, 0 = U 1 (g) U 0 (g) U 1 (g)..., U(g) = p U p (g) U p (g)u q (g) U p+q (g), [U p (g), U q (g)] U p+q 1 (g). U(g) PBW,. Associated graded algebra gr U
|
|
|
- ときな よどぎみ
- 5 years ago
- Views:
Transcription
1 W ( ) 1. ( )W Kac-Moody Virasoro,,,,, 4, Langlands.,, W., W, W ([A1, A2, A3, A7]). Premet[Pre] W ( )W, Kostant[Kos]. W Slodowy, primitive ideal. Premet Losev[Los2]. primitive ideal. W. ( )W Losev. Kac-Moody ĝ ( primitive ideal, primitive ideal Kac-Moody kλ 0 (k ) g., Kac-Moody ĝ ĝ ĝ g (Feigin-Frenkel [A5]). ĝ [A4] g,, ([A5])., Kac-Moody [A6], Losev, Kac-Moody W, 20 W [FKW] ([A7]).. 2. g, U(g) g : U(g) = T (g)/ x y x x [x, y] x, y g., T (g) g., g ( )U(g). U p (g) := ( p g ) U(g)
2 , 0 = U 1 (g) U 0 (g) U 1 (g)..., U(g) = p U p (g) U p (g)u q (g) U p+q (g), [U p (g), U q (g)] U p+q 1 (g). U(g) PBW,. Associated graded algebra gr U(g) = p U p (g)/u p 1 (g) Poisson. σ p (a)σ q (b) = σ ab, {σ p (a), σ q (b)} = σ p+q 1 ([a, b])., σ p : U p (g) U p (g)/u p 1 (g) (symbol map). 2.1 (Poincaré-Birkoff-Witt).. gr U(g) = S(g) = C[g ]., g Kirillov-Kostant Poisson ( x, y g C[g ] {x, y} = [x, y]). Z(g) U(g), Z p (g) = Z(g) U p (g) Z(g), gr Z(g) = p Z p(g)/z p 1 (g) gr U(g) = C[g ] Poisson., gr Z(g) = C[g ] G. Lie G = g. I U(g), I p = I U p (g) associated graded gr I = p I p/i p 1 C[g ] C[g ] G. Var(I) := {λ g f(λ) = 0, f gr I} I. Var(I) g G,. I U(g) primitive ideal, M 1. Shur, M γ : Z(g) C, (1) z χ(z) I = Ann U(g) M, z Z(g). N g. N =: {x g (ad x) r = 0, r 0} g. g g g ( ), N = {λ g p(λ) = 0, p C[g ] G +} g (Duflo 77). primitigve ideal.
3 . C[g ] G + C[g ] G argumentation ideal. C[g ] = C[p 1,..., p l ], deg p i > 0, C[g ] G + = l i=1 C[g ]p i.) (1), primitive ideal I 2. Var(I) N 2.3 (Joseph[Jos]). U(g) primitive ideal I., 3 O. Var(I) = O 3. W f N. Jacobson-Morozov f sl 2 {e, f, h} : [h, e] = 2e, [h, f] = 2f, [e, f] = h. f g Slodowy S f := f + g e g = g. S f g G ([GG]). γ : C G, t γ t, h g G 1, C t : x t 2 Ad(γt 1 )(x) S f C. C f. S f. χ g x (f x)., g., g = j Z g j, g j = {x g ad h(x) = jx} g 1 g 1 C (x, y) χ([x, y]). g 1 l, m = l j 2 g j g, m g., χ m (, χ([m, m]) = 0) M m G µ : g m M, χ m M. µ 1 (χ) = χ + m S f, S f. 2, U(g). 3 O = Ad G.x, x N ( )G.
4 3.1 ([Kos, GG]). (i) χ µ. (ii). M S f µ 1 (χ), (g, x) Ad g.x. 3.1 S f = µ 1 (χ)/m. S f ([Kos, Pre]). BRST ([KS]). Cl m m m, deg m = 1, deg m = 1 : Cl m = p Z Clp m. Cl m Cl m,p = Λ p (m)λ (m ) gr Cl m = p Cl m,p/cl m,p 1.., gr Cl m = Λ(m) Λ(m ) {x, y} = 0 {f, g} = 0 {f, x} = f(x) (x, y m, f, g m ). U(g) Cl m, Lie. θ : m U(m) Cl 0 U(g) Cl m, ad : m Cl 0 m Lie ([KS, Akm, BD] x x ad(x) (i) x m [Q, 1 x] = θ(x) + χ(x) U(g) Cl 1 m Q. (ii) Q Q 2 = 0., Q. m {x i }, {x i }, {c k ij } Q = (x i + χ(x i )) x i 1 1 (2) c k 2 ijx i x j x k. i Q odd Q 2 = 0 (ad Q) 2 = 0. (C (g), ad Q) dga(differential graded algebra). i,j,k H f (U(g)) := H (U(g) Cl m, ad Q). U(g) Cl m F p (U(g) Cl m ) = i+j p U i(g) Cl m,j associated graded gr F (U(g) Cl m ) = gr U(g) gr Cl m = C[g ] gr Cl m 4 m.
5 . Q (F 1 U(g) Cl m) gr F (U(g) Cl m ) = C[g ] gr Cl m Q (ad Q) 2 = 0. (C[g ] gr Cl m, ad Q) dgpa(differential graded Poisson algebra). ( ). 3.3 ([KS]). i Z H i (C[g ] gr Cl m, ad Q) = 0,. H 0 (C[g ] gr Cl m, ad Q) = C[S f ] 3.4 ([Kos, Pre, GG]). H i 0 f (U(g)) = 0 and gr Hf 0(U(g)) = C[S f ]. gr Hf 0(U(g)) U(g) Cl m H f (U(g)) associated graded. S f (3) U(g, f) := H 0 f (U(g)) (g, f) W 5. W, ([A2, DSK, BGK]) g = gl N. (gl N. ) f = f prin := f = F prin., N = Ad G.f prin y 1 y 2 y 3 y N 1 y 1 y 2... y N 1 S f = 0 1 y 1... y N 2 y ,..., y N C y 1, C[S f ] C[g ] GL N = C[h ] S N Chevalley. h g Cartan. (4) U(g, f prin ). Z(g) = S(h) S N, Kostant Whitttaker [Kos] Harish-Chandra 5 U(g, f) l.
6 W. (cf. [A2, A7]) U(g) Harish-Chandra HC : HC = {g U(g) }. M HC M Cl m U(g) Cl m, ad Q. H f (M) = H0 (M Cl m, ad Q) U(g, f), (5). HC { U(g, f) }, M H 0 f (M), 3.6 ([Los2, Gin]). M HC H i 0 f (M) = 0. (5). I U(g) I U(g)/I HC H 0 f (I) U(g, f) H 0 f (U(g)/I) 0. Hf 0 (I) (graded ). U(g, f) J associated variety Var(J) S f, Hf 0 (I) graded Var(Hf 0(I)) S f C. 3.7 ([Los1], [Gin]). U(g) I. Var(H 0 f (I)) = Var(I) S f I U(g) primitive ideal, O Var(I) = O g. (i) Hf 0 (U(g)/I) 0 Ad G.f O. (ii) Hf 0 (U(g)/I) ( ) ( ) f O. Hf 0(U(g)/I) U(g, f), H0 f (U(g)/I) U(g, f). 3.9 ([Los2]). I U(g) primitive ideal, O Var(I) = O g, f O, Hf 0 (U(g)/I). U(g, f). 4. ĝ g Kac-Moody. ĝ = g C[t, t 1 ] CK. [xt m, yt n ] = [x, y]t m+n + (x y)δ m+n,0 K, [K, ĝ] = 0 (x, y g). g (θ, θ) = 2. θ g. k C V k (g) := U(ĝ) U(g[t] CK) C k
7 ., C k g[t], K k g[t] CK. V k (g), V k (g) = k ĝ ([Kac, FBZ] ). ĝ M ĝ k K k, x g, m M xt r m = 0 r. V k (g) g k. N k (g) V k (g), N k (g) V k (g). L k (g) = V k (g)/n k (g). L k (g) L k (g) -Mod V k (g) V k (g) -Mod(= k ĝ ). L k (g)-mod = {M V k (g)-mod a (n) M = 0 a N k (g), n Z}, V V -Mod V, V (End M)[[z, z 1 ]], a a(z) = n Z a (n) z n 1 V M. L k (g)-mod., k, L k (g)-mod ĝ k. V k (g), L k (g) deg xt n = n. Zhu[FZ] { } {C }, V A(V ),,. {Z 0 V } { A(V ) } M M top., M top M., A(V k (g)) = U(g) k, U(g) E V k (g) U(g) U(g[t] CK) E. V V A(V ) A(V ). L k (g) Zhu U(g) : A(L k (g)) = U(g)/I k I k. L k (g) (U(g) modulo ) I k U(g) primitive ideal. Primitive ideal J I k Var(J) Var(I k ), I k associated variety. A(V ),.
8 , V. Zhu[Zhu], R V := V/C 2 (V ), C 2 (V ) = {a ( 2) b a, b V }.. ā b = a ( 1) b, {ā, b} = a (0) b. R V Zhu C 2. V X V X V = Specm R V. V V k (g),, C 2 (V ) = g[t 1 ]t 2 V., C[g ] = S(g) = R V k (g), x xt 1 1 X V k (g) = g. X Lk (g) g G. X V V. 4.2 ([A4]).. (i) dim X V = 0, (ii) dim Spec(gr V ) = 0. gr V V associated graded vertex algebra 6. X V 0 C , C 2, ([ABD, H2, H1, Miy] ) (i) dim X Lk (g) = 0, (ii) L k (g). (iii) k Z 0., C 2 Kac-Moody. Zhu Zhu C ([ALY]). R V gr A(V ). gr A(V ) V A(V ) associated graded Poisson algebra. 6 Poisson vertex algebra, C.
9 (6) Var(I k ) X Lk (g). R Lk (g) gr A(L k (g)) 7,. 4.5 ([A7]). k Var(I k ) = X Lk (g). U(g) primitive associated variety, X Lk (g) N., k L k (g) = V k (g) X Lk (g) = g., 4.3 k Z 0 X Lk (g) = {0}. X Lk (g) N h g Cartan, ĥ = h CK ĝ Cartan. ĥ+ = h CΛ 0., Λ 0 (K) = 1, Λ 0 (h) = 0. re re ĝ, + ĝ. λ ĥ L(λ) λ ĝ. ĝ, L k (g) = L(kΛ 0 ). L(λ). (i) λ regular dominant., λ + ρ, α {0, 1, }, α re + ĝ. (ii) Q (λ) = Q re., Q (λ) λ integral root system: (λ) = {α re λ, α Z}., { } { }..,. (i) Weyl-Kac. (7) ch L(λ) = w c W (λ) re + ( 1) lλ(w) e w λ α b + (1 e α ) dim bg. α Ŵ (λ) re (λ) ĝ Ŵ. (ii) ch L(λ), SL 2 (Z) ( [KW1]). 2 ([GK]). ĝ., ( ) 8,. (, (. 7 g E8 k = 1, (vareity {0} ). 8 (7) c W (λ).
10 k L(kΛ 0 ).. (1) k Q (2) kλ 0 ĝ regular dominant. ([KW2]). { k + h = p q, (p, q) N, (p, q) = 1, p h (r, q) = 1, h (r, q) = r 1 g ADE, h, h ĝ Coxter Coxeter, r = 2 g BCF 3 g G 2. Feigin-Frenkel, g = sl 2 Feigin-Malikov [FM]. 4.6 ([A5]). k X Lk (g) N. (Joseph ). 4.7 ([A5]). k X Lk (g) k q N N., q N O q, q k X Lk (g) = O q. O q. { {x g (ad x) 2q = 0}, (q, r ) = 1, O q = {x g π θs (x) 2q/r = 0}, (q, r ) = r. θ s g, π θs θ s g. { (n) n (q n) 4.8. g = sl n (C), O q (q, q,..., q, s) n (0 s < n) (q < n) k Specm(gr L k (g)) = JO q. JX X [A7] k 4.5., q N k Var(I k ) = O q ( )W. Duflo, I k U(g) primitive ideal ([A6]). λ ĥ k(i.e. λ(k) = k) L(λ) L k (g) L(λ) (λ) = (kλ 0 ). 9 C X, JX C A Hom(Spec A, JX) = Hom(Spec A[[t]]], X) (cf. citeeinmus).
11 5. ( )W f N, W k (g, f) (g, f) k C W ([FF, KRW]). W k (g, f),. X Wk (g,f) = S f, gr W k (g, f) = C[JS f ] ([DSK, A5]). W k (g, f) W U(g, f) 10 ([[DSK, A2]]). A(W k (g, f)) = U(g, f) W k (g, f) Kac-Moody Virasoro. W k (g, 0) = V k (g), W k (sl 2, f) (k 2 ) 1 6(k+1) 2 /(k+2) Virasoro. W k (g, f) Drinfeld-Sokolov. BRST H f (?) W k (g, f) = H 0 f (V k (g)) ((3) ). W k (g, f) 11 W k (g, f). X WK (g,f) = S f X Wk (g,f) S f C., W k (g, f) C 2 X Wk (g,f) = {f} g = gl N, f = f prin. (4), U(g, f prin ) S(h) S N S(h). h Heisenberg. h ( κ = k + N 12 ) H κ. x i (z) = n Z x i,nz n 1 (i = 1,..., N), OPE x i (z)x j (w) κδ i,j (z w) 2 ( [x i,n, x j,m ] = κnδ n+m,0 δ i,j ). Harish-Chandra, W k (g, f prin ) H κ.. H κ W k (g, f prin ) W (1) (z), W (2) (z),..., W (N) (z) 10 W k (g, f) 1 2 Z 0-graded Zhu Ramond twisted Zhu. 11 k = h graded simple quotient 12 κ 0.
12 . N W (j) (z)(ν ) N j =: (ν z + x 1 (z))(ν z + x 1 (z))... (ν z + x N (z)) : j=0, z x i (z) = d dz x i(z) + x i (z) z, ν = κ 1 = k + N 1. W (r) (z) = : x i1 (z)x i2 (z)... x ir (z) : +lower i 1 <i 2 < <i r ([FL], [AM] )., W (r) )(z) OPE( ) closed formula ( ).,,. V k (g)-mod W k (g, f)-mod, M H 0 f (M) KL k ĝ k Harish-Chandra (ĝ, G[[t]]). KL k G[[t]], g[[t]] g[t] ĝ k ĝ. 5.2 ([A5]). k C, M KL k H i 0 f (M) = 0.. KL k W k (g, f)-mod, M H 0 f (M),, V k (g) L k (g) W k (g, f) = H 0 f (V k (g)) H 0 f (L k(g)). H 0 f (L k(g)) W k (g, f). W W k (g, f). 5.3 ([FKW, KRW]). H 0 f (L k(g)) W k (g, f). [A1, A2, A3]. 5.4 ([A5]). k C, f N. X H 0 f (L k (g)) = X L(kΛ0 ) S f H 0 f (L k(g)) 0 f X L(kΛ0 ) k q N, f O q. X H 0 f (L(kΛ 0 )) = {f}. H 0 f (L(kΛ 0)) C 2. W k (g, f) C W V V C 2, CFT. (i) ([Zhu], Dong-Lin-Ng]) {M 1,..., M r } V, v V k {q c v 24 tr Mi (o(v)q L 0 )} k., o(v) V v(z). (ii) ([H1]) V. Reshetikhin- Turaev 3.
13 O prin N G. k. X Lk (g) = O prin (= N ). q N k. { h ((q, r ) = 1 ), q r L h ((q, r ) = r ). 6.2 ([FKW]). k W k (g, f prin ) ( C 2 ). g = sl 2, W k (g, f prin ) Virasoro ([BPZ]). 6.2 Kac- [KW2]. O q ([A5]). 6.3 ([KW2]). k q N. W k (g, f). f O q. f Levi type. (q, r ) = 1., (A.). k q N. f O q W k (g, f) (i) ([A7]) 6.2. (ii) ([A8]) g = sl n, ). (iii) ([A8]) f ADE Kac- 6.4 (DE Levi type ). References [ABD] Toshiyuki Abe, Geoffrey Buhl, and Chongying Dong. Rationality, regularity, and C 2 - cofiniteness. Trans. Amer. Math. Soc., 356(8): (electronic), [Akm] Füsun Akman. A characterization of the differential in semi-infinite cohomology. J. Algebra, 162(1): , [ALY] Tomoyuki Arakawa, Ching Hung Lam, and Hiromichi Yamada. Zhu s algebra, C 2 -algebra and C 2 -cofiniteness of parafermion vertex operator algebras. Adv. Math., 264: , [AM] Tomoyuki Arakawa and Alexander Molev. Explicit generators in rectangular affine W -algebras o type A. arxiv: [math.rt]. [A1] Tomoyuk Arakawa. Representation theory of superconformal algebras and the Kac-Roan- Wakimoto conjecture. Duke Math. J., 130(3): , [A2] Tomoyuki Arakawa. Representation theory of W -algebras. Invent. Math., 169(2): , [A3] Tomoyuki Arakawa. Representation theory of W -algebras, II. In Exploring new structures and natural constructions in mathematical physics, volume 61 of Adv. Stud. Pure Math., pages Math. Soc. Japan, Tokyo, [A4] Tomoyuki Arakawa. A remark on the C 2 cofiniteness condition on vertex algebras. Math. Z., 270(1-2): , [A5] Tomoyuki Arakawa. Associated varieties of modules over Kac-Moody algebras and C 2 - cofiniteness of W-algebras v2.
14 [A6] T. Arakawa. Rationality of admissible affine vertex algebras in the category O. arxiv: [math.qa]. [A7] Tomoyuki Arakawa. Rationality of W-algebras; principal nilpotent cases. arxiv: [math.qa]. [A8] Tomoyuki Arakawa. in preparation. [BD] Alexander Beilinson and Vladimir Drinfeld. Quantization of hitchin s integrable system and hecke eigensheaves. preprint. [BGK] Jonathan Brundan, Simon M. Goodwin, and Alexander Kleshchev. Highest weight theory for finite W -algebras. Int. Math. Res. Not. IMRN, (15):Art. ID rnn051, 53, [BPZ] A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov. Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B, 241(2): , [DSK] Alberto De Sole and Victor G. Kac. Finite vs affine W -algebras. Japan. J. Math., 1(1): , [FBZ] 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, [FF] Boris Feigin and Edward Frenkel. Quantization of the Drinfel d-sokolov reduction. Phys. Lett. B, 246(1-2):75 81, [FKW] Edward Frenkel, Victor Kac, and Minoru Wakimoto. Characters and fusion rules for W - algebras via quantized Drinfel d-sokolov reduction. Comm. Math. Phys., 147(2): , [FL] V. A. Fateev and S. L. Lykyanov. The models of two-dimensional conformal quantum field theory with Z n symmetry. Internat. J. Modern Phys. A, 3(2): , [FM] Boris Feigin and Fyodor Malikov. Modular functor and representation theory of sl b 2 at a rational level. In Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), volume 202 of Contemp. Math., pages , Providence, RI, Amer. Math. Soc. [FZ] Igor B. Frenkel and Yongchang Zhu. Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J., 66(1): , [GG] Wee Liang Gan and Victor Ginzburg. Quantization of Slodowy slices. Int. Math. Res. Not., (5): , [Gin] Victor Ginzburg. Harish-Chandra bimodules for quantized Slodowy slices. Represent. Theory, 13: , [GK] Maria Gorelik and Victor Kac. On complete reducibility for infinite-dimensional Lie algebras. Adv. Math., 226(2): , [H1] Yi-Zhi Huang. Rigidity and modularity of vertex tensor categories. Commun. Contemp. Math., 10(suppl. 1): , [H2] Yi-Zhi Huang. Vertex operator algebras and the Verlinde conjecture. Commun. Contemp. Math., 10(1): , [Jos] Anthony Joseph. On the associated variety of a primitive ideal. J. Algebra, 93(2): , [Kac] Victor Kac. Vertex algebras for beginners, volume 10 of University Lecture Series. American Mathematical Society, Providence, RI, second edition, [Kos] Bertram Kostant. On Whittaker vectors and representation theory. Invent. Math., 48(2): , [KRW] Victor Kac, Shi-Shyr Roan, and Minoru Wakimoto. Quantum reduction for affine superalgebras. Comm. Math. Phys., 241(2-3): , [KS] Bertram Kostant and Shlomo Sternberg. Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras. Ann. Physics, 176(1):49 113, [KW1] V. G. Kac and M. Wakimoto. Classification of modular invariant representations of affine algebras. In Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), volume 7 of Adv. Ser. Math. Phys., pages World Sci. Publ., Teaneck, NJ, [KW2] Victor G. Kac and Minoru Wakimoto. On rationality of W -algebras. Transform. Groups, 13(3-4): , [Los1] Ian Losev. 1-dimensional representations and parabolic induction for w-algebras. preprint, arxiv:math/ [math.rt]. [Los2] Ivan Losev. Finite-dimensional representations of W -algebras. Duke Math. J., 159(1):99 143, 2011.
15 [Miy] Masahiko Miyamoto. Modular invariance of vertex operator algebras satisfying C 2 - cofiniteness. Duke Math. J., 122(1):51 91, [Pre] Alexander Premet. Special transverse slices and their enveloping algebras. Adv. Math., 170(1):1 55, With an appendix by Serge Skryabin. [Zhu] Yongchang Zhu. Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc., 9(1): , 1996.
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
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],
1 Affine Lie 1.1 Affine Lie g Lie, 2h A B = tr g ad A ad B A, B g Killig form., h g daul Coxeter number., g = sl n C h = n., g long root 2 2., ρ half
Wess-Zumino-Witten 1999 3 18 Wess-Zumino-Witten., Knizhnik-Zamolodchikov-Bernard,,. 1 Affine Lie 2 1.1 Affine Lie.............................. 2 1.2..................................... 3 2 WZW 4 3 Knizhnik-Zamolodchikov-Bernard
i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,.
R-space ( ) Version 1.1 (2012/02/29) i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,. ii 1 Lie 1 1.1 Killing................................
平成 15 年度 ( 第 25 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 ~8 15 月年 78 日開催月 4 日 ) X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = (
1 1.1 X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = (X 1,..., X n ) ( ) X 1,..., X n f 1,..., f r A T X + XA XBR 1 B T X + C T QC = O X 1.2 X 1,..., X n X i X j X j X i = 0, P i
0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo t
e-mail: koyama@math.keio.ac.jp 0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo type: diffeo universal Teichmuller modular I. I-. Weyl
. Mac Lane [ML98]. 1 2 (strict monoidal category) S 1 R 3 A S 1 [0, 1] C 2 C End C (1) C 4 1 U q (sl 2 ) Drinfeld double. 6 2
![. Mac Lane [ML98]. 1 2 (strict monoidal category) S 1 R 3 A S 1 [0, 1] C 2 C End C (1) C 4 1 U q (sl 2 ) Drinfeld double. 6 2](/thumbs/91/105488587.jpg ". Mac Lane [ML98]. 1 2 (strict monoidal category) S 1 R 3 A S 1 [0, 1] C 2 C End C (1) C 4 1 U q (sl 2 ) Drinfeld double. 6 2")
2014 6 30. 2014 3 1 6 (Hopf algebra) (group) Andruskiewitsch-Santos [AFS09] 1980 Drinfeld (quantum group) Lie Lie (ribbon Hopf algebra) (ribbon category) Turaev [Tur94] Kassel [Kas95] (PD) x12005i@math.nagoya-u.ac.jp
2018 : msjmeeting-2018mar-02i003 : Demazure ( ) 1. Macdonald Weyl Demazure. g, h Cartan., Q := i I Zα i h root lattice, Q + := i I Z 0α
2018 : 2018 21 msjmeeting-2018mar-02i003 : Demazure ( ) 1. Macdonald 1.1. Weyl Demazure. g, h Cartan, Q := i I Zα i h root lattice, Q + := i I Z 0α i Q, P := i I Zϖ i h g weight lattice ;, ϖ i h, i I,
2 2 Belavin Polyakov Zamolodchikov (BPZ) 1984 [13] 2 BPZ BPZ Virasoro [16][18] [20], [30], [47] [1][6] [8][10], [11], [12] Affine [6],GKO [2] W
![2 2 Belavin Polyakov Zamolodchikov (BPZ) 1984 [13] 2 BPZ BPZ Virasoro [16][18] [20], [30], [47] [1][6] [8][10], [11], [12] Affine [6],GKO [2] W](/thumbs/91/105488419.jpg "2 2 Belavin Polyakov Zamolodchikov (BPZ) 1984 [13] 2 BPZ BPZ Virasoro [16][18] [20], [30], [47] [1][6] [8][10], [11], [12] Affine [6],GKO [2] W")
SGC -83 2 2 Belavin Polyakov Zamolodchikov (BPZ) 1984 [13] 2 BPZ BPZ 1 3 4 Virasoro [16][18] [20], [30], [47] [1][6] [8][10], [11], [12] Affine [6],GKO [2] W [14] c = 1 CFT [8] Rational CFT [15], [56]
Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P
![Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P](/thumbs/94/118231293.jpg "Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P")
Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, PC ( 4 5 )., 5, Milnor Milnor., ( 6 )., (I) Z modulo
R R P N (C) 7 C Riemann R K ( ) C R C K 8 (R ) R C K 9 Riemann /C /C Riemann 10 C k 11 k C/k 12 Riemann k Riemann C/k k(c)/k R k F q Riemann 15
(Gen KUROKI) 1 1 : Riemann Spec Z 2? 3 : 4 2 Riemann Riemann Riemann 1 C 5 Riemann Riemann R compact R K C ( C(x) ) K C(R) Riemann R 6 (E-mail address: kuroki@math.tohoku.ac.jp) 1 1 ( 5 ) 2 ( Q ) Spec
1
1 Borel1956 Groupes linéaire algébriques, Ann. of Math. 64 (1956), 20 82. Chevalley1956/58 Sur la classification des groupes de Lie algébriques, Sém. Chevalley 1956/58, E.N.S., Paris. Tits1959 Sur la classification
Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ
![Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ](/thumbs/78/78261740.jpg "Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ")
Galois ) 0 1 1 2 2 4 3 10 4 12 5 14 16 0 Galois Galois Galois TaylorWiles Fermat [W][TW] Galois Galois Galois 1 Noether 2 1 Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R
2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c
GL 2 1 Lie SL(2, R) GL(2, A) Gelbart [Ge] 1 3 [Ge] Jacquet-Langlands [JL] Bump [Bu] Borel([Bo]) ([Ko]) ([Mo]) [Mo] 2 2.1 H = {z C Im(z) > 0} Γ SL(2, Z) Γ N N Γ (N) = {γ SL(2, Z) γ = 1 2 mod N} g SL(2,
I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = { z = x
![I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = { z = x](/thumbs/103/158868201.jpg "I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = { z = x")
I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ). 1.1. modular symbol., notation. H = z = x iy C y > 0, cusp H = H Q., Γ = PSL 2 (Z), G Γ [Γ : G]
Dynkin Serre Weyl
Dynkin Naoya Enomoto 2003.3. paper Dynkin Introduction Dynkin Lie Lie paper 1 0 Introduction 3 I ( ) Lie Dynkin 4 1 ( ) Lie 4 1.1 Lie ( )................................ 4 1.2 Killing form...........................................
1 Part I (warming up lecture). (,,...) 1.1 ( ) M = G/K :. M,. : : R-space. R-space..
( ) ( ) 2012/07/14 1 Part I (warming up lecture). (,,...) 1.1 ( ) M = G/K :. M,. : : R-space. R-space.. 1.2 ( ) ( ): M,. : (Part II). 1 (Part III). : :,, austere,. :, Einstein, Ricci soliton,. 1.3 : (S,
Step 2 O(3) Sym 0 (R 3 ), : a + := λ 1 λ 2 λ 3 a λ 1 λ 2 λ 3. a +. X a +, O(3).X. O(3).X = O(3)/O(3) X, O(3) X. 1.7 Step 3 O(3) Sym 0 (R 3 ),
1 1 1.1,,. 1.1 1.2 O(2) R 2 O(2).p, {0} r > 0. O(3) R 3 O(3).p, {0} r > 0.,, O(n) ( SO(n), O(n) ): Sym 0 (R n ) := {X M(n, R) t X = X, tr(x) = 0}. 1.3 O(n) Sym 0 (R n ) : g.x := gxg 1 (g O(n), X Sym 0
1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civi
1 Surveys in Geometry 1980 2 6, 7 Harmonic Map Plateau Eells-Sampson [5] Siu [19, 20] Kähler 6 Reports on Global Analysis [15] Sacks- Uhlenbeck [18] Siu-Yau [21] Frankel Siu Yau Frankel [13] 1 Surveys
1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe
![1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe](/thumbs/86/93804893.jpg "1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe")
3 del Pezzo (Hirokazu Nasu) 1 [10]. 3 V C C, V Hilbert scheme Hilb V [C]. C V C S V S. C S S V, C V. Hilbert schemes Hilb V Hilb S [S] [C] ( χ(s, N S/V ) χ(c, N C/S )), Hilb V [C] (generically non-reduced)
1 1, 2016 D B. 1.1,.,,. (1). (2). (3) Milnor., (1) (2)., (3). 1.2,.,, ( )..,.,,. 1.3, webpage,.,,.
1 1, 2016 D B. 1.1,.,,. (1). (2). (3) Milnor., (1) (2)., (3). 1.2,.,, ( )..,.,,. 1.3, 2015. webpage,.,,. 2 1 (1),, ( ). (2),,. (3),.,, : Hashinaga, T., Tamaru, H.: Three-dimensional solvsolitons and the
k + (1/2) S k+(1/2) (Γ 0 (N)) N p Hecke T k+(1/2) (p 2 ) S k+1/2 (Γ 0 (N)) M > 0 2k, M S 2k (Γ 0 (M)) Hecke T 2k (p) (p M) 1.1 ( ). k 2 M N M N f S k+
1 SL 2 (R) γ(z) = az + b cz + d ( ) a b z h, γ = SL c d 2 (R) h 4 N Γ 0 (N) {( ) } a b Γ 0 (N) = SL c d 2 (Z) c 0 mod N θ(z) θ(z) = q n2 q = e 2π 1z, z h n Z Γ 0 (4) j(γ, z) ( ) a b θ(γ(z)) = j(γ, z)θ(z)
数学メモアール 第4巻, （2004）
6 9 7 i 2002 5 6 0, [TUY] Lie, P ŝl 2, 3 ) OPE) 2) 3) factorization property ), 2, 4 2) 4, 3) 7, 2, OPE 3, P n 3, 4, 5 6, 7 6, 7 factorization property,,, Lie Lie 2, [K] 5.2, 7.9 5.2 3.4, 7.9, 6.2, 6,
Macdonald, ,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdona
Macdonald, 2015.9.1 9.2.,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdonald,, q., Heckman Opdam q,, Macdonald., 1 ,,. Macdonald,
Gauss Fuchs rigid rigid rigid Nicholas Katz Rigid local systems [6] Fuchs Katz Crawley- Boevey[1] [7] Katz rigid rigid Katz middle convolu
![Gauss Fuchs rigid rigid rigid Nicholas Katz Rigid local systems [6] Fuchs Katz Crawley- Boevey[1] [7] Katz rigid rigid Katz middle convolu](/thumbs/93/114422882.jpg "Gauss Fuchs rigid rigid rigid Nicholas Katz Rigid local systems [6] Fuchs Katz Crawley- Boevey[1] [7] Katz rigid rigid Katz middle convolu")
rigidity 2014.9.1-2014.9.2 Fuchs 1 Introduction y + p(x)y + q(x)y = 0, y 2 p(x), q(x) p(x) q(x) Fuchs 19 Fuchs 83 Gauss Fuchs rigid rigid rigid 7 1970 1996 Nicholas Katz Rigid local systems [6] Fuchs Katz
Chern-Simons Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q
![Chern-Simons Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q](/thumbs/94/118766307.jpg "Chern-Simons Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q")
Chern-Simons E-mail: fuji@th.phys.nagoya-u.ac.jp Jones 3 Chern-Simons - Chern-Simons - Jones J(K; q) []Jones q J (K + ; q) qj (K ; q) = (q /2 q /2 )J (K 0 ; q), () J( ; q) =. (2) K Figure : K +, K, K 0
Siegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo
![Siegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo](/thumbs/96/128156654.jpg "Siegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo")
Siegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo] 2 Hecke ( ) 0 1n J n =, Γ = Γ n = Sp(n, Z) = {γ GL(2n,
K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X
![K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X](/thumbs/91/106734137.jpg "K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X")
2 E 8 1, E 8, [6], II II, E 8, 2, E 8,,, 2 [14],, X/C, f : X P 1 2 3, f, (O), f X NS(X), (O) T ( 1), NS(X), T [15] : MWG(f) NS(X)/T, MWL(f) 0 (T ) NS(X), MWL(f) MWL(f) 0, : {f λ : X λ P 1 } λ Λ NS(X λ
2 (March 13, 2010) N Λ a = i,j=1 x i ( d (a) i,j x j ), Λ h = N i,j=1 x i ( d (h) i,j x j ) B a B h B a = N i,j=1 ν i d (a) i,j, B h = x j N i,j=1 ν i
1. A. M. Turing [18] 60 Turing A. Gierer H. Meinhardt [1] : (GM) ) a t = D a a xx µa + ρ (c a2 h + ρ 0 (0 < x < l, t > 0) h t = D h h xx νh + c ρ a 2 (0 < x < l, t > 0) a x = h x = 0 (x = 0, l) a = a(x,
『共形場理論』
T (z) SL(2, C) T (z) SU(2) S 1 /Z 2 SU(2) (ŜU(2) k ŜU(2) 1)/ŜU(2) k+1 ŜU(2)/Û(1) G H N =1 N =1 N =1 N =1 N =2 N =2 N =2 N =2 ĉ>1 N =2 N =2 N =4 N =4 1 2 2 z=x 1 +ix 2 z f(z) f(z) 1 1 4 4 N =4 1 = = 1.3
Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p
![Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p](/thumbs/98/138321532.jpg "Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p")
Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara 80 1963 Sp(2, R) p L holomorphic discrete series Eichler Brandt Eichler
コホモロジー的AGT対応とK群類似
AGT K ( ) Encounter with Mathematics October 29, 2016 AGT L. F. Alday, D. Gaiotto, Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010), arxiv:0906.3219.
Affine Hecke ( A ) Irreducible representations of affine Hecke algebras (survey talk with emphasis on type A) (Syu Kato) Recently, there are
Affine Hecke ( A ) Irreducible representations of affine Hecke algebras (survey talk with emphasis on type A) (Syu Kato) 20 10 29 Recently, there are several successful attempts on the classification of
λ n numbering Num(λ) Young numbering T i j T ij Young T (content) cont T (row word) word T µ n S n µ C(µ) 0.2. Young λ, µ n Kostka K µλ K µλ def = #{T
0 2 8 8 6 3 0 0 Young Young [F] 0.. Young λ n λ n λ = (λ,, λ l ) λ λ 2 λ l λ = ( m, 2 m 2, ) λ = n, l(λ) = l {λ n n 0} P λ = (λ, ), µ = (µ, ) n λ µ k k k λ i µ i λ µ λ = µ k i= i= i < k λ i = µ i λ k >
2016 Course Description of Undergraduate Seminars (2015 12 16 ) 2016 12 16 ( ) 13:00 15:00 12 16 ( ) 1 21 ( ) 1 13 ( ) 17:00 1 14 ( ) 12:00 1 21 ( ) 15:00 1 27 ( ) 13:00 14:00 2 1 ( ) 17:00 2 3 ( ) 12
(2) Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [2], [13]) Poincaré e m Poincaré e m Kähler-like 2 Kähler-like
![(2) Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [2], [13]) Poincaré e m Poincaré e m Kähler-like 2 Kähler-like](/thumbs/97/133596579.jpg "(2) Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [2], [13]) Poincaré e m Poincaré e m Kähler-like 2 Kähler-like")
() 10 9 30 1 Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [], [13]) Poincaré e m Poincaré e m Kähler-like Kähler-like Kähler M g M X, Y, Z (.1) Xg(Y, Z) = g( X Y, Z) + g(y, XZ)
SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T
SAMA- SUKU-RU Contents 1. 1 2. 7.1. p-adic families of Eisenstein series 3 2.1. modular form Hecke 3 2.2. Eisenstein 5 2.3. Eisenstein p 7 3. 7.2. The projection to the ordinary part 9 3.1. The ordinary
,2,4
2005 12 2006 1,2,4 iii 1 Hilbert 14 1 1.............................................. 1 2............................................... 2 3............................................... 3 4.............................................
φ 4 Minimal subtraction scheme 2-loop ε 2008 (University of Tokyo) (Atsuo Kuniba) version 21/Apr/ Formulas Γ( n + ɛ) = ( 1)n (1 n! ɛ + ψ(n + 1)
φ 4 Minimal subtraction scheme 2-loop ε 28 University of Tokyo Atsuo Kuniba version 2/Apr/28 Formulas Γ n + ɛ = n n! ɛ + ψn + + Oɛ n =,, 2, ψn + = + 2 + + γ, 2 n ψ = γ =.5772... Euler const, log + ax x
v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i
1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [
2 Three-wave Painlevé VI 21 -Wilson three-wave Painlevé VI Gauss -Wilson [KK3] n 3 3 t = t 1 t 2 t 3 -Wilson W z; t := I + W 1 z + W 2 z 2 + z; t := 0
![2 Three-wave Painlevé VI 21 -Wilson three-wave Painlevé VI Gauss -Wilson [KK3] n 3 3 t = t 1 t 2 t 3 -Wilson W z; t := I + W 1 z + W 2 z 2 + z; t := 0](/thumbs/59/43209884.jpg "2 Three-wave Painlevé VI 21 -Wilson three-wave Painlevé VI Gauss -Wilson [KK3] n 3 3 t = t 1 t 2 t 3 -Wilson W z; t := I + W 1 z + W 2 z 2 + z; t := 0")
1473 : de nouvelles perspectives 2006 2 pp 102 119 VI q 1 Tetsuya Kikuchi Sabro Kakei Drinfel d-sokolov Painlevé [KK1] [KK2] [KK3] [KIK] [ ] [ ] [KK3] three-wave equation Painlevé VI q q Drinfel d-sokolov
compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) a, Σ a {0} a 3 1
014 5 4 compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) 1 1.1. a, Σ a {0} a 3 1 (1) a = span(σ). () α, β Σ s α β := β α,β α α Σ. (3) α, β
欧州特許庁米国特許商標庁との共通特許分類 CPC (Cooperative Patent Classification) 日本パテントデータサービス ( 株 ) 国際部 2019 年 7 月 31 日 CPC 版が発効します 原文及び詳細はCPCホームページのCPC Revision
欧州特許庁米国特許商標庁との共通特許分類 CPC (Cooperative Patent Classification) 日本パテントデータサービス ( 株 ) 国際部 2019 年 7 月 31 日 CPC 2019.08 版が発効します 原文及び詳細はCPCホームページのCPC Revisions(CPCの改訂 ) をご覧ください https://www.cooperativepatentclassification.org/cpcrevisions/noticeofchanges.html
untitled
Lie L ( Introduction L Rankin-Selberg, Hecke L (,,, Rankin, Selberg L (GL( GL( L, L. Rankin-Selberg, Fourier, (=Fourier (= Basic identity.,,.,, L.,,,,., ( Lie G (=G, G.., 5, Sp(, R,. L., GL(n, R Whittaker
1 Slodowy 005 3 9 3 1 ADE ADE (0) ADE (1) SL(, C), R 3 (), (3) ADE, II 1 (4) SL(, Z)- (5) F 4, B, M McKay E 6, E 7, E 8 4 A n 5 5 5... 5 5 5 D n 5 5 5... 5 5 5 E 6 5 5 5 5 5 5 E 7 5 5 5 5 5 5 5 E 8 5
( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )
( 3 7 4 ) 2 2 ) 8 2 954 2) 955 3) 5) J = σe 2 6) 955 7) 9) 955 Statistical-Mechanical Theory of Irreversible Processes 957 ) 3 4 2 A B H (t) = Ae iωt B(t) = B(ω)e iωt B(ω) = [ Φ R (ω) Φ R () ] iω Φ R (t)
all.dvi
72 9 Hooke,,,. Hooke. 9.1 Hooke 1 Hooke. 1, 1 Hooke. σ, ε, Young. σ ε (9.1), Young. τ γ G τ Gγ (9.2) X 1, X 2. Poisson, Poisson ν. ν ε 22 (9.) ε 11 F F X 2 X 1 9.1: Poisson 9.1. Hooke 7 Young Poisson G
( ) (, ) ( )
( ) (, ) ( ) 1 2 2 2 2.1......................... 2 2.2.............................. 3 2.3............................... 4 2.4.............................. 5 2.5.............................. 6 2.6..........................
2017 : msjmeeting-2017sep-00f003 ( ) 1. 1 = A 1.1. / R T { } Λ R = a i T λ i a i R, λ i R, lim λ i = + i i=0 R Λ R v T ( a i T λ i ) = inf λ i, v T (0
2017 : msjmeeting-2017sep-00f003 ( ) 1. 1 = A 1.1. / R T { } Λ R = a i T λ i a i R, λ i R, lim λ i = + i i=0 R Λ R v T ( a i T λ i ) = inf λ i, v T (0) = + a i 0 i=0 v T Λ R 0 = {x Λ R v T (x) 0} R Remark
E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1
E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1 (4/12) 1 1.. 2. F R C H P n F E n := {((x 0,..., x n ), [v 0 : : v n ]) F n+1 P n F n x i v i = 0 }. i=0 E n P n F P n
tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p.
tomocci 18 7 5...,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. M F (M), X(F (M)).. T M p e i = e µ i µ. a a = a i
数学Ⅱ演習（足助・09夏）
II I 9/4/4 9/4/2 z C z z z z, z 2 z, w C zw z w 3 z, w C z + w z + w 4 t R t C t t t t t z z z 2 z C re z z + z z z, im z 2 2 3 z C e z + z + 2 z2 + 3! z3 + z!, I 4 x R e x cos x + sin x 2 z, w C e z+w
CAPELLI (T\^o $\mathrm{r}\mathrm{u}$ UMEDA) MATHEMATICS, KYOTO UNIVERSITY DEPARTMENT $\mathrm{o}\mathrm{p}$ $0$:, Cape i,.,.,,,,.,,,.
1508 2006 1-11 1 CAPELLI (T\^o $\mathrm{r}\mathrm{u}$ UMEDA) MATHEMATICS KYOTO UNIVERSITY DEPARTMENT $\mathrm{o}\mathrm{p}$ $0$: Cape i Capelli 1991 ( ) (1994 ; 1998 ) 100 Capelli Capelli Capelli ( ) (
D 24 D D D
5 Paper I.R. 2001 5 Paper HP Paper 5 3 5.1................................................... 3 5.2.................................................... 4 5.3.......................................... 6
X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
II R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k
II 231017 1 1.1. R n k +1 v 0,, v k k v 1 v 0,, v k v 0 1.2. v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ kσ dimσ = k 1.3. k σ {v 0,...,v k } {v i0,...,v il } l σ τ < τ τ σ 1.4.
2 R U, U Hausdorff, R. R. S R = (S, A) (closed), (open). (complete projective smooth algebraic curve) (cf. 2). 1., ( ).,. countable ( 2 ) ,,.,,
15, pp.1-13 1 1.1,. 1.1. C ( ) f = u + iv, (, u, v f ). 1 1. f f x = i f x u x = v y, u y = v x.., u, v u = v = 0 (, f = 2 f x + 2 f )., 2 y2 u = 0. u, u. 1,. 1.2. S, A S. (i) A φ S U φ C. (ii) φ A U φ
QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1
QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 (vierbein) QCD QCD 1 1: QCD QCD Γ ρ µν A µ R σ µνρ F µν g µν A µ Lagrangian gr TrFµν F µν No. Yes. Yes. No. No! Yes! [1] Nash & Sen [2] Riemann
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy
1 Abstract 2 3 n a ax 2 + bx + c = 0 (a 0) (1) ( x + b ) 2 = b2 4ac 2a 4a 2 D = b 2 4ac > 0 (1) 2 D = 0 D < 0 x + b 2a = ± b2 4ac 2a b ± b 2
1 Abstract n 1 1.1 a ax + bx + c = 0 (a 0) (1) ( x + b ) = b 4ac a 4a D = b 4ac > 0 (1) D = 0 D < 0 x + b a = ± b 4ac a b ± b 4ac a b a b ± 4ac b i a D (1) ax + bx + c D 0 () () (015 8 1 ) 1. D = b 4ac
Λ (Kyo Nishiyama) 1 p q r ( determinantal variety) n n r Kostant ( Rallis, Steinberg ) D 1980 Borho-Brylinski Vogan Springer theta theta theta
Λ (Kyo Nishiyama) 1 p q r ( determinantal variety) n n r 1960 70 Kostant ( Rallis, Steinberg ) D 1980 Borho-Brylinski Vogan Springer theta theta theta theta ( ) Λ ( ) August 9, 2000 Theta lifting of representations
Note.tex 2008/09/19( )
1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................
main.dvi
SGC - 70 2, 3 23 ɛ-δ 2.12.8 3 2.92.13 4 2 3 1 2.1 2.102.12 [8][14] [1],[2] [4][7] 2 [4] 1 2009 8 1 1 1.1... 1 1.2... 4 1.3 1... 8 1.4 2... 9 1.5... 12 1.6 1... 16 1.7... 18 1.8... 21 1.9... 23 2 27 2.1
非可換Lubin-Tate理論の一般化に向けて
Lubin-Tate 2012 9 18 ( ) Lubin-Tate 2012 9 18 1 / 27 ( ) Lubin-Tate 2012 9 18 2 / 27 Lubin-Tate p 1 1 ( ) Lubin-Tate 2012 9 18 2 / 27 Lubin-Tate p 1 1 Lubin-Tate GL n n 1 Lubin-Tate ( ) Lubin-Tate 2012
) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
![) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4](/thumbs/92/107866707.jpg ") ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4")
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
( 3) b 1 b : b b f : a b 1 b f = f (2.7) g : b c g 1 b = g (2.8) 1 b b (identity arrow) id b f a b g f 1 b b c g (2.9) 3 C C C a, b a b Hom C (a, b) h
2011 9 5 1 Lie 1 2 2.1 (category) (object) a, b, c, a b (arrow, morphism) f : a b (2.1) f a b (2.2) ( 1) f : a b g : b c (composite) g f : a c ( 2) f f a b g f g c g h (2.3) a b c d (2.4) h (g f) = (h
TOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17
, CH n. CH n, CP n,,,., CH n,,. RH n ( Cartan )., CH n., RH n CH n,,., RH n, CH n., RH n ( ), CH n ( 1.1 (v), (vi) )., RH n,, CH n,., CH n,. 1.2, CH n
( ), Jürgen Berndt,.,. 1, CH n.,,. 1.1 ([6]). CH n (n 2), : (i) CH k (k = 0,..., n 1) tube. (ii) RH n tube. (iii). (iv) ruled minimal, equidistant. (v) normally homogeneous submanifold F k tube. (vi) normally
セアラの暗号
1 Cayley-Purser 1 Sarah Flannery 16 1 [1] [1] [1]314 www.cayley-purser.ie http://cryptome.org/flannery-cp.htm [2] Cryptography: An Investigation of a New Algorithm vs. the RSA(1999 RSA 1999 9 11 2 (17
0406_total.pdf
59 7 7.1 σ-ω σ-ω σ ω σ = σ(r), ω µ = δ µ,0 ω(r) (6-4) (iγ µ µ m U(r) γ 0 V (r))ψ(x) = 0 (7-1) U(r) = g σ σ(r), V (r) = g ω ω(r) σ(r) ω(r) (6-3) ( 2 + m 2 σ)σ(r) = g σ ψψ (7-2) ( 2 + m 2 ω)ω(r) = g ω ψγ
006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................
keisoku01.dvi
2.,, Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 4 Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 5 Mon, 2006, 401, SAGA, JAPAN Dept.
講義ノート 物性研究 電子版 Vol.3 No.1, (2013 年 T c µ T c Kammerlingh Onnes 77K ρ 5.8µΩcm 4.2K ρ 10 4 µωcm σ 77K ρ 4.2K σ σ = ne 2 τ/m τ 77K
2 2 T c µ T c 1 1.1 1911 Kammerlingh Onnes 77K ρ 5.8µΩcm 4.2K ρ 1 4 µωcm σ 77K ρ 4.2K σ σ = ne 2 τ/m τ 77K τ 4.2K σ 58 213 email:takada@issp.u-tokyo.ac.jp 1933 Meissner Ochsenfeld λ = 1 5 cm B = χ B =
A11 (1993,1994) 29 A12 (1994) 29 A13 Trefethen and Bau Numerical Linear Algebra (1997) 29 A14 (1999) 30 A15 (2003) 30 A16 (2004) 30 A17 (2007) 30 A18
2013 8 29y, 2016 10 29 1 2 2 Jordan 3 21 3 3 Jordan (1) 3 31 Jordan 4 32 Jordan 4 33 Jordan 6 34 Jordan 8 35 9 4 Jordan (2) 10 41 x 11 42 x 12 43 16 44 19 441 19 442 20 443 25 45 25 5 Jordan 26 A 26 A1
Twist knot orbifold Chern-Simons
Twist knot orbifold Chern-Simons 1 3 M π F : F (M) M ω = {ω ij }, Ω = {Ω ij }, cs := 1 4π 2 (ω 12 ω 13 ω 23 + ω 12 Ω 12 + ω 13 Ω 13 + ω 23 Ω 23 ) M Chern-Simons., S. Chern J. Simons, F (M) Pontrjagin 2.,
p.2/76
kino@info.kanagawa-u.ac.jp p.1/76 p.2/76 ( ) (2001). (2006). (2002). p.3/76 N n, n {1, 2,...N} 0 K k, k {1, 2,...,K} M M, m {1, 2,...,M} p.4/76 R =(r ij ), r ij = i j ( ): k s r(k, s) r(k, 1),r(k, 2),...,r(k,
D-brane K 1, 2 ( ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane
![D-brane K 1, 2 ( ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane](/thumbs/94/118614691.jpg "D-brane K 1, 2 ( ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane")
D-brane K 1, 2 E-mail: sugimoto@yukawa.kyoto-u.ac.jp (2004 12 16 ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane D-brane RR D-brane K D-brane K D-brane K K [2, 3]
( ),.,,., C A (2008, ). 1,, (M, g) (Riemannian symmetric space), : p M, s p : M M :.,.,.,, (, ).,, (M, g) p M, s p : M M p, : (1) p s p, (
( ),.,,., C A (2008, ). 1,,. 1.1. (M, g) (Riemannian symmetric space), : p M, s p : M M :.,.,.,, (, ).,,. 1.2. (M, g) p M, s p : M M p, : (1) p s p, (2) s 2 p = id ( id ), (3) s p ( )., p ( s p (p) = p),,
Z: Q: R: C:
0 Z: Q: R: C: 3 4 4 4................................ 4 4.................................. 7 5 3 5...................... 3 5......................... 40 5.3 snz) z)........................... 4 6 46 x
²ÄÀÑʬΥ»¶ÈóÀþ·¿¥·¥å¥ì¡¼¥Ç¥£¥ó¥¬¡¼ÊýÄø¼°¤ÎÁ²¶á²òÀÏ Asymptotic analysis for the integrable discrete nonlinear Schrödinger equation
Asymptotic analysis for the integrable discrete nonlinear Schrödinger equation ( ) ( ) 2016 12 17 1. Schrödinger focusing NLS iu t + u xx +2 u 2 u = 0 u(x, t) =2ηe 2iξx 4i(ξ2 η 2 )t+i(ψ 0 +π/2) sech(2ηx
newmain.dvi
数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published
SUSY DWs
@ 2013 1 25 Supersymmetric Domain Walls Eric A. Bergshoeff, Axel Kleinschmidt, and Fabio Riccioni Phys. Rev. D86 (2012) 085043 (arxiv:1206.5697) ( ) Contents 1 2 SUSY Domain Walls Wess-Zumino Embedding
医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 2 版 1 刷発行時のものです.
医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
等質空間の幾何学入門
2006/12/04 08 tamaru@math.sci.hiroshima-u.ac.jp i, 2006/12/04 08. 2006, 4.,,.,,.,.,.,,.,,,.,.,,.,,,.,. ii 1 1 1.1 :................................... 1 1.2........................................ 2 1.3......................................
1 [BPZ] model Wess-Zumino-Witten model,, compact Riemann, principal G-bunlde quasi parabolic structure) family, family base space twisted D-module) 11
![1 [BPZ] model Wess-Zumino-Witten model,, compact Riemann, principal G-bunlde quasi parabolic structure) family, family base space twisted D-module) 11](/thumbs/93/111456459.jpg "1 [BPZ] model Wess-Zumino-Witten model,, compact Riemann, principal G-bunlde quasi parabolic structure) family, family base space twisted D-module) 11")
2003 12 26 ) 71 1995 11 2 ) 1 2 2 Twisted diffrential operator tdo) 6 21 compact Riemann quasi parabolic G-bundle 6 22 compact Riemann quasi parabolic G-bundle family 6 23 Lie algebroid dg Lie algebroid
II (No.2) 2 4,.. (1) (cm) (2) (cm) , (
II (No.1) 1 x 1, x 2,..., x µ = 1 V = 1 k=1 x k (x k µ) 2 k=1 σ = V. V = σ 2 = 1 x 2 k µ 2 k=1 1 µ, V σ. (1) 4, 7, 3, 1, 9, 6 (2) 14, 17, 13, 11, 19, 16 (3) 12, 21, 9, 3, 27, 18 (4) 27.2, 29.3, 29.1, 26.0,
4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,
![1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,](/thumbs/91/105754557.jpg "1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,")
2005 4 1 1 2 2 6 3 8 4 11 5 14 6 18 7 20 8 22 9 24 10 26 11 27 http://matcmadison.edu/alehnen/weblogic/logset.htm 1 1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition)
The Physics of Atmospheres CAPTER :
The Physics of Atmospheres CAPTER 4 1 4 2 41 : 2 42 14 43 17 44 25 45 27 46 3 47 31 48 32 49 34 41 35 411 36 maintex 23/11/28 The Physics of Atmospheres CAPTER 4 2 4 41 : 2 1 σ 2 (21) (22) k I = I exp(
0. Intro ( K CohFT etc CohFT 5.IKKT 6.
E-mail: sako@math.keio.ac.jp 0. Intro ( K 1. 2. CohFT etc 3. 4. CohFT 5.IKKT 6. 1 µ, ν : d (x 0,x 1,,x d 1 ) t = x 0 ( t τ ) x i i, j, :, α, β, SO(D) ( x µ g µν x µ µ g µν x ν (1) g µν g µν vector x µ,y
Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib
![Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib](/thumbs/78/77523767.jpg "Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib")
( ) Donaldson Seiberg-Witten Witten Göttsche [GNY] L. Göttsche, H. Nakajima and K. Yoshioka, Donaldson = Seiberg-Witten from Mochizuki s formula and instanton counting, Publ. of RIMS, to appear Donaldson
S K(S) = T K(T ) T S K n (1.1) n {}}{ n K n (1.1) 0 K 0 0 K Q p K Z/pZ L K (1) L K L K (2) K L L K [L : K] 1.1.
![S K(S) = T K(T ) T S K n (1.1) n {}}{ n K n (1.1) 0 K 0 0 K Q p K Z/pZ L K (1) L K L K (2) K L L K [L : K] 1.1.](/thumbs/91/105754396.jpg "S K(S) = T K(T ) T S K n (1.1) n {}}{ n K n (1.1) 0 K 0 0 K Q p K Z/pZ L K (1) L K L K (2) K L L K [L : K] 1.1.")
() 1.1.. 1. 1.1. (1) L K (i) 0 K 1 K (ii) x, y K x + y K, x y K (iii) x, y K xy K (iv) x K \ {0} x 1 K K L L K ( 0 L 1 L ) L K L/K (2) K M L M K L 1.1. C C 1.2. R K = {a + b 3 i a, b Q} Q( 2, 3) = Q( 2
反D中間子と核子のエキゾチックな 束縛状態と散乱状態の解析
.... D 1 in collaboration with 1, 2, 1 RCNP 1, KEK 2 . Exotic hadron qqq q q Θ + Λ(1405) etc. uudd s? KN quasi-bound state? . D(B)-N bound state { { D D0 ( cu) B = D ( cd), B = + ( bu) B 0 ( bd) D(B)-N
t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z
I 1 m 2 l k 2 x = 0 x 1 x 1 2 x 2 g x x 2 x 1 m k m 1-1. L x 1, x 2, ẋ 1, ẋ 2 ẋ 1 x = 0 1-2. 2 Q = x 1 + x 2 2 q = x 2 x 1 l L Q, q, Q, q M = 2m µ = m 2 1-3. Q q 1-4. 2 x 2 = h 1 x 1 t = 0 2 1 t x 1 (t)
Îã³°·¿¤Î¥·¥å¡¼¥Ù¥ë¥È¥«¥êto=1=¡á=1=¥ë¥�¥å¥é¥¹
(kaji@math.sci.fukuoka-u.ac.jp) 2009 8 10 R 3 R 3 ( wikipedia ) (Schubert, 19 ) (= )(Ehresmann, 20 ) (Chevalley, 20 ) G/P: ( : ) W : ( : ) X w : W X w W G: B G: Borel P B: G/P: 1 C n ( ) Fl n := {0 V
ADM-Hamiltonian Cheeger-Gromov 3. Penrose
ADM-Hamiltonian 1. 2. Cheeger-Gromov 3. Penrose 0. ADM-Hamiltonian (M 4, h) Einstein-Hilbert M 4 R h hdx L h = R h h δl h = 0 (Ric h ) αβ 1 2 R hg αβ = 0 (Σ 3, g ij ) (M 4, h ij ) g ij, k ij Σ π ij = g(k
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
( ) Lemma 2.2. X ultra filter (1) X = X 1 X 2 X 1 X 2 (2) X = X 1 X 2 X 3... X N X 1, X 2,..., X N (3) disjoint union X j Definition 2.3. X ultra filt
NON COMMTATIVE ALGEBRAIC SPACE OF FINITE ARITHMETIC TYPE ( ) 1. Introduction (1) (2) universality C ( ) R (1) (2) ultra filter 0 (1) (1) ( ) (2) (2) (3) 2. ultra filter Definition 2.1. X F filter (1) F
( ) 1., ([SU] ): F K k., Z p -, (cf. [Iw2], [Iw3], [Iw6]). K F F/K Z p - k /k., Weil., K., K F F p- ( 4.1).,, Z p -,., Weil..,,. Weil., F, F projectiv
![( ) 1., ([SU] ): F K k., Z p -, (cf. [Iw2], [Iw3], [Iw6]). K F F/K Z p - k /k., Weil., K., K F F p- ( 4.1).,, Z p -,., Weil..,,. Weil., F, F projectiv](/thumbs/102/156263797.jpg "( ) 1., ([SU] ): F K k., Z p -, (cf. [Iw2], [Iw3], [Iw6]). K F F/K Z p - k /k., Weil., K., K F F p- ( 4.1).,, Z p -,., Weil..,,. Weil., F, F projectiv")
( ) 1 ([SU] ): F K k Z p - (cf [Iw2] [Iw3] [Iw6]) K F F/K Z p - k /k Weil K K F F p- ( 41) Z p - Weil Weil F F projective smooth C C Jac(C)/F ( ) : 2 3 4 5 Tate Weil 6 7 Z p - 2 [Iw1] 2 21 K k k 1 k K
ｎｓｇ０２－１３／ｋｙ０４５０５９３０１６０００３３２１０
φ φ φ φ κ κ α α μ μ α α μ χ et al Neurosci. Res. Trpv J Physiol μ μ α α α β in vivo β β β β β β β β in vitro β γ μ δ μδ δ δ α θ α θ α In Biomechanics at Micro- and Nanoscale Levels, Volume I W W v W