) ξ : I Z ξ a = ξ a + 1 a Ω ADE Dynkin ξ Q (repetition quiver) Q = (Î, Ω) Î := {(i, p) I Z p ξ i 2Z}, Ω := {(i, p) (j, p + 1) (i, p) Î, i j} *8. *6 2 C *7 section retraction 2 *8 Q Q Ω
A 3 (3, 2) (3, 0) (3, 2) (3, 4) (2, 3) (2, 1) (2, 1) (2, 3) (1, 2) (1, 0) (1, 2) (1, 4) 3.3 (cf. [4]). ξ ϕ : Q = Γ(D b (CQ-mod fd )) i I ϕ(i i [0]) = (i, ξ i ) I i CQ-mod fd M αi (injective hull) t CQ-mod fd D b (CQ-mod fd ) R + α ϕ(α) := ϕ(m α [0]) Î 3.4. A 3 R + = {α 1, α 2, α 3, (α 1 + α 2 ), (α 2 + α 3 ), (α 1 + α 2 + α 3 )} (1) Q = (1 2 3) (ξ 1, ξ 2, ξ 3 ) = (2, 1, 0) ϕ α 1 + α 2 + α 3 α 2 + α 3 α 1 + α 2 α 3 α 2 α 1 ϕ (3, 0) (2, 1) (2, 1) (1, 2) (1, 0) (1, 2) (2) Q = (1 2 3) (ξ 1, ξ 2, ξ 3 ) = (2, 1, 2) ϕ α 1 + α 2 α 3 α 2 α 1 + α 2 + α 3 α 2 + α 3 α 1 ϕ (3, 0) (3, 2) (2, 1) (2, 1) (1, 0) (1, 2) 3.3 Hernandez-Leclerc C Q C C Q Î P + := Z 0 Î ϕ(r + ) Î P0 + P + ( ) P + (1 + uc[u]) I ; l i,p (i, p) (1 q p u) li,p (i,p) Î p i I Drinfeld (1 + uc[u]) I 2.2 Irr C = (1 + uc[u]) I C Serre C Z (resp. C Q ) Irr C Z = P + (resp. Irr C Q = P+) 0 C Z C D b (CQ-mod fd ) C Q C Z CQ-mod fd D b (CQ-mod fd )
C Z C 3.5 (cf. [5]). C Z C K(C) = a C /q 2Z K(τ a C Z ) τ a U q (Lg) τ a L(π(u)) = L(π(au)) Lie n + g Lie N + C Q N + C[N + ] (categorification) 3.6 (Hernandez-Leclerc [6]). C Q C Z K(C Q ) Z C = C[N + ] C Q C[N + ] (dual canonical basis) *9 1 : 1 C Q C[N + ] C[N + ] (cluster algebra) 4 4.1 C Q KP(β) (m α ) α m αϕ(α) P 0 + KP(β) P0 + P 0 + = β Q + KP(β) KP(β) + KP(β ) KP(β + β ) C Q,β Irr C Q,β = KP(β) C Q Serre C Q = β Q + C Q,β C β C β C β+β * 10 β Q + C Q,β β = i I d iα i Q + ϕ π := i I d iϕ(α i ) P 0 + (1 + uc[u]) I π Drinfeld (π i (u)) i I λ := i I (deg π i)ϖ i P + 2.3 Φ λ : U q (Lg) K G(λ) (Z(λ)) R β := lim R λ /r k k π Φ λ Φ β : U q (Lg) K G(λ) (Z(λ)) K G(λ) (Z(λ)) Rλ Rβ =: K β R λ = R(G(λ)) A C G(λ) 1 T ( = C ) G(λ) r π r π M 0 (λ) T M 0 (λ) T T G(λ) G d T 4.1 (Hernandez-Leclerc [6]). G d M 0 (λ) T = E d E d G d Gabriel 3.2 Φ β *9 U q (g) Lusztig, *10 Grothendiek C[N + ] = β Q + C[N + ] β
4.2 (F. [3]). (1) Φ β : U q (Lg) K β K β -mod fd = CQ,β C Q,β ĈQ,β := K β -mod fg (2) ĈQ,β M(λ) T E d * 11 ĈQ,β Weyl 4.2 Dynkin Schur-Weyl A Schur-Weyl U q (Lsl n ) GL Hecke Schur-Weyl ADE Kang- -Kim [7] Dynkin Q Hecke * 12 U q (Lg) C Q 1 : 1 4.2 Kang- -Kim ([3]) [1] V. Chari and A. Pressley. Quantum affine algebras and their representations. Representations of groups (Banff, AB, 1994), 59 78, CMS Conf. Proc., 16, Amer. Math. Soc., Providence, RI, 1995. [2] V. Chari and A. Pressley. Weyl modules for classical and quantum affine algebras. Represent. Theory, 5:191 223, 2001. [3] R. Fujita. Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types. preprint. arxiv:1710.11288. [4] D. Happel. Triangulated categories in the representation theory of finite-dimensional algebras. London Mathematical Society Lecture Note Series, 119. Cambridge University Press, 1988. [5] H. Hernandez and B. Leclerc. Cluster algebras and quantum affine algebras. Duke Math. J., 154(2):265 341, 2010. [6] H. Hernandez and B. Leclerc. Quantum Grothendieck rings and derived Hall algebras. J. Reine Angew. Math., 701:77 126, 2015. [7] S.-J. Kang, M. Kashiwara, and M. Kim. Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras, II. Duke Math. J., 164(8):1549 1602, 2015. [8] A. Kleshchev. Affine highest weight categories and affine quasi-hereditary algebras. Proc. Lond. Math. Soc. (3), 110(4):841 882, 2015. [9] H. Nakajima. Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Amer. Math. Soc., 14(1):145 238, 2001. *11 [8] *12 GL Hecke Khovanov-Lauda-Rouquier