. 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

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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 1

. Mac Lane [ML98]. 1 2 (strict monoidal category) 2 2 2 2 3 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

1 1.1 k (U k ) k U k ( k ), k k k k. (a b) c a (b c) 1 a a a 1 k (category) (monoidal category) C (tensor product) : C C C (unit object) 1 C (1.1) ( ) Z ( Z), 1 1 (,, Z C) ( ) Z ( Z) (1.1) (1.2) a,,z : ( ) Z ( Z), l : 1, r : 1. (,, Z) ( ) Z (,, Z) ( Z) C C C C a,,z,, Z l r (1.2) 1,..., n ( 1 (( 2 1) 3 )) (((1 4 ) 5 ) ) 1,..., n 1 n 5 (1.3) A ( 1 2 ) ( 3 ( 4 5 )), B ( 1 (( 2 3 ) 4 )) 5 3

((12)(34))5 1((23)(45)) 1 a 2,34,5 a 12,34,5 1 a 2,3,45 1(2((34)5)) a 1,2,(34)5 1 (2 a 3,4,5 ) (12)((34)5) 1(2(3(45))) (1 2) a 3,4,5 a 1,2,3(45) (12)(3(45)) a 12,34,5 1 a 2,3,45 ((12)(34))5 a 12,3,45 1((23)(45)) a 12,3,4 5 ((12)3)(45) a 1,2,3 (4 5) a 1,23,45 a (12)3,4,5 (((12)3)4)5 (1(23))(45) (a 1,2,3 4) 5 a 1(23),4,5 ((1(23))4)5 a 12,3,4 5 a 1,23,45 ((12)(34))5 a 1,23,4 5 1((23)(45)) a 1,2,34 5 (1((23)4))5 (1 a 2,3,4) 5 a 1,(23)4,5 1 a 23,4,5 (1(2(34)))5 1(((23)4)5) a 1,2(34),5 1 (a 2,3,4 5) 1((2(34))5) a 1,2,34 5 1 a 23,4,5 ((12)(34))5 1((23)(45)) (( 1 2 ) ( 3 4 )) 5 ((12)(34))5 i i 5 14 1 3 6 14 1 4

1 2 3 4 5 1 (1.2) A B (1.2). (1.2) (1.3) (coherence theorem) 1.1 (). (1),,, Z C (( ) ) Z a,, id Z a,,z ( ( )) Z ( ) ( Z) a,,z (( ) Z) id a,,z a,, Z ( ( Z)) (2), C a,1, ( 1) (1 ) r id id l 1.2. : C C C, 1 C a,,z : ( ) Z ( Z), l : 1, r : 1. C 1 [ML98, II.2] 1.7 1.3. k ec k k (tensor category) 5

1.4. Set { } 1.5. ( ) Z ( Z) 1 1 a, l, r (strict monoidal category) Set ec C C End(C) C C End(C) id C : C C 1.2. C D C D (monoidal functor) φ, : F () F ( ) F ( ) φ 0 : 1 F (1) F : C D (1),, Z C (F () F ( )) F (Z) a F (),F ( ),F (Z) F () (F ( ) F (Z)) φ, id F (Z) F ( ) F (Z) id F () φ,z F () F ( Z) φ,z F (( ) Z) F (a,,z ) φ, Z F ( ( Z)) (2) C 2 1 F () l F () F () F () 1 r F () F () φ 0 id F () F (l ) id F () φ 0 F (r ) F (1) F () φ 1, F (1 ) F () F (1) φ,1 F ( 1) lax monoidal functor φ φ 0 F (strong monoidal functor) ψ : F ( ) F () F ( ) (, C) ψ 0 : F (1) 1 comonoidal functor colax monodal functor φ φ 0 (monoidal natural transformation) 6

F : C D (monoidal equivalence) G F id C, F G id D (as monoidal functors) G : D C C D 2-1.6. F F 1.7 ().. C C str C str C 0 S C str F (S) C F (( 1,..., n )) F (( 1,..., n 1 )) n (n 2), F (( 1 )) 1, F ( ) 1. Hom C str(s, S ) : Hom C (F (S), F (S )) C C str C str S, S C str S S S, S C str φ S,S : F (S) F (S ) F (S S ) C str f : S T g : S T f g φ T,T (f g) φ 1 S,S f g Hom C str(s S, T T ) ( Hom C (F (S S ), F (T T ))) C str C str C S F (S) F : C C str φ φ 0 id 1 : 1 F ( ) 1 1.6 C C str [Kas95, I] φ S,S C str C 1.8. C C 0 C 0 C C C 0 C 0 C ( ) Z ( Z) 1 1,, Z C 0 7

C 0 a, l, r Isbell C Set [ML98, II.1] 1.9. (C,, 1, a, l, r) [Sch01b, Definition 4.1] : C C C (C,, 1, a, l, r) (C,, 1, id, id, id) [Sch01b, Theorem 4.3] C 2 2.1 C C f : 2 f : g : 3 f id f id f 4, C 5 C f : g : 6 7 8 7 8 C C C (2.1) (g 1 f 1 ) (g 2 f 2 ) (g 1 g 2 ) (f 1 f 2 ) 8 7 well-defined 5 5 id id id 1 1 1 9 C p q C p id id q p p 1 p q 8

f g f f g id f f f id 2 f : 3 4 f id f id f f g f g f 1 g 1 f 2 g 2 5 6 f g : 7 (1) f 1 f 2 g 1 f 1 g 1 g 2 g 2 f 2 p q 8 (2) 9 1 1 p id id q 1 2.2 2.1. C e : 1 c : 1 C (2.2) (id e) (c id ) id, (e id ) (id c) id (, e, c) (left dual object) (, e, c) (right dual object) C C (left rigid) C 2.2 9

( i, e i, c i ) (i 1, 2) C e 2 e 1 (ϕ 1 id ) c 2 (id ϕ) c 1 ϕ : 1 2 [BK01, 2.1] C C (, ev, coev ) ev : 1 coev : 1 ev coev id (2.2) (2.3) C f : f : (2.4) f : f ( ) : C C (f g) g f 2.2 (). (2.2) Bakalov-Kirillov [BK01] right dual object Kassel [Kas95] left duality Kerler-Lyubashenko [KL01] dual object Majid [Maj95] left dual duality functor Kassel [Kas95] 2.3 (). C (2.2) (id e) (c id ) id 1 c id ( ), ( ) id e 1 a, l, r 10

(2.2) r (id e) a,, (c id ) l 1 id, l (e id ) a 1,, (id c) r 1 id 2.4. R R M R M R M R R- M : Hom(M R, R R ) (a f b)(m) : f(b m a) (f M, a, b R, m M) R- M m 1,..., m r M f 1,..., f r M m M m 1 f 1 (m) + + m r f r (m) m ev : M R M R coev : R M R M ev(f m) f(m), coev(a) a m i f i (f M, m M, a R) (M, ev, coev) M M M R- 2.5. C End(C) ε : F G id C η : id C G F End(C) (2.2) counit-unit identity End(C) 3 3.1 S 1 R 3 (link) (oriented link) 1 (knot) Reidemeister I Reidemeister II Reidemeister III 11

(3.1) C C id, id, coev, ev C C (3.1) C 3.2 3.1 (). C (braiding),, Z C σ, : (, C) (3.2) σ,z (σ,z id ) (id σ,z ), σ, Z (id σ Z, ) (σ, id Z ) (braided monoidal category) (braided category) C σ σ, σ 1, σ, σ 1, (3.2) Z Z Z Z (3.3) Z Z Z Z 12

II III II σ 1, σ, id III f : g : σ, (f g) (g f) σ, (3.4) f g g f f g III Z Z Z Z (3.3) σ, (3.4) σ, (3.3) C σ, σ, σ, 3.3 3.2. C (twist) θ : (3.5) θ σ, σ, (θ θ ), (θ ) θ (, C) (ribbon category) (3.5) A : S 1 [0, 1] R 3 13

S 1 {1/2} A (blackboard framing) (3.6) II III I Reidemeister I : I, II, III C C ev ev σ, (θ id ), coev (id θ ) σ, coev ev coev (3.7) θ θ 1 14

1 (3.7) (3.6) (3.5) 3.4 C C L D D (3.1) L D C 1 1 F (D) End C (1) F ( ) coev coev id σ, id σ 1, id id id σ, id ev ev (ev ev ) (id σ, id ) (σ 1, id id ) (id σ, id ) (coev coev ). F (D) D II III (3.7) I L : F (D) 3.3. L L *1 *1 L : F (D) L (3.1) F (D) Turaev [Tur94] C C-colored ribbon graph I, II, III 15

C C L F (D) C k (linear category) Hom C (, ) k k- C (3.8) θ λ id ( λ k) C k L D F (D) End C (1) D II III I w(d) ( D ) () D D F (D ) λ F (D), w(d ) w(d) + 1 F (D) w(d) I 3.4. C L D P (L) : λ w(d) F (D) End C (1) L 4 4.1 k H k : H H H ε : H k (h) h (1) h (2) (Sweedler notation) H- h (v w) h (1) v h (2) w (h H, v, w ) 16

H- k ε H- h H (h (1) ) h (2) h (1) (h (2) ), ε(h (1) )h (2) h ε(h (2) )h (1) H (H,, ε) (bialgebra) H H- H M k H S : H H h H S(h (1) )h (2) ε(h)1 H h (1) S(h (2) ) H (Hopf algebra) H H- H M fd H M fd : Hom k (, k) (h f)(v) f(s(h)v) (h H, f, v ) H- H R a i b i H H H- (4.1) σ, :, σ, (v w) b i w a i v (v, w ) σ {σ, } H M R (h (1) h (2) ) R 1 h (2) h (1) ( h H), ( id)(r) R 13 R 23, (id )(R) R 13 R 12 R 12 a i b i 1, R 13 a i 1 b i, R 23 1 a i b i R R- (universal R-matrix) (ribbon Hopf algebra) R- R a i b i H θ H (4.2) S(θ) θ, (θ) R 21 R(θ θ) (R 21 b i a i ). H- H M fd (4.1) (4.3) θ (v) θ v (v H M fd ) (4.2) (4.3) (3.5) H 4.1. q U q (sl 2 ) K, K 1, E, F KK 1 1 K 1 K, KEK 1 q 2 E, KF K 1 q 2 F, EF F E (q q 1 ) 1 (K K 1 ) C(q) (E) E K + 1 E, (F ) F 1 + K 1 F, (K) K K, ε(e) ε(f ) 0, ε(k) 1 U q (sl 2 ) q e h/2 h R- well-defined U q (sl 2 )- 2 3 17

4.2. Lie gl(1 1) U q (gl(1 1)) 3 U q (gl(1 1)) U q (gl(1 1)) [Sar14] self-contained 4.2 4.1 4.2. C C Z(C) C e : ( C) (, e) e( ) (id e ) (e id ) (, C) (cf. (3.2)) Z(C) f : (, e) (, e ) C f : e e C (, e) (, e ) : (, ẽ), ẽ() (id e ) (e id ) ( C) C Z(C) σ (,e),(,e ) e k- R R- ec (Schauenburg [Sch01a]) Drifeld double. C H Drinfeld double D(H) H H D(H) R- H M D(H) M D(H) 4.3. 4.1 q q C \ {0, ±1} U q (sl 2 ) q C 1 N N > 1 U q (sl 2 ) U q : U q (sl 2 )/(K N 1, E N, F N ) (K N 1, E N, F N ) U q U q (sl 2 ) K E U q A U q D(A) D(A) U q U q R- [Kas95] I.6 I.6 18

4.4. k 0 H Drinfeld double D(H) D(H) [S13] D(H) [Maj95, 9.3] (3.8) P ( ) dim k (H) (H ) [AFS09] N. Andruskiewitsch and. Ferrer Santos. The beginnings of the theory of Hopf algebras. Acta Appl. Math., 108(1):3 17, 2009. [BK01] B. Bakalov and A. Kirillov, Jr. Lectures on tensor categories and modular functors, volume 21 of University Lecture Series. American Mathematical Society, Providence, RI, 2001. [Kas95] C. Kassel. Quantum groups, volume 155 of Graduate Texts in Mathematics. Springer-erlag, New ork, 1995. [KL01] T. Kerler and.. Lyubashenko. Non-semisimple topological quantum field theories for 3- manifolds with corners, volume 1765 of Lecture Notes in Mathematics. Springer-erlag, Berlin, 2001. [Maj95] S. Majid. Foundations of quantum group theory. Cambridge University Press, Cambridge, 1995. [ML98] S. Mac Lane. Categories for the working mathematician, volume 5 of Graduate Texts in Mathematics. Springer-erlag, New ork, second edition, 1998. [Sar14] A. Sartori. The Alexander polynomial as quantum invariant of links. Arkiv för Matematik, 2014. [Sch01a] P. Schauenburg. The monoidal center construction and bimodules. J. Pure Appl. Algebra, 158(2-3):325 346, 2001. [Sch01b] P. Schauenburg. Turning monoidal categories into strict ones. New ork J. Math., 7:257 265 (electronic), 2001. [S13] K. Shimizu and M. akui. Schrödinger representations from the viewpoint of monoidal categories. ariv:1312:5037. [Tur94]. G. Turaev. Quantum invariants of knots and 3-manifolds, volume 18 of de Gruyter Studies in Mathematics. alter de Gruyter & Co., Berlin, 1994. 19

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