2 TOMOYUKI ARAKAWA 2. Beilinson-Drinfeld W W. Weyl. g C Lie, G, W Weyl, h Cartan. S(h) W S(h) W. S(h) 3 Heisenberg( ) (free boson). Fateev-Lukyanov [F
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1 PRINCIPAL AFFINE W -ALGEBRAS: AN OVERVIEW TOMOYUKI ARAKAWA ( ). Borcherds [Bor86] (vertex algebra),,. W. W Virasoro ([KRW03]),,. W Zamolodchikov[Zam85]. Feigin-Frenkel[FF90], Kac-Roan-Wakimoto[KRW03], W g BRST., W, W, BRST W., BRST, W. ( Z-algebra) W 2., W, W. W (Zhu ) W ([dbt93]), (Lie g )Gelfand-Graev End ([DSK06])., W Kostant-Lynch [Kos78, Lyn79]., Premet[Pre02], W ( ) Lie, 0 W Slodowy ([GG02] )., Brundan-Kleshchev[BK06, BK05], A W A Yangian ( )., W. W.,, ( )W. W,.,,., C W 90 [BS95]. 2,.
2 2 TOMOYUKI ARAKAWA 2. Beilinson-Drinfeld W W. Weyl. g C Lie, G, W Weyl, h Cartan. S(h) W S(h) W. S(h) 3 Heisenberg( ) (free boson). Fateev-Lukyanov [FL88] S(h) W, Heisenberg W. W 4,, ( )A, D. ( ) 5,. Chevalley () C[h] W = C[g] G., C[h] W C[g] G. g exponent = d < d 2 < < d l (l = rank g), C[g] G d i + P i (i =, 2,..., l) C[g]., C[g] G augmentation ideal C[g] G + g N : (2) N = Spec C[g]/C[g] G + = Spec(C[g]/ P,..., P l )., G, g, N G, g, N ( ) 6. (6) (7) G = G[[t]], g = g[[t]] = lim n. {x i } g g n, g n = g[t]/(t n+ ) C[g ] = C[x i,(n) ; n ] (x i = x i,( ) ). C[g ] T. (8) (9) T x i,( n) = nx i,( n ). N = Spec(C[g ]/ T n P i ; i =,..., l, n 0 ). C[N ]. P i C[g] G T n P i C[g ] G C X X, X m- X m. (3) X = lim m X m. X m. (4) Hom(Spec R, X m ) = Hom(Spec R[z]/(z m+ ), X) X, X (5) C[X] C[[z]], a X a (n) z n.. n R.
3 PRINCIPAL AFFINE W -ALGEBRAS: AN OVERVIEW 3 2. (Beilionson-Drinfeld[BD]). :, C[g ] G = C[T n P i ; i =,..., l, n 0]. N = Spec C[g ]/(C[g ] G + ). Z(g) U(g)., U(g), Z(g) C[g], C[g] G., C[g ], C[g ] G?, C[g ] = S(g[t ]t ), C[g ] ( ) U(g[t ]t ), g V k (g) (k 7 ) 8., V U(g) standard filtration, gr V. gr V (differential algebra). V k (g), (0) gr V k (g) = C[g ]., g ( ) W W k (g), () gr W k (g) = C[g ] G 9., k C. ( k = h, h g Coxeter ), W h (g) V h (g) [FF92, Fre07],., V k (g)., W k (g) V k (g)., W k (g) V k (g) Whittaker., W k (g), ( ). W 0, V. V =Virasoro ( ), Virasoro.,, g = sl 2 W, Virasoro( ) 2., g = sl 2, (OPE), W k (g) Lie. W. W 3,. Lie Virasoro,,, W. 7 g Lie gaff. 8 V k (g) g g aff ( 3.3 ). 9 C[g ] G, unique. 0 Wakimoto.,. 2, W Virasoro. 3 W k (sl 3 ), [BMP96]. k k
4 4 TOMOYUKI ARAKAWA 3. Vertex algebra basics. [Kac98, MN99, FBZ04] V, V [[z, z ]] = { n Z v n z n ; v n V }. V [[z, z ]] V C[[z, z ]]., V [[z, w, z, w ]] = { v m,n z m w n ; v m,n V, m, n Z}, m,n Z V ((z)) = { n Z v n z n V [[z, z ]]; v n = 0 (n 0)}. V ((z)) C((z))., a(z) (End V )[[z, z ]] (2) a(z) = n Z a (n) z n. a(z), v V a(z)v V ((z)), a (n) v = 0 n 0, (quantum) field. field a(z), b(z), a(z)b(z) well-defined., a(z)b(w) (End V )[[z, w, z, w ]] well-defined, v V (3) a(z)b(w)v V ((z))((w)), b(w)a(z)v V ((w))((z))., C[z, w, z, w, z w ] ( ) C[z, w, z, w ] z w.. (4) (5) t z,w : C[z, w, z, w, t w,z : C[z, w, z, w, f ] C((z))((w)), z w f ] C((z))((w)), z w z w f n 0 z w f n 0 z (w z )n, w ( z w )n. τ z,w z > w, τ w,z w > z,,. (6) δ(z w) := τ z,w ( z w ) τ w,z( z w ) = z (w z )n C[[z, w, z, w ]]. n Z f(z) (End V )[[z, z ]] f(z)δ(z w), f(w)δ(z w) well-defined. 3.. f(z) (End V )[[z, z ]] f(z)δ(z w) = f(w)δ(z w).
5 PRINCIPAL AFFINE W -ALGEBRAS: AN OVERVIEW 5. f C[z, w, z, w ] τ z,w (f) = τ w,z (f). (z n w n )δ(z w) = (z n w n )(τ z,w ( z w ) τ w,z( z w )) n Z. = τ z,w ( zn w n z w ) τ w,z( zn w n z w ) = OPE., Res z f f z, [j] w = j j! w j 3.2. (i) (z w) N w [j] δ(z w) = 0 (N j + ). (ii) Res z (z w) N δ w [j] δ(z w) = δ N,j V field a(z), b(z) local, N (z w) N [a(z), b(w)] = V field a(z), b(z). (7) (i) a(z), b(z) local. (ii) V field c 0 (z), c (z),... c N (z),. [a(z), b(w)] = N j=0 c j (w) [j] w δ(z w). (iii) V field c 0 (z), c (z),... c N (z),. N ( ) a(z)b(w) =: a(z)b(w) : + c j (w)τ z,w (z w) j+,, j=0 N ( b(w)a(z) =: a(z)b(w) : + c j (w)τ w,z (z w) j+ j=0 : a(z)b(w) := a(z) b(w) + b(w)a(z) +, a(z) = n<0 a (n)z n, a(z) + = n 0 a (n)z n., 3.4 (ii) ( (iii)), a(z)b(w) N j=0 c j (w) (z w) j+, a(z) b(w) OPE(operator product expansion). 3.2 (ii) (8). n (9) c j (w) = Res z (z w) j [a(z), b(w)] a(w) (n) b(w) = Res z (z w) n [a(z), b(w)], a(w) b(w) n. Dong. 3.5 (Li[Li96]). a(z), b(z), c(z) local, n Z a(z) (n) b(z) c(z) local. ).
6 6 TOMOYUKI ARAKAWA (20) OPE a(z)b(w) j 0 a(w) (j) b(w) (z w) j , V. V, T End V, V field {a α (z); α A} ( ).. (i) α, β A a α (z), a β (z) local. (ii) a α (n )... aα r (n r ) (α i A, n i Z) V. (iii) α A, a α (n) = 0 (n 0). aα (z) V [[z]]. (iv) T = 0. (v) α A, [T, a α (z)] = z a α (z). 3.7 (state-field correspondence). V, Y (?, z) : V (End V )[[z, z ]], a Y (a, z) = a(z) = n Z a (n) z n ( state-field correspondence ). (i) a α (z) Y (a α ( ), z) = aα (z). (ii) a V Y (a, z) V field. (iii) a, b V Y (a, z) Y (b, z) local. (iv) a V, [T, Y (a, z)] = z Y (a, z). (v) a V. Y (a, z) V [[z]] lim z 0 Y (a, z) = a., state-field correspondence V a field a(z) = Y (a, z). V. (2) Y (a, z)y (b, w) j 0 Y (a (j) b, w) (z w) j+, (22) Y (a (n) b, w) = Res z (z w) n [Y (a, z), Y (b, w)]... [a (m), b (n) ] = ( ) m (23) (a j (j) b) (m+n j), j 0 (a (m) b) (n) = ( ) m (24) ( ) j (a j (m j) b (n+j) ( ) m b (m+n j) a (j) ). j 0 (23), (24) m.
7 PRINCIPAL AFFINE W -ALGEBRAS: AN OVERVIEW ( ) g. (g, ( )) Lie g aff. (25). g aff = g[t, t ] CK CD. [x(m), y(n)] = [x, y](m + n) + m(x y)δ m+n,0 K [D, x(m)] = mx(m), [K, g aff ] = 0., x(m) = x t m. C k (26) V k (g) = U(g aff ) U(g[t] CK CD) C k (x, y g),. C k g[t] CD, K k. V k (g) field x(z) (x g), (27) x(z) = n Z x(n)z n 4., x(z), y(z) (x, y g) local, OPE. (28) x(z)y(w) [x, y](w) z w + k(x y) (z w) 2. V k (g) =, {x(z); x g}. V k (g) (g, ( )) k M V, (29),. (30) (3) (32) Y M (?, z) : V (End M)[[z, z ]], a Y M (a, z) Y M (a, z) M field Y M (a, z)y M (b, w) j 0 Y M (a (j) b, w) (z w) j+, Y M (a (n) b, w) = Res z (z w) n [Y M (a, z), Y M (b, w)]. V V. N V V/N Zhu., V H 5 : V = V, V = {v V : Hv = v}. Z 0, V. (33) [H, Y (a, z)] = Y (Ha, z) + zy (Ha, z) a V. 4 ( ) x(n) = x(n). 5 VOA( ),.
8 8 TOMOYUKI ARAKAWA H a a., (34) [H, a (n) ] = (n a + )a (n) V k (g) field S(z). (35) S(z) = a : J a (z)j a (z) :., {J a } g. State-field correspondence S(z) = Y (S ( ), z). k h, (36), L(z) OPE L(z) = n Z L(n)z n 2 = L(z)L(w) z w wl(w) + 2(k + h ) S(z) k dim g (z w) 2 L(w) + k+h 2(z w) 4., {L(n); n Z} k dim g k+h Virasoro ( )., L(0) V k (g). (37) [L(0), xt n ] = nxt n, L(0) = 0, L(0) = D. (L(0) ) k = h D V h (g). (38), Lie V = V [t, t ]/ Im(T + d/dt), Lie V Lie ([Bor86]). [at m, bt n ] = ( ) m (39) (a j (j) b)t m+n j. j 0, a t m + Im(T + d/dt) at m. [H, at m ] = (n a +)at m, V Lie V : Lie V = d Z (Lie V ) d., (Lie V ) d = {x Lie V ; Hx = dx}. (23) V Lie V., (24), Lie V V. U(Lie V ) = d Z U(Lie V ) d,. Ũ(Lie V ) = d Z Ũ(Lie V ) d, Ũ(Lie V ) d = lim N U(Lie V ) d / r>n U(Lie V ) d r U(Lie V ) r. ( [MNT05] standard degreewise filtration.) (24) t n δ n, Ũ(Lie V ) graded ideal I. Ũ(Lie V ) I U(V ), V 6. U(V ) : U(V ) = d Z U(V ) d. 6 ( ) [FZ92] V.
9 PRINCIPAL AFFINE W -ALGEBRAS: AN OVERVIEW 9 U(V ) 0 U(V ), U(V ) 0 Zh(V ) (40) Zh(V ) = U(V ) 0 / p>0 U(V ) p U(V ) p, V Zhu 7. C A, Zh(V ) = A V A (, V unique ) (4) (42) U(V k (g)) = Ũk(g aff ), Zh(V k (g)) = U(g)., Ũk(g aff ) U k (g aff ) = U(g aff )/ K k standard degreewise completion. (Ũk(g aff ) ad D. ) V k (g) U(g). V -Mod M = d C M d graded V. C d,..., d r, d i d i Z 0 M d = 0. V -Mod V positive energy representation ( admissible representation). Zh(V ). 3. (Zhu[Zhu96]). V -Mod Zh(V ). V k (g) g aff V k (g),. k Z 0, Zh(V k (g)) = U(g)/ e k+ θ. Open Problem. k 8, Zh(V k (g)) standard filtration. C 2 (V ) a ( 2) b (a, b V ) V. V/C 2 (V ) ([Zhu96]). (43) (44) ā b = a ( ) b, {ā, b} = a (0) b V/C 2 (V ), V 20., V field. V C 2 (V k (g)) = g[t ]t 2 V k (g). V k (g)/c 2 (V k (g)) = S(g t ) = S(g) = C[g ] V. 7 Zhu[Zhu96] original, ([FZ92, NT05]) 8 k Q>0 \Z >0 9 sl2 ( ). 20.
10 0 TOMOYUKI ARAKAWA (i) V (OPE 0). (ii) a V, a (n) = 0 (n 0). 3.3 V (45) Y (a, z) = n a (n) z n (End V )[[z]]., V. (46) a b = a ( ) b. T,. 3.4 ([Li04]). V (47) F p V = span{a ( n )... ar ( n r ) ; n i, a + + a r p}., {F p V } V quasi-commutative filtration. gr V., standard filtration V compatible, (48) V = F V F V F V = F V /F V gr V V/C 2 (V ). 0, gr V V/C 2 (V ). (49) C[(Spec V/C 2 (V )) ] gr V V k (g) = U(g[t ]t ), V k (g) standard filtration U(g[t ]t ) standard filtration. (50) gr V k (g) = S(g[t ]t ) = C[g ]. (49). 4. Feigin-Frenkel [FBZ04, Fre07]. 4.. BRST Z(ḡ) Whittaker. e g, sl 2 {e, h, f}. g j = {x g; [h, x] = 2jx}, n + = j>0 g j, h = g 0, n = j<0 ḡj, g = n h n + g. b ± = n ± h. N n (Kostant [Kos78]).. N (f + g e ) f + b +.
11 PRINCIPAL AFFINE W -ALGEBRAS: AN OVERVIEW, (5) µ : g = g n + = n, µ N. χ = (f?) n + χ n +. χ N. µ (χ) = f + b +, Hamiltonian reduction f + g e = (f + b + )/N reduced Poisson variety ([GG02] )., C[g] G C[f + g e ] = C[f + b + ] N. C[g] G = C[f + g e ], C[g] G BRST [KS87]. C[g] U(g). Cl. ( ) : ψ α (α ), : [ψ α, ψ β ] = δ α+β,0. g. +, +, ψ α (α ± ) Cl Λ(n ± ), Cl = Λ(n ) Λ(n + ). U(g) purely even, U(g) Cl. (.) U(g) Cl odd Q. (52) Q = (x α + χ(x α ))ψ α c γ α,β 2 ψ αψ β ψ γ α + α,β,γ +, x α α, c γ α,β. (53) Q 2 = 0. Q odd, ( ) (54) (ad Q) 2 = 0., deg u = 0 (u U(g)), deg ψ α = (α ), deg ψ α = (α + ) U(g) Cl, (U(g) Cl, ad Q). (55) H (U(g) Cl, ad Q) = i Z H i (U(g) Cl, ad Q). 4.3 ([Kos78, KS87]). (i) H i 0 (U(g) Cl, ad Q) = 0. (ii) Z(g) H 0 (U(g) Cl, ad Q).
12 2 TOMOYUKI ARAKAWA 4.2. W : Whittaker. k C, V k (g) U(g). OPE odd field ψ α (α ) ( ) 2 +. (56) (57) (58) ψ α (z)ψ β (w) δ α+β,0 z w. ψ α (z) = n Z ψ α (n)z n (α + ), ψ α (z) = n Z ψ α (n)z n (α ) (59) [ψ α (m), ψ α (n)] = δ α+β,0 δ m+n,0. odd ψ α (n) (α, n Z) (59) 2 Cl aff. + 2 Cl aff. + [H, ψ α (n)] = nψ α (n), H = 0., H, deg = 0, deg ψ α (n) = (α ), deg ψ α (n) = (α + ) : 2 + = 2 +i. i Z 4.4. U( 2 + ) = Cl aff, Zh( 2 + ) = Cl., Cl aff Cl aff standard degreewise completion. V k (g), 2 +, V k (g) U(V k (g) 2 + ) = Uk (g aff ) Cl aff, Zh(V k (g) 2 + ) = U(g) Cl., Uk (g aff ) Cl aff U k (g aff ) Cl aff standard degreewise completion. (60) V k (g) 2 + field Q(z), Q(z) = α (x α (z) + χ(x α ))ψ α (z) 2 α,β,γ c γ α,β ψ α(z)ψ β (z)ψ γ (z). State-field correspondence, Q(z) = Y (Q ( ), z).. (6) Q(z)Q(w) 0. Q(z) odd field, (6). (62) Q 2 (0) = 0, [Q (m), Q (n) ] = 0 for all m, n Z. Q (0) V k (g) 2 +i V k (g) 2 +i+. (V k (g) 2 +, Q (0) ). (63) [Q (0), Y (a, z)] = Y (Q (0) a, v)
13 PRINCIPAL AFFINE W -ALGEBRAS: AN OVERVIEW 3, H (V k (g) 2 +, Q (0) )., (64) W k (g) := H 0 (V k (g) 2 +, Q (0) ), H (V k (g) 2 +, Q (0) ) (purely even ). W k (g) g ( ) k ( ) W Clasical Hamiltonian reduction. C = V k (g) 2 +. C /C 2 (C ) (C, Q (0) ), Q (0) C /C 2 (C ) Q ( )., (65) C /C 2 (C ) = C[g ] Λ(n +) Λ(n + ), C /C 2 (C ) H 0 (C /C 2 (C ), Q (0) ) (BRST ) (5) Hamiltonian reduction ([Kos78, KS87]). (i) H i 0 (C /C 2 (C ), Q (0) ) = 0, (ii) C[g ] G H 0 (C /C 2 (C ), Q (0) )., Feigin-Frenkel, Frenkel-Ben-Zvi. Q (0) Uk (g aff ) Cl aff. Q (0) = x α ( n)ψ α (n) c γ α,β 2 ψ α( k)ψ β ( l)ψ γ (k + l) α + n Z α,β,γ + k,l Z + α χ(x α )ψ α (), Q (0) V k (g) 2 +. H new [H new, x α (n)] = (n ht α)x α (n), [H new, x α (n)] = (n + ht α)x α (n) (α + ), [H new, J(n)] = nj(n) (J h), [H new, ψ α (n)] = (n ht α)ψ α (n), [H new, ψ α (n)] = (n + ht α)ψ α (n) (α + ), H new = 0, H new Q (0) V k (g) 2 +., H new W k (g) V k (g) 2 + : 4.8 (Feigin-Frenkel[FF90], Frenkel-Ben-Zvi [FBZ04]). k C. (i) H i 0 (V k (g) 2 +, Q (0) ) = 0. (ii) W k (g)/c 2 (W k (g)) = H i 0 (V k (g)/c 2 (V k (g)), Q (0) ) = C[g] G. (iii) W k (g) = 0 ( < 0)., gr W k (g) = C[g ] G.
14 4 TOMOYUKI ARAKAWA W i P i (mod C 2 (W k (g))) W k (g) W i, 4.8 gr W k (g) = C[T j Wj ; j =, 2,..., l, j 0]. W k (g) l W (z), W 2 (z),... W l (z). W i d i W : W (z) = S(z) + 2(k + h )( α + : z ψ α (z)ψ α (z) : + z ρ (z)),,. k h, ρ (z) = ρ (z) + W ( ) S ( ) L(z) = α + ht α : ψ α (z)ψ α (z) :. (mod C 2 (V k (g))) 2(k + h ) W (z) c(k) = l 2(κ ρ 2 2 ρ, ρ + ρ 2 /κ), κ = k + h Virasoro, L(0) = H, L( ) = T. W k (g). W k (sl 2 ) c(k) 2 Virasoro W. V Z(V ) Z(V ) = {a V : [a (m), b (n) ] = 0 for all m, n Z, b V }. Z(V ) V. 4.0 (Feigin-Frenkel [FF92]). (i) k h Z(V k (g)) = C. (ii) Z(V h (g)) W h (g) W Langlands. Feigin-Frenkel [FF92] k generic. 4.. k h. W k (g) = W Lk ( L g)., L g g Langlands, r (k + h )( L k + L h ) =. 4. k, Feigin-Frenkel Drinfeld. (66) C[Op G (D)](= W (g)) = Z(V h ( L g))., Op G (D) Disk D oper [BD05] ([FBZ04] ). 2 sl2 k + 2 = p/q c(k) = 6(p q) 2 /pq
15 PRINCIPAL AFFINE W -ALGEBRAS: AN OVERVIEW 5 5. BRST W M V k (g). M 2 + V k (g) 2 +., (67) H (M 2 +, Q (0) ) W k (g).,, (68) H (M 2 +, Q (0) ) = H 2 + (n + [t, t ], M C χaff )., Feigin [Feĭ84] semi-infinite, χ aff n + [t, t ]., (69) χ aff (x(n)) = δ n, χ(x) M H 2 +0 (L n +, M C χaff ) V k (g) W k (g) g aff. h aff = h CK CD g aff Cartan., O k g aff k BGG. k λ h aff, M(λ), L(λ) λ g aff Verma,. (4), O k V k (g). O KL k D M d M, O k. λ λ h aff h, L(λ) O k Ok KL, λ P + = {λ h ; λ, α Z 0 α + }. + - k + h Q >0 O KL k. 5. (A., to appear). k + h Q >0, M O KL k O KL k H 2 +i (n + [t, t ], M C χ+ ) = 0 i 0., W k (g) -Mod, M H 2 +0 (n + [t, t ], M C χ+ )., g aff., 4.0, W. 5.2 (A., to appear). O KL h L(λ), dim H 2 +0 (n + [t, t ], L(λ) C χaff ) =. 5., 5.2 ( ). 5.3 ([Ara07a]). O h KL. ch L(λ) =, re aff,+ g aff. L(λ), ch L(λ) w W ( )l(w) e w λ α + ( e λ+ρ,α δ ) α ( e α ). re aff,+
16 6 TOMOYUKI ARAKAWA k + h Q >0 Lie (, sl 2, [Ara05] ). Frenkel-Kac-Wakimoto [FKW92], W k (g) Zhu. Uk (g aff ) Cl aff Q aff. Q aff = x α ( n)ψ α (n) c γ α,β 2 ψ α( k)ψ β ( l)ψ γ (k + l) α n Z α,β,γ k,l Z + α χ(x α )ψ α (0). (Q aff = w 0 t ρ (Q (0) )., w 0 W, t ρ ρ extended Weyl )., (70) Q 2 aff = 0, (ad Q aff ) 2 = 0. Q aff Uk (g aff ) Cl aff ( ) ([Ara07b]). k C. (i) U(W k (g)) = H 0 ( Uk (g aff ) Cl aff, ad Q aff ) (ii) Zh(W k (g)) = H 0 (U(g) Cl, ad Q) = Z(g). 5.4 (ii) W k (g) Z(g)., 3., W k (g) positive energy representation Z(g). γ W k (g) L(γ). k = h L(γ),, L(γ) M O k aff M 2 + Uk (g aff ) Cl aff, (7) Q aff (M 2 +i ) M 2 +i, Q 2 aff = 0., 5.4 (i), (72) H (M 2 +, Q aff ) W k (g).,, (73) 24., χ aff H (M 2 +, Q aff ) = H 2 + (n [t, t ], M C χ ) aff χ aff (x α(n)) = δ n,0 χ(x α ) n [t, t ]. M H 2 + (n [t, t ], M C χ ) aff Oaff k W k (g) -Mod.. +, D. 23, Q(0). 24 semi-infinite cohomology semi-infinite homology grading.
17 PRINCIPAL AFFINE W -ALGEBRAS: AN OVERVIEW W k (g). λ h, M g (λ) g Verma, γ λ : Z(g) C M g (λ) evaluation. 5.5 ([Ara07b]). k C. (i) O k aff M, H 2 +i (n [t, t ], M C χ ) = 0 (i 0). aff (ii) k λ h aff H 2 +i (n [t, t ], M C χ ) = aff { L(γ λ) λ anti-dominant λ g anti-dominant, λ g aff generic (!) ( 5.7 ). γ Z(g), γ = γ λ anti-dominant λ h. λ = λ+kλ 0, 5.5 (iii) L(γ λ) = H 2 +0 (n [t, t ], L(λ) C χ ) aff. W k (g). L(λ) (74) ch L(λ) = µ h [L(λ) : M(µ))] ch M(µ), [L(λ) : M(µ))] Z, Euler-Poincare, 5.5 (i) (75) ch L(γ λ) = i ( qi ) l [L(λ) : M(µ)]q µ,d µ h. ch L(γ) = tr L(γ) q H new. k h, [L(λ) : M(µ)] (Kazhdan-Lusztig ) ([KT00]). (75) W k (g). 5.7., (75) [Hay88] Lie Kac-Kazhdan [KK79] ([Ara06]) 5.7. Frenkel-Kac-Wakimoto.,. Lie g aff,., 5.5,. Kac-Wakimoto. Kac-Wakimoto SL 2 (Z), ([KW89]) λ h aff,. (i) λ regular dominant., λ + ρ aff, α 0,, 2,..., ( α re aff,+ ). (ii) λ integral root system (λ) g root system : (λ) = {α re aff; λ + ρ aff, α Z} = re aff.
18 8 TOMOYUKI ARAKAWA,. Kac-Wakimoto [KW89], k (76) k + h = q p Q >0, p, q, (p, q) =, p h, (q, r ) =. λ anti-dominant, (76) (77). λ, (78) ch L(λ) = w W (λ) q h = g Coxeter ( ) l(w) e w λ α aff ( e α ) dim(g aff ) α, λ integral Weyl group W (λ) = s α ; α (λ) g aff W aff., SL 2 (Z). (75,78), λ anti-dominant λ, (79) ch L(γ λ) = w W (λ) ( )l(w) q w λ,d i ( qi ) l, SL 2 (Z). Frenkel-Kac-Wakimoto[FKW92], λ H 2 +i ( n [t, t ], L(λ) C χ ) = 0 (i 0) H aff 2 +0 ( n [t, t ], L(λ) C χ ) aff, SL 2 (Z) W k (g) sl 2 ( ) Virasoro W., W g O. W k (g) O W. O, e O, {e, h, f} sl 2, W W k (g, O) (80) gr W k (g, O) = C[(f + g e ) ]. O ([Ara05] ). g 26. Richardson, [Mat90],., W. W A ([BK05]), W. ( ). Richardson, ( )., W ([Pre06, Los07]),. 26 [Ara05] Lie [GK07].
19 PRINCIPAL AFFINE W -ALGEBRAS: AN OVERVIEW 9 References [Ara05] Tomoyuki Arakawa. Representation theory of superconformal algebras and the Kac- Roan-Wakimoto conjecture. Duke Math. J., 30(3): , [Ara06] Tomoyuki Arakawa. A new proof of the Kac-Kazhdan conjecture. Int. Math. Res. Not., pages Art. ID 2709, 5, [Ara07a] Tomoyuki Arakawa. Characters of representations of affine Kac-Moody Lie algebras at the critical level. preprint, arxiv: v2[math.aq]. [Ara07b] Tomoyuki Arakawa. Representation theory of W -algebras. Invent. Math., 69(2):29 320, [BD] A. Beilison and V. Drinfeld. Quantization of Hitchin s integral system and hecke eigensheaves. preprint. [BD05] Alexander Beilinson and Vladimir Drinfeld. Opers. preprint, arxiv:math/050398v [math.ag]. [BK05] Jonathan Brundan and Alexander Kleshchev. Representations of shifted Yangians and finite W -algebras. preprint, math.rt/ , to appear in Mem. Amer. Math. Soc. [BK06] Jonathan Brundan and Alexander Kleshchev. Shifted Yangians and finite W -algebras. Adv. Math., 200():36 95, [BMP96] Peter Bouwknegt, Jim McCarthy, and Krzysztof Pilch. The W 3 algebra, volume 42 of Lecture Notes in Physics. New Series m: Monographs. Springer-Verlag, Berlin, 996. Modules, semi-infinite cohomology and BV algebras. [Bor86] Richard E. Borcherds. Vertex algebras, Kac-Moody algebras, and the Monster. Proc. Nat. Acad. Sci. U.S.A., 83(0): , 986. [BS95] P. Bouwknegt and K. Schoutens, editors. W-symmetry, volume 22 of Advanced Series in Mathematical Physics. World Scientific Publishing Co. Inc., River Edge, NJ, 995. [dbt93] Jan de Boer and Tjark Tjin. Quantization and representation theory of finite W algebras. Comm. Math. Phys., 58(3):485 56, 993. [DSK06] Alberto De Sole and Victor G. Kac. Finite vs affine W -algebras. Japan. J. Math., ():37 26, [FBZ04] Edward Frenkel and David Ben-Zvi. Vertex algebras and algebraic curves, volume 88 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, second edition, [Feĭ84] B. L. Feĭgin. Semi-infinite homology of Lie, Kac-Moody and Virasoro algebras. Uspekhi Mat. Nauk, 39(2(236)):95 96, 984. [FF90] Boris Feigin and Edward Frenkel. Quantization of the Drinfel d-sokolov reduction. Phys. [FF92] Lett. B, 246(-2):75 8, 990. Boris Feigin and Edward Frenkel. Affine Kac-Moody algebras at the critical level and Gel fand-dikiĭ algebras. In Infinite analysis, Part A, B (Kyoto, 99), volume 6 of Adv. Ser. Math. Phys., pages World Sci. Publ., River Edge, NJ, 992. [FKW92] Edward Frenkel, Victor Kac, and Minoru Wakimoto. Characters and fusion rules for W -algebras via quantized Drinfel d-sokolov reduction. Comm. Math. Phys., 47(2): , 992. [FL88] [Fre07] [FZ92] [GG02] [GK07] [Hay88] [Kac98] V. A. Fateev and S. L. Lykyanov. The models of two-dimensional conformal quantum field theory with Z bn symmetry. Internat. J. Modern Phys. A, 3(2): , 988. Edward Frenkel. Langlands correspondence for loop groups, volume 03 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, Igor B. Frenkel and Yongchang Zhu. Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J., 66():23 68, 992. Wee Liang Gan and Victor Ginzburg. Quantization of Slodowy slices. Int. Math. Res. Not., (5): , Maria Gorelik and Victor Kac. On simplicity of vacuum modules. Adv. Math., 2(2):62 677, Takahiro Hayashi. Sugawara operators and Kac-Kazhdan conjecture. Invent. Math., 94():3 52, 988. Victor Kac. Vertex algebras for beginners, volume 0 of University Lecture Series. American Mathematical Society, Providence, RI, second edition, 998.
20 20 TOMOYUKI ARAKAWA [KK79] V. G. Kac and D. A. Kazhdan. Structure of representations with highest weight of infinite-dimensional Lie algebras. Adv. in Math., 34():97 08, 979. [Kos78] Bertram Kostant. On Whittaker vectors and representation theory. Invent. Math., 48(2):0 84, 978. [KRW03] Victor Kac, Shi-Shyr Roan, and Minoru Wakimoto. Quantum reduction for affine superalgebras. Comm. Math. Phys., 24(2-3): , [KS87] Bertram Kostant and Shlomo Sternberg. Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras. Ann. Physics, 76():49 3, 987. [KT00] Masaki Kashiwara and Toshiyuki Tanisaki. Characters of irreducible modules with noncritical highest weights over affine Lie algebras. In Representations and quantizations (Shanghai, 998), pages China High. Educ. Press, Beijing, [KW89] V. G. Kac and M. Wakimoto. Classification of modular invariant representations of affine algebras. In Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 988), volume 7 of Adv. Ser. Math. Phys., pages World Sci. Publ., Teaneck, NJ, 989. [Li96] Hai-Sheng Li. Local systems of vertex operators, vertex superalgebras and modules. J. Pure Appl. Algebra, 09(2):43 95, 996. [Li04] Haisheng Li. Vertex algebras and vertex Poisson algebras. Commun. Contemp. Math., 6():6 0, [Los07] Ivan V. Losev. Quantized symplectic actions and W-algebras. preprint, arxiv: v2[math.rt]. [Lyn79] T. E. Lynch. Generalized Whittaker vectors and representation theory. PhD thesis, M.I.T., 979. [Mat90] Hisayosi Matumoto. Whittaker modules associated with highest weight modules. Duke Math. J., 60():59 3, 990. [MN99] Atsushi Matsuo and Kiyokazu Nagatomo. Axioms for a vertex algebra and the locality of quantum fields, volume 4 of MSJ Memoirs. Mathematical Society of Japan, Tokyo, 999. [MNT05] Atsushi Matsuo, Kiyokazu Nagatomo, and Akihiro Tsuchiya. Quasi-finite algebras graded by Hamiltonian and vertex operator algebras. math.qa/050507, [NT05] Kiyokazu Nagatomo and Akihiro Tsuchiya. Conformal field theories associated to regular chiral vertex operator algebras. I. Theories over the projective line. Duke Math. J., 28(3):393 47, [Pre02] Alexander Premet. Special transverse slices and their enveloping algebras. Adv. Math., 70(): 55, With an appendix by Serge Skryabin. [Pre06] Alexander Premet. Primitive ideals, non-restricted representations and finite W- algebras. preprint, arxiv:math/062465v2[math.rt]. [Zam85] A. B. Zamolodchikov. Infinite extra symmetries in two-dimensional conformal quantum field theory. Teoret. Mat. Fiz., 65(3): , 985. [Zhu96] Yongchang Zhu. Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc., 9(): , 996.
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