Quantum invariants of 3-manifolds associated to restricted quantum groups

QUANTUM INV ARIANTS OF 3-MANIF OLDS ASSOCIA TED TO RESTRICTED QUANTUM GR OUPS QI CHEN, CHIH-CHIEN YU AN D YU ZHAN G Abstra ct. W e sho w that the Witten-Reshetikhin-T uraev S U (2) in v ari- ant and the Hennings inv a riant associated to the restricted quantum sl 2 are essential ly the same for rational homology 3-spheres. 1. Int roduction After the disco v ery of the Jones p olynomial, Witten prop osed in [W] an in - v a riant of 3-manifolds using the Chern -Simons theory . The fi rst ‘mathemat- ically rigorous’ construction of this in v arian t w as obtained b y Reshetikhin and T uraev in [R T ] using th e representati on theory of the quantum group U ζ ( sl 2 ) at a ro ot of unity ζ . T h is inv arian t, denoted τ ζ , is no w known as the Witten-Reshetikhin-T uraev S U (2) in v arian t, or the WR T S U (2) inv arian t in sh ort. On the other hand Hennin gs s h o w ed in [He] that one can d efine a 3-manifold inv ariant ψ ζ , called the Hennings in v arian t, indep end en t of the represent ation theory . In th is note w e will sho w that these t wo inv ariants are essentiall y the same. More precisely we hav e Theorem 1. L et M b e a close d 3-manifold and ζ a r o ot of unity of or der ℓ > 1 then ψ ζ ( M ) = h ( M ) τ ζ ( M ) , (1) wher e h( M ) is the or der of the first homolo gy gr oup if it is finite and ze r o otherwise. This theorem will b e p ro v ed in Section 5 . The same relation was sho wn in [CKS] for the WR T S O (3) inv ariant and th e Hennings inv ariant asso ciated to the small qu an tum sl 2 . In general the r estricted quantum groups are harder to deal with than the corresp ond ing small ones. One new obstacle in the pro of of the S U (2) case is to sh o w Lemma 3. W e will also give a muc h simplified pro of of a k ey lemma, Lemm a 8 in [CKS]. R emark 1.1 . Kauffman and Radford compared ψ ζ and τ ζ in [KR2] when ℓ = 8 and M is the Len s space L ( k , 1 ). Their calculation confirms th e ab o v e theorem. Note that in their corollary on page 154 I N V ( L ( k , 1)) should b e equal to | k | . As a consequence of the ab o ve theorem we h a v e the f ollo wing corollary . Corollary 2. If ( ℓ, h ( M )) = 1 then the WR T S U (2) invariant τ ζ ( M ) is always an algebr aic inte ger. 1 2 QI CHEN, CHIH-CHIEN YU AND YU ZHANG Sketch of P r o of. Let B ζ b e th e Borel sub algebra of U ζ ( sl 2 ). Its quan tum double D ζ := D ( B ζ ) is a ribb on Hopf algebra, c.f. [KR1]. Let ˆ ψ ζ b e the Hennings in v ariant constructed from D ζ . It is p r o ved in [CK] that ˆ ψ ζ ( M ) is an algebraic in teger for any closed 3-manifold M . F urthermore ˆ ψ ζ ( M ) = η ψ ζ ( M ) for some eigh th r o ot of unit y . Therefore ψ ζ ( M ) is also an algebraic intege r. Theorem 1 then sa ys that the only p ossible denomin ator for τ ζ ( M ) is h ( M ). But according to [KM] the only p ossible d enominators for τ ζ ( M ) are factors of ℓ . Since ℓ and M are coprime w e ha ve τ ζ ( M ) is an algebraic int eger.  R emark 1.2 . S p ecial cases of the corollary were stud ied extensiv ely by man y authors. Murak ami p ro v ed it in [M] w hen ℓ is a pr ime. His pro of w as simp li- fied by Masbaum and Rob erts in [MR] using the Kauffman Brack et. Late r Habiro pro v ed it in [H2] for all integ ral homology 3-spheres without any restriction on ℓ . Integral it y is a very imp ortant p rop erty for quan tum in- v arian ts b ecause int egral in v arian ts can b e used to extract top ologi cal infor- mation, c.f. [GKP, CL1], and to construct TQ FT s o ve r Dedekind domains, c.f. [G, CL2]. W e b eliev e that some similar relatio n b et w een the WR T and Hennings inv ariants exists f or h igher ranked quan tum groups. Our pro of of the in tegralit y can then b e used to prov e the int egralit y for all WR T in v arian ts. A cknow le dgment. The first auth or would like to thank Thang Le for his help. 2. The restricte d quantum group Fix a ro ot of unit y ζ of order ℓ > 1. T o simplify the argumen ts we will just tak e ζ = e 2 π i/ℓ and ζ a = e 2 π ia/ℓ . Also set θ = ζ 1 / 2 . The restricted quan tum group U ζ = U ζ ( sl 2 ) is a C -algebra generated by E , F , K and K − 1 with the relations: K K − 1 = K − 1 K = 1 , K E = θ E K, K F = θ − 1 F K, E F − F E = K 2 − K − 2 θ − θ − 1 and E ℓ = F ℓ = 0 , K 4 ℓ = 1 . (2) It is a Hopf algebra with the com ultiplication ∆, an tip o de S an d counit ǫ giv en b y ∆( K ) = K ⊗ K, ∆( E ) = 1 ⊗ E + E ⊗ K 2 , ∆( F ) = F ⊗ 1 + K − 2 ⊗ F , S ( K ) = K − 1 , S ( E ) = − E K − 2 , S ( F ) = − K 2 F , ǫ ( K ) = 1 , ǫ ( E ) = ǫ ( F ) = 0 . F or i ∈ Z / 4 ℓ let π i := 4 ℓ X j =1 θ ij K j . QUANTUM I NV ARIANTS 3 Then one has K π i = θ − i π i , E π i = π i − 1 E , F π i = π i +1 F . Recall that an elemen t x in a Hopf algebra H is said to b e a c ointe gr al if xy = y x = ǫ ( y ) x , ∀ y ∈ H . An element f in H ∗ is said to b e a left inte gr al if g f = g (1) f , ∀ g ∈ H ∗ . It is known that U ζ con tains a non zero coint egral Λ = F ℓ − 1 π ℓ − 1 E ℓ − 1 , and a nonzero left inte gral λ ( F i K j E m ) = δ i,ℓ − 1 δ j, 2( ℓ − 1) δ m,ℓ − 1 . (3) Denote the quantum int eger b y [ n ] = ( θ n − θ − n ) / ( θ − θ − 1 ) an d th e quan- tum factorial by [ n ]! = [ n ][ n − 1] · · · [1]. The Hopf algebra U ζ is quasi- triangular with the u niv ersal R -matrix in U ζ ⊗ U ζ R = D ℓ − 1 X n =0 ( θ − θ − 1 ) n [ n ]! θ n ( n − 1) 2 E n ⊗ F n . Here D is the d iagonal p art w ith D = 1 4 ℓ 4 ℓ − 1 X m,n =0 θ − mn 2 K m ⊗ K n . F urthermore U ζ is a r ibb on Hopf algebra whose r ib b on elemen t r and its in v erse r − 1 , s ee (5) b elo w, b elong to the Hopf su balgebra U ev ζ generated by E , F and K 2 . W e f ollo w [F] to describ e the cen ter Z ev of U ev ζ . Note that their K and q are equal to our K 2 and θ resp ectiv ely . T h e dimension of Z ev is 3 ℓ − 1 with a basis e i , w ± j , 0 ≤ i ≤ ℓ and 1 ≤ j ≤ ℓ − 1. These elemen ts satisfy e i e j = δ ij e i , e i w ± j = δ ij w ± j , w ± i w ± j = w ± i w ∓ j = 0 . (4) T o simp lify notation w e will consider w ± 0 = w ± ℓ = 0. The r ibb on elemen t r and its in v erse r − 1 are in Z ev : r ± 1 = ℓ X m =0 a ± ,m e m + b ± ,m w m + c ± ,m w + m (5) where w m := w + m + w − m and a ± ,m = ( − 1) m +1 θ ± 1 − m 2 2 , b ± ,m = ± ( − 1) ℓ − 1 ( θ − θ − 1 ) θ ± 1 − m 2 2 m [ m ] , c ± ,m = − ℓ b ± ,m m . 4 QI CHEN, CHIH-CHIEN YU AND YU ZHANG P S f r a g r e p l a c e m e n t s R 1 R 2 S ( R 1 ) R 2 K − 2 K 2 Figure 1. Basic d iagram lab els. 3. The universal inv ariant of bo ttom t angles The d efinition of the u niv ersal in v arian t is th e same as Section 2.2 in [CKS]. See also [He, H1, O1]. W e include it here for ease of r eading. A b ottom tangle is an orient ed framed tangle prop erly embed ded in R 2 × [0 , 1) suc h that its i -th comp onent starts from (0 , 2 i, 0) and ends at (0 , 2 i − 1 , 0). Note th at b ottom tangles do not ha ve circle comp onent s. Botto m tangles are considered equiv alent up to ambien t isotop y relativ e to b oundary . The unive rsal U ζ -in v arian t Γ ζ of b ottom tangles can b e calculated as follo ws. Let T b e a b ottom tangle. Cho ose a generic d iagram of T and lab el it according to Figure 1, where R = P R 1 ⊗ R 2 is the u niv ersal R -matrix and S is th e an- tip o de. Multiply th e lab els on eac h comp onent, opp osite to the orien tation, to obtain an element in U ζ . The tensor p ro duct of the elemen ts from all comp onent s is the universal inv ariant Γ ζ ( T ). Clearly if T has m -comp onent then Γ ζ ( T ) is in U ⊗ m ζ . But one can say a little more ab out the v alue of Γ ζ ( T ). Let’s fir st recall that f or a Hopf algebra A and an A -mo d ule W , the in v arian t su bmo du le Inv( W ) is equal to { w ∈ W | a ( w ) = ǫ ( a ) w , ∀ a ∈ A } . The adjoint action mak es A to b e an A -mo du le, i.e. ad a ( b ) = P ( a ) a ′ bS ( a ′′ ). Lemma 3. L et T b e an m -c omp onent b ottom tangle such that ˆ T ha s 0 linking matrix. Then Γ ζ ( T ) ∈ Inv(( U ev ζ ) ⊗ m ) . Pr o of. By Corollary 12 in [K1], Γ ζ ( T ) ∈ Inv(( U ζ ) ⊗ m ). So it is enough to sho w Γ ζ ( T ) ∈ ( U ev ζ ) ⊗ m . This will follo w from [H1] C orollary 9.15, i.e. w e need to show that (a) if x ⊗ y is in ( U ev ζ ) ⊗ 2 then so are P ad R 2 ( y ) ⊗ ad R 1 ( x ) and P ad S ( R 1 ) ( y ) ⊗ ad R 2 ( x ); (b) if x is in U ev ζ then so are P R 2 S (ad R 1 ( x )) and P S − 1 (ad R 1 ( x )) R 2 ; (c) if x is in U ev ζ then P x ′ S ( R 2 ) ⊗ (ad R 1 ( x ′′ )) is in ( U ev ζ ) ⊗ 2 and (d) Γ ζ ( B ) is in ( U ev ζ ) ⊗ 3 where B ’s n atural closure is the Borromean r ing. QUANTUM I NV ARIANTS 5 These four statemen ts f ollo w from direct calculation. W e will only sho w detail for (c, d ). S et | F m K n E p | = p − m . Let x ∈ U ev ζ , X x ′ S ( R 2 ) ⊗ ad R 1 ( x ′′ ) = 1 4 ℓ 4 ℓ − 1 X m,n,j =0 ( x ) θ m ( m − 1) − nj 2 ( θ − θ − 1 ) m [ m ]! x ′ S ( K j F m ) ⊗ ad K n E m ( x ′′ ) = 1 4 ℓ 4 ℓ − 1 X m,j =0 ( x ) θ m ( m − 1) 2 ( θ − θ − 1 ) m [ m ]! x ′ S ( K j F m ) ⊗ 4 ℓ − 1 X n =0 θ n 2 (2 m +2 | x ′′ |− j ) ad E m ( x ′′ ) . It b elongs to U ev ζ b ecause P 4 ℓ − 1 n =0 θ n 2 (2 m +2 | x ′′ |− j ) v anishes when j is o dd. The other half of (c) can b e calculated similarly . As for (d) we n ote that only the d iagonal p art D of R contributes to Γ ζ p ossible elemen ts outside of U ev ζ . One can slide the diagonal parts to the same p lace using: D ( x ⊗ y ) = ( K 2 | y | x ⊗ y K 2 | x | ) D . (6) Therefore the sliding only inserts ev en p ow ers of K in some places. Since the Borromean ring has 0 linking matrix th e diagonal p arts got canceled after they are slided to the same p lace. T h erefore Γ ζ ( B ) is in U ev ζ .  4. The 3-manifold inv ariants The WR T in v ariant an d the Hennings inv arian t can b e b oth calculated from Γ ζ in Secti on 3. L et M b e a closed 3- manifold and T b e an m - comp onent b ottom tangle suc h that M is the result of surgery on ˆ T , the natural closure of T . The Henn ings in v ariant ψ ζ ( M ) = λ ⊗ m (Γ ζ ( T )) λ ( r − 1 ) σ + λ ( r ) σ − , (7) where σ ± is the n um b er of p ositiv e/negativ e eigenv alues of the linking matrix of ˆ T . The discrepancy in sign is due to the fact that the v alue of Γ ζ at the trivial b ottom tangle with a p ositiv e t wist is r − 1 . T o define the WR T in v ariant we n eed to consider the represent ations of U ζ . F or 1 ≤ n ≤ ℓ − 1 let V n b e th e irr educible repr esen tation of U ζ of dimension n . Recall that for an y U ζ -mo dule V th e quan tum trace tr V q : U ζ → C is defined b y tr V q ( x ) = tr V ( K 2 x ), where tr V is the ordinary trace on V . T he WR T in v arian t of M is τ ζ ( M ) = (tr ω q ) ⊗ m (Γ ζ ( T )) tr ω q ( r − 1 ) σ + tr ω q ( r ) σ − , where tr ω q = P ℓ − 1 n =1 [ n ]tr V n q . 6 QI CHEN, CHIH-CHIEN YU AND YU ZHANG 5. Proof of the main theorem The pro of is d ivided into three cases: (i) h( M ) = 0, i.e. M has infi n ite fi rst homology; (ii) h( M ) 6 = 0 and M is the result of sur gery on a link with diagonal linking matrix; (iii) h( M ) 6 = 0 and M can n ot b e obtained by su r gery on a link with diagonal linking matrix. Case (i) follo ws from th e fact ψ ζ ( M ) = 0 if M h as infin ite fi r st homology according to [O1, K2]. Case (ii) will b e pro ve d in 5.3. T o sh ow (iii) we recall from [O2] that in this case there exist lens s p aces L ( n i , 1), i = 1 . . . m su c h that M ′ = M # L ( n 1 , 1)# · · · # L ( n m , 1) is the result of s urgery on a link w ith diagonal linking m atrix. By (ii) we ha v e ψ ζ ( M ′ ) = h( M ′ ) τ ( M ′ ), w hic h is the same as (b ecause ψ ζ , τ ζ and h are m ultiplicativ e with r esp ect to connected sum): ψ ζ ( M ) m Y i =1 ψ ζ ( L ( n i , 1)) = h( M ) τ ζ ( M ) m Y i =1 | n i | τ ζ ( L ( n i , 1)) . It remains to note th at τ ζ ( L ( n i , 1)) 6 = 0, c.f. [LL], and ψ ζ ( L ( n i , 1)) = | n i | τ ζ ( L ( n i , 1)) by (ii). 5.1. Some lemmas ab out the cen ter. W e will need some preparation lemmas th at will lead to a pro of of (ii) in 5.3. Denote by ˜ Z ev the s ubset of Z ev spanned by e m , 0 ≤ m ≤ ℓ , and w n , 1 ≤ n < ℓ . The follo wing lemma follo ws from [F] Prop osition D.1.1. Lemma 4. ˜ Z ev = C [ C ] wher e C = F E + K 2 θ + K − 2 θ − 1 ( θ − θ − 1 ) 2 . The next lemma deals w ith the v alue of the left integral λ , c.f. (3), on the cen ter. Lemma 5. The left inte gr al λ vanishes on ˜ Z ev and λ ( w + m ) = ( − 1) m − 1 θ − 2 [ m ] 3 2 ℓ ([ ℓ − 1]!) 2 . (8) Pr o of. By Lemma 4 the fi rst half is equiv alent to λ ( C i ) = 0 , i = 0 , 1 , 2 , . . . (9) Because C i = F i E i + terms with low er degree of E QUANTUM I NV ARIANTS 7 w e see that (9) holds for 0 ≤ i < ℓ − 1. It is kn o w n, c.f. (3.6) in [F], that K 2 ℓ = 1 2 ⌊ p 2 ⌋ X i =0 ( − 1) i − 1 ℓ ℓ − i  ℓ − i i  ( θ − θ − 1 ) 2( ℓ − 2 i ) C ℓ − 2 i . (10) This implies (9) by indu ction. Equation (8) follo ws from (4.19) in [F].  Next w e d iscu ss the restriction of tr ω q on ˜ Z ev . Lemma 6. We have tr ω q ( w ± m ) = tr ω q ( w m ) = 0 , 1 ≤ m < ℓ , (11) and tr ω q ( e m ) = [ m ] 2 . (12) Pr o of. Recal l th at V n is the n -dimen s ional U ζ -mo dule such that C | V n = θ n + θ − n ( θ − θ − 1 ) 2 Id | V n . This lemma follo w s easily fr om the ab ov e equation an d (D.3-5 ) in [F].  It turn s out that λ and tr ω q are closely related: Lemma 7. F or any x ∈ ˜ Z ev and n ∈ N , λ ( x r ± n ) λ ( r ± 1 ) = n tr ω q ( x r ± n ) tr ω q ( r ± 1 ) (13) Pr o of. F rom (4, 5) w e ha ve r ± n = ℓ X m =0 a n ± ,m e m + n a n − 1 ± ,m  b ± ,m w m + c ± ,m w + m  . W e only need to prov e (13) f or x = e j and x = w j . If x = e j then λ ( x r ± n ) λ ( r ± 1 ) = n a n − 1 ± ,j c ± ,j λ ( w + j ) P ℓ m =0 c ± ,m λ ( w + m ) = n ( − 1) ( j − 1) n θ ± (1 − j 2 ) n 2 [ j ] 2 P ℓ m =0 ( − 1) m − 1 θ ± 1 − m 2 2 [ m ] 2 = n a n ± ,j tr ω q ( e j ) P ℓ m =0 a ± ,m tr ω q ( e m ) = n tr ω q ( x r ± n ) tr ω q ( r ± 1 ) . If x = w j then b oth sid es of (13) is 0.  5.2. An impro v emen t of Lemma 3. F or any A -mod ule W of a Hopf algebra A set ¯ W = W / { ax − ǫ ( a ) x, ∀ a ∈ A, x ∈ W } . It is clear that ¯ W inherits a tr ivial A -mo du le str ucture from W . Since U ζ con tains a coint egral Λ, λ factors through ¯ U ζ , c.f. Prop osition 8 in [LS]. I t is kno wn that tr V q also factors through ¯ U ζ for any U ζ -mo dule V , c.f. Section 7.2 in [H1]. W e will also need the follo wing lemma whose pro of is immediate. 8 QI CHEN, CHIH-CHIEN YU AND YU ZHANG Lemma 8. F or any z ∈ Z ev , λ z and (tr ω q ) z b oth factor thr ough ¯ U ev ζ , wher e λ z ( x ) := λ ( xz ) , and (tr ω q ) z ( x ) := tr ω q ( xz ) , ∀ x ∈ U ev ζ . The follo wing k ey lemma is an improv emen t of Lemma 3. It is similar to Lemma 8 in [CKS], w hic h wa s pro ved in a m u c h more complicated wa y . Lemma 9. L et T b e an m -c omp onent b ottom tangle whose natur al closur e ˆ T has 0 link i ng matrix. If χ i : U ev ζ → C f actors thr ough ¯ U ev ζ then (Id ⊗ χ 2 ⊗ · · · ⊗ χ m )Γ ζ ( T ) b elongs to ˜ Z ev . Pr o of. W e n eed another version of quantum sl 2 . Let ˆ U ζ b e the C -algebra generated by the same generators with the same relations as U ζ but omitting (2). Denote by p the canonical pro jection of ˆ U ζ to U ζ . T he algebra ˆ U ζ itself is not quasi-triangular b ut there exists in some completion of ˆ U ζ ⊗ ˆ U ζ a unive rsal R -matrix ˆ R = ˆ D ∞ X n =0 ( θ − θ − 1 ) n [ n ]! θ n ( n − 1) 2 E n ⊗ F n . Here ˆ D is the diagonal part, whic h satisfies the same relation as D in (6): ˆ D ( x ⊗ y ) = ( K 2 | y | x ⊗ y K 2 | x | ) ˆ D . (14) One can u s e Fig. 1 to define the u niv ersal inv ariant ˆ Γ ζ asso ciated to ˆ U ζ . Since ˆ T has 0 linking matrix one can cancel the d iagonal parts by sliding them to the same place u sing (14). Comparing the form ulas of R and ˆ R it is then clear th at p ◦ ˆ Γ ζ ( T ) = Γ ζ ( T ) . (15) The p ro of of Lemma 3 can b e used word for wo rd to s h o w that ˆ Γ ζ ( T ) ∈ ( ˆ U ev ζ ) ⊗ m . S ince ¯ ˆ U ev ζ inherits a trivial ˆ U ev ζ -mo dule stru ctur e from the adjoint action an d ˆ χ i := χ ◦ p factors thr ough ¯ ˆ U ev ζ w e hav e (Id ⊗ ˆ χ 2 ⊗ · · · ⊗ ˆ χ m ) ˆ Γ ζ ( T ) ∈ Inv( ˆ U ev ζ ) = C en ter( ˆ U ev ζ ) . According to [DK] Theorem 4.2, Cen ter ( ˆ U ev ζ ) is generated b y E ℓ , F ℓ , K ± ℓ and C . Note that their K and ǫ are equal to our K 2 and θ resp ectiv ely . F rom (15 ) w e hav e (Id ⊗ χ 2 ⊗ · · · ⊗ χ m )Γ ζ ( T ) = p ◦ (Id ⊗ ˆ χ 2 ⊗ · · · ⊗ ˆ χ m ) ˆ Γ ζ ( T ) , whic h is a p olynomial in K ± ℓ and C . It remains to note th at K ± ℓ can b e expressed as a p olynomial in C , c.f. (4.2.8) in [DK] and (10).  QUANTUM I NV ARIANTS 9 5.3. Pro of of ( ii). Let T b e an m -comp onen t b ottom tangle whose natural closure ˆ T has diagonal linking matrix diag( f 1 , f 2 , · · · , f m ). S upp ose M is the result of sur gery on ˆ T . Then h( M ) = | f 1 · · · f m | . Also assume that f 1 , . . . , f i > 0 and f i +1 , . . . , f m < 0. Let T 0 b e the b ottom tangle obtained from T b y c hanging the framing on eac h comp onent to 0. W e h a ve ψ ζ ( M ) = λ ⊗ m (Γ ζ ( T )) λ ( r − 1 ) i λ ( r ) m − i = λ r − f 1 ⊗ · · · ⊗ λ r f m (Γ ζ ( T 0 )) λ ( r − 1 ) i λ ( r ) m − i = λ r − f 1 λ ( r − 1 ) Id ⊗ λ r − f 2 ⊗ · · · ⊗ λ r f m λ ( r − 1 ) i − 1 λ ( r ) m − i (Γ ζ ( T 0 )) !  b y Lemmas 7 and 9  = | f 1 | (tr ω q ) r − f 1 tr ω q ( r − 1 ) Id ⊗ λ r − f 2 ⊗ · · · ⊗ λ r f m λ ( r − 1 ) i − 1 λ ( r ) m − i (Γ ζ ( T 0 )) ! = · · · = | f 1 · · · f m | (tr ω q ) r − f 1 ⊗ · · · ⊗ (tr ω q ) r f m (Γ ζ ( T 0 )) tr ω q ( r − 1 ) i tr ω q ( r ) m − i = h ( M ) τ ζ ( M ) . This ends the pro of of (ii) and hence the p ro of of Theorem 1. Referen ces [CKS] Q. Chen, S. Kuppum and P . Sriniv asan. On the relation b etw een the WR T inv ariant and the Hennings inv arian t. M ath. Pr o c. Cambridge Phi los. So c. , 146(1):151–16 3, 2009. [CK] Q. Chen and T. Kerler. I ntegra l TQFTs from the q uantum double construction. Pr eprint . 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