Hopf-cyclic Cohomology of Quantum Enveloping Algebras
aa r X i v : . [ m a t h . K T ] S e p Hopf-cyclic Cohomology of Quantum Enveloping Algebras
Atabey Kaygun and Serkan S¨utl¨u
Abstract
In this paper we calculate both the periodic and non-periodic Hopf-cyclic cohomology ofDrinfeld-Jimbo quantum enveloping algebra U q ( g ) for an arbitrary semi-simple Lie algebra g with coefficients in a modular pair in involution. We show that its Hochschild cohomologyis concentrated in a single degree determined by the rank of the Lie algebra g . In this paper we calculate the Hopf-cyclic cohomology of Drinfeld-Jimbo quantum envelopingalgebra U q ( g ) for an arbitrary semi-simple Lie algebra g with coefficients in a modular pairin involution (MPI) σ k ε . This cohomology was previously calculated only for sℓ by Crainicin [7]. We also verified the original calculations of Moscovici-Rangipour [26] of the Hopf-cycliccohomology of the Connes-Moscovici Hopf algebras H and H with coefficients in the trivialMPI k ε using our new cohomological machinery.The calculation of the Hopf-cyclic cohomology for Connes-Moscovici Hopf algebras H n is a bigchallenge. These Hopf algebras are designed to calculate the characteristic classes of codimension- n foliations [4]. The intricate calculations of Hopf-cyclic cohomology of these Hopf algebras byMoscovici and Rangipour used crucially the fact that these Hopf algebras are bicrossed productHopf algebras [24, 13, 26, 27, 28]. On top of previously calculated explicit classes of H , ex-plicit representatives of Hopf-cyclic cohomology classes of H are recently obtained in [32] viaa cup product with a SAYD-twisted cyclic cocycle. More recently, Moscovici gave a geometricapproach for H n in [25] using explicit quasi-isomorphisms between the Hopf-cyclic complex of H n , Dupont’s simplicial de Rham DG-algebra [9], and the Bott complex [1].The cohomological machinery we developed in this paper, allowed us to replicate the results of[26] on the Hopf-cyclic cohomology of the Hopf algebra H , and its Scwarzian quotient H . Tothis end, we start by calculating in Proposition 4.3 the cohomologies of the weight 1 subcomplexof the coalgebra Hochschild complexes of these Hopf algebras using their canonical grading,but without appealing their bicrossed product structure. Then we use the Cartan homotopyformula for H , as developed in [26], to obtain the periodic Hopf-cyclic cohomology groups withcoefficients in the trivial, which happens to be the only finite dimensional, SAYD module forthese Hopf algebras [31]. 1n the quantum enveloping algebra side there are several computations [29, 30, 11, 17] on theExt-groups. However, the literature on Hopf-cyclic cohomology, or even coalgebraic cohomologyof any variant, of quantum enveloping algebras is rather meek. The only results we are awareof are both for U q ( sℓ ): one for the ordinary Hopf-cyclic cohomology by Crainic [7], and one forthe dual Hopf-cyclic cohomology by Khalkhali and Rangipour [21].The central result we achieve in this paper is the computation of both the periodic and non-periodic Hopf-cyclic cohomology of the quantized enveloping algebras U q ( g ) in full, for an ar-bitrary semisimple Lie algebra g . In Theorem 4.8 we first calculate the coalgebra Hochschildcohomology of U q ( g ) with coefficients in the comodule σ k ε of Klimyk-Schm¨udgen [22, Prop. 6.6],which is in fact an MPI over U q ( g ). We observe that the Hochschild cohomology is concentratedin a single degree determined by the rank of the Lie algebra g , and finally we calculate theperiodic and non-periodic Hopf-cyclic cohomology groups of U q ( g ) in Theorem 4.10.One of the important implications of Theorem 4.8 is that we now have candidates for non-commutative analogues of the Haar functionals for U q ( g ). The fact that coalgebra Hochschildcohomology of U q ( sℓ ) is concentrated only in a single degree was first observed by Crainic in [7].The dual version of the statement, that is the algebra Hochschild homology of k q [ SL ( N )] thequantized coordinate ring of SL ( N ) with coefficients twisted by the modular automorphism σ ofthe Haar functional is also concentrated in a single degree, is proven by Hadfield and Kraehmerin [12]. Kraehmer used this fact to prove an analogue of the Poicare duality for Hochschildhomology and cohomology for k q [ SL ( N )] in [23]. We plan on investigating the ramification ofthe fact that Hochschild cohomology of U q ( g ) is concentrated in a single degree, and its connec-tions with the dimension-drop phenomenon and twisted Calabi-Yau coalgebras [16], in a futurepaper. In this section we recall basic material that will be needed in the sequel. More explicitly, inthe first subsection our objective is to recall the coalgebra Hochschild cohomology. To this endwe also bring the definitions of the cobar complex of a coalgebra, and hence the Cotor-groups.The second subsection, on the other hand, is devoted to a very brief summary of the Hopf-cycliccohomology with coefficients.
In this subsection we recall the definition of the cobar complex of a coalgebra C , and it isfollowed by the definition of the Cotor-groups associated to a coalgebra C and a pair ( V, W ) of C -comodules of opposite parity.Let C be a coassociative coalgebra. Following [3, 8] and [20], the cobar complex of C is defined2o be the differential graded space CB ∗ ( C ) := M n ≥ C ⊗ n +2 with the differential d : CB n ( C ) −→ CB n +1 ( C ) d ( c ⊗ · · · ⊗ c n +1 ) = n X j =0 ( − j c ⊗ · · · ⊗ ∆( c j ) ⊗ · · · ⊗ c n +1 . Let C e := C ⊗ C cop be the enveloping coalgebra of C . In case C is counital, the cobar complex CB ∗ ( C ) yields a C e -injective resolution of the (left) C e -comodule C , [8].Following the terminology of [19], for a pair ( V, W ) of two C -comodules of opposite parity (say, V is a right C -comodule, and W is a left C -comodule), we call the complex( CB ∗ ( V, C , W ) , d ) , CB ∗ ( V, C , W ) := V ✷ C CB ∗ ( C ) ✷ C W where d : CB n ( V, C , W ) −→ CB n +1 ( V, C , W ) ,d ( v ⊗ c ⊗ . . . ⊗ c n ⊗ w ) = v < > ⊗ v < > ⊗ c ⊗ · · · ⊗ c n ⊗ w + n X j =1 ( − j c ⊗ · · · ⊗ ∆( c j ) ⊗ · · · ⊗ c n ⊗ w + ( − n +1 v ⊗ c ⊗ · · · ⊗ c n ⊗ w < − > ⊗ w < > , (2.1)the two-sided (cohomological) cobar complex of the coalgebra C .The Cotor-groups of a pair ( V, W ) of C -comodules of opposite parity are defined byCotor ∗C ( V, W ) := H ∗ ( V ✷ C Y ( C ) ✷ C W, d ) , where Y ( C ) is an injective resolution of C via C -bicomodules. In case C is a counital coalgebraone has Cotor ∗C ( V, W ) = H ∗ ( CB ∗ ( V, C , W ) , d ) . (2.2)We next recall the Hochschild cohomology of a coalgebra C with coefficients in the C -bicomodule(equivalently C e -comodule) V , from [8], as the homology of the complex CH ∗ ( C , V ) = M n ≥ CH n ( C , V ) , CH n ( C , V ) := V ⊗ C ⊗ n b : CH n ( C , V ) → CH n +1 ( C , V ) b ( v ⊗ c ⊗ · · · ⊗ c n ) = v < > ⊗ v < > ⊗ c ⊗ · · · ⊗ c n + n X k =1 ( − k c ⊗ · · · ⊗ ∆( c k ) ⊗ · · · ⊗ c n + ( − n +1 v < > ⊗ c ⊗ · · · ⊗ c n ⊗ v < − > . (2.3)Identification CB n ( C ) ∼ = C e ⊗ C ⊗ n , n > , c ⊗ · · · ⊗ c n +1 −→ ( c ⊗ c n +1 ) ⊗ c ⊗ · · · ⊗ c n as left C e -comodules, where the left C e -comodule structure on C ⊗ n +2 is given by ∇ ( c ⊗ · · · ⊗ c n +1 ) = ( c (1) ⊗ c n +1 (2) ) ⊗ ( c (2) ⊗ c ⊗· · ·⊗ c n ⊗ c n +1 (1) ), and on C e ⊗C ⊗ n by ∇ (( c ⊗ c ′ ) ⊗ ( c ⊗· · ·⊗ c n )) =( c (1) ⊗ c ′ (2) ) ⊗ ( c (2) ⊗ c ′ (1) ) ⊗ ( c ⊗ · · · ⊗ c n ), yields( CH ∗ ( C , V ) , b ) ∼ = ( V ✷ C e CB ∗ ( C ) , d ) . Hence, in case C is counital one can interpret the Hochschild cohomology of C , with coefficientsin V , in terms of Cotor-groups as HH ∗ ( C , V ) = H ∗ ( CH ∗ ( C , V ) , b ) = Cotor ∗C e ( V, C ) , or more generally, HH ∗ ( C , V ) = H ∗ ( V ✷ C e Y ( C ) , d )for any injective resolution Y ( C ) of C via left C e -comodules. In this subsection we recall the basics of the Hopf-cyclic cohomology theory for Hopf algebrasfrom [4, 6]. To this end we start with the coefficient spaces for this homology theory, the stableanti-Yetter-Drinfeld (SAYD) modules.Let H be a Hopf algebra. A right H -module, left H -comodule V is called an anti-Yetter-Drinfeld(AYD) module over H if H ( v · h ) = S ( h (3) ) v < − > h (1) ⊗ v < > · h (2) , for any v ∈ V , and any h ∈ H , and V is called stable if v < > · v < − > = v for any v ∈ V . In particular, the field k , regarded as an H -module by a character δ : H −→ k ,and a H -comodule via a group-like σ ∈ H , is an AYD module over H if S δ = Ad σ , S δ ( h ) = δ ( h (1) ) S ( h (2) ) , δ ( σ ) = 1 . Such a pair ( δ, σ ) is called a modular pair in involution (MPI), [5, 15].Let V be a right-left SAYD module over a Hopf algebra H . Then C ∗ ( H , V ) = M n ≥ C n ( H , V ) , C n ( H , V ) := V ⊗ H ⊗ n is a cocyclic module [14], via the face operators d i : C n ( H , V ) → C n +1 ( H , V ) , ≤ i ≤ n + 1 d ( v ⊗ h ⊗ · · · ⊗ h n ) = v ⊗ ⊗ h ⊗ · · · ⊗ h n ,d i ( v ⊗ h ⊗ · · · ⊗ h n ) = v ⊗ h ⊗ · · · ⊗ h i (1) ⊗ h i (2) ⊗ · · · ⊗ h n ,d n +1 ( v ⊗ h ⊗ · · · ⊗ h n ) = v < > ⊗ h ⊗ · · · ⊗ h n ⊗ v < − > , the degeneracy operators s j : C n ( H, V ) → C n − ( H, V ) , ≤ j ≤ n − s j ( v ⊗ h ⊗ · · · ⊗ h n ) = v ⊗ h ⊗ · · · ⊗ ε ( h j +1 ) ⊗ · · · ⊗ h n , and the cyclic operator t : C n ( H, V ) → C n ( H, V ) ,t ( v ⊗ h ⊗ · · · ⊗ h n ) = v < > · h (1) ⊗ S ( h (2) ) · ( h ⊗ · · · ⊗ h n ⊗ v < − > ) . The total cohomology of the associated first quadrant bicomplex ( CC ∗ , ∗ ( H , V ) , b, B ), [27], where CC p,q ( H , V ) := (cid:26) C q − p ( H , V ) if q ≥ p ≥ , if p > q, with the coalgebra Hochschild coboundary b : CC p,q ( H , V ) −→ CC p,q +1 ( H , V ) , b := q X i =0 ( − i d i , and the Connes boundary operator B : CC p,q ( H , V ) −→ CC p − ,q ( H , V ) , B := p X i =0 ( − ni t i ! s p − t, is called the Hopf-cyclic cohomology of the Hopf algebra H with coefficients in the SAYD module V , and is denoted by HC ( H , V ).Finally, the periodic Hopf-cyclic cohomology is defined similarly as the total complex of thebicomplex CC p,q ( H , V ) := (cid:26) C q − p ( H , V ) if q ≥ p if p > q, and is denoted by HP ( H , V ). 5 The cohomological machinery
This section contains the main computational tool of the present paper, namely, given a coal-gebra coextension C −→ D with a coflatness condition, we compute the coalgebra Hochschildcohomology of C by means of the Hochschild cohomology of D - on the E -term of a spectralsequence.Let a coextension π : C −→ D be given. We first introduce the auxiliary coalgebra Z := C ⊕ D with the comultiplication∆( y ) = y (1) ⊗ y (2) and ∆( x ) = x (1) ⊗ x (2) + π ( x (1) ) ⊗ x (2) + x (1) ⊗ π ( x (2) )and the counit ε ( x + y ) = ε ( y ) , for any x ∈ C and y ∈ D .Next, let V be a C -bicomodule, and let C be coflat both as a left and a right D -comodule. Thenconsider the decreasing filtration G p + qp = (cid:26) L n + ··· + n p = q V ⊗ Z ⊗ n ⊗ D ⊗ · · · ⊗ Z ⊗ n p − ⊗ D ⊗ Z ⊗ n p , p ≥ , p < . In the associated spectral sequence we get E i,j = G i + ji /G i + ji +1 = M n + ··· + n i = j V ⊗ C ⊗ n ⊗ D ⊗ · · · ⊗ C ⊗ n i − ⊗ D ⊗ C ⊗ n i , which gives us E ,j = HH j ( C, V ) . On the horizontal differential however, by the definition of the filtration we use only the D -bicomodule structure on C . Hence, by the coflatness assumption E i,j = 0 , i > . As a result, the spectral sequence collapses and we get HH n ( Z, V ) ∼ = HH n ( C, V ) , n ≥ . (3.1)Alternatively, one can use the short exact sequence0 → D i −→ Z p −−→ C → i : y (0 , y ) p : ( x, y ) x of coalgebras, and [10, Lemma 4.10], to conclude (3.1).Now consider CH ∗ ( Z, V ), this time with the decreasing filtration F n + pp = (cid:26) L n + ··· + n p = n V ⊗ Z ⊗ n ⊗ C ⊗ · · · ⊗ Z n p − ⊗ C ⊗ Z ⊗ n p , p ≥ , p < . E i,j = F i + ji /F i + ji +1 = M n + ··· + n i = j V ⊗ D ⊗ n ⊗ C ⊗ · · · ⊗ D n i − ⊗ C ⊗ D ⊗ n i , and by the coflatness assumption, on the vertical direction it computes HH j ( D, C ✷ D i ✷ D V ) . We can summarize our discussion in the following theorem.
Theorem 3.1.
Let π : C −→ D be a coalgebra projection, V a C -bicomodule, and C be coflatboth as a left and a right D -comodule. Then there is a spectral sequence, whose E -term is E i,j = HH j ( D, C ✷ D i ✷ D V ) , converging to HH i + j ( C, V ) . In this section we will apply the cohomological machinery developed in Section 3 to computethe Hopf-cyclic cohomology groups of quantized enveloping algebras, Connes-Moscovici Hopfalgebra H and its Schwarzian quotient H .To this end, we will compute the Cotor-groups with MPI coefficients, from which we will obtaincoalgebra Hochschild cohomology groups in view of [7, Lemma 5.1]. Therefore we note thefollowing analogue of Theorem 3.1. Theorem 4.1.
Let π : C −→ D be a coalgebra projection, and V = V ′ ⊗ V ′′ a C -bicomodulesuch that the left C -comodule structure is given by V ′ and the right C -comodule structure isgiven by V ′′ . Let also C be coflat both as a left and a right D -comodule. Then there is a spectralsequence, whose E -term is of the form E i,j = Cotor jD ( V ′′ , C ✷ D i ✷ D V ′ ) , converging to HH i + j ( C, V ) .Proof. The proof follows from Theorem 3.1 in view of the definition (2.1) of the coboundarymap of a cobar complex, and (2.2).Let us also recall the principal coextensions from [33], see also [2]. Let H be a Hopf algebra witha bijective antipode, and C a left H -module coalgebra. Moreover, H + being the augmentationideal of H (that is, H + := ker ε ), define the (quotient) coalgebra D := C/H + C .Then, by [33, Theorem II], 7a) C is a projective left H -module,(b) can : H ⊗ C −→ C ✷ D C , h ⊗ c h · c (1) ⊗ c (2) is injective,if and only if(a) C is faithfully flat left (and right) D -comodule,(b) can : H ⊗ C −→ C ✷ D C is an isomorphism.We will use this set-up to meet the hypothesis of Theorem 3.1 (and Theorem 4.1). In this subsection we will compute the periodic Hopf-cyclic cohomology groups of the Connes-Moscovici Hopf algebra H and its Schwarzian quotient H using the spectral sequence intro-duced in Theorem 4.1, and the Cartan homotopy formula developed in [26].We will use our main machinery to recover the results of [26] on the periodic Hopf-cyclic coho-mology of the Connes-Moscovici Hopf algebra H . Therefore, in this subsection we will take aquick detour to the Hopf algebra H of codimension 1, and its Schwarzian quotient H from[4, 6, 26].Let F R −→ R be the frame bundle over R , equipped with the flat connection whose fundamentalvertical vector field is Y = y ∂∂y , and the basic horizontal vector field is X = y ∂∂x , in local coordinates of F R . They act on the crossed product algebra A = C ∞ c ( F R ) ⋊ Diff( R ), atypical element of which is written by f U ∗ ϕ := f ⋊ ϕ − , via Y ( f U ∗ ϕ ) = Y ( f ) U ∗ ϕ , X ( f U ∗ ϕ ) = X ( f ) U ∗ ϕ . Then H is the unique Hopf algebra that makes A to be a (left) H -module algebra. To thisend one has to introduce the further differential operators δ n ( f U ∗ ϕ ) := y n ddx n (log ϕ ′ ( x )) f U ∗ ϕ , n ≥ , H is given by[ Y, X ] = X, [ Y, δ n ] = nδ n , [ X, δ n ] = δ n +1 , [ δ n , δ m ] = 0 , ∆( Y ) = Y ⊗ ⊗ Y, ∆( δ ) = δ ⊗ ⊗ δ , ∆( X ) = X ⊗ ⊗ X + δ ⊗ Y,ε ( X ) = ε ( Y ) = ε ( δ n ) = 0 ,S ( X ) = − X + δ Y, S ( Y ) = − Y, S ( δ ) = − δ . The ideal generated by the Schwarzian derivative δ ′ := δ − δ , is a Hopf ideal (an ideal, a coideal and is stable under the antipode), therefore the quotient H becomes a Hopf algebra, called the Schwarzian Hopf algebra. As an algebra H is generated by X, Y, Z , and the Hopf algebra structure is given by[
Y, X ] = X, [ Y, δ n ] = nδ n , [ X, Z ] = 12 Z , ∆( Y ) = Y ⊗ ⊗ Y, ∆( Z ) = Z ⊗ ⊗ Z, ∆( X ) = X ⊗ ⊗ X + Z ⊗ Y,ε ( X ) = ε ( Y ) = ε ( Z ) = 0 ,S ( X ) = − X + ZY, S ( Y ) = − Y, S ( Z ) = − Z. Hence F := Span n δ n α . . . δ n p α p | p, α , . . . α p ≥ , n , . . . n p ≥ o ⊆ H is a Hopf subalgebra of H . Finally let us note, by [31], that the only modular pair in involution(MPI) on H is ( δ,
1) of [4], where δ : gℓ aff1 −→ k is the trace of the adjoint representation of gℓ aff1 on itself.Let C := H , and let us consider the Hopf subalgebra F ⊆ C . Then D = C/ F + C = U := U ( gℓ aff1 ) , see [27, Lemma 3.19], or [6, Section 5]. Hence, by [33, Thm. II] we conclude that C is (faithfully)coflat as left and right D -comodule. Therefore, the hypothesis of Theorem 4.1 is satisfied.Since σ k = k , we have C ✷ D i ✷ D k = F ⊗ i , and hence by Theorem 3.1, E i,j = HH j ( U , F ⊗ i ) ⇒ HH i + j ( H , k ) . U -coaction on F is trivial, we have E i,j = HH j ( U , k ) ⊗ F ⊗ i ⇒ HH i + j ( H , k ) . As a result of the Cartan homotopy formula [26, Coroll. 3.9] for H , one has [26, Coroll. 3.10]as recalled below. We adopt the same notation from [26], and we denote by HP ( H ♮ [ p ] , k ) theperiodic Hopf-cyclic cohomology, with trivial coefficients, of the weight p subcomplex of H ,with respect to the grading given by the adjoint action of Y ∈ H . Corollary 4.2.
The periodic Hopf-cyclic cohomology groups of H are computed by the weight1 subcomplex, i.e. HP ( H ♮ [1] , k ) = HP ( H , k ) , HP ( H ♮ [ p ] , k ) = 0 , p = 1 . Since our spectral sequence respects the weight, in view of [18, Thm. 18.7.1] we check only E , = (cid:10) ⊗ δ ⊗ (cid:11) ∈ k ⊗ C ✷ D k,E , = (cid:10) ⊗ Y ⊗ δ ⊗ (cid:11) ∈ k ⊗ D ⊗ C ✷ D k,E , = (cid:10) ⊗ X ⊗ (cid:11) ∈ k ⊗ D ⊗ k,E , = (cid:10) ⊗ X ∧ Y ⊗ (cid:11) ∈ k ⊗ D ⊗ D ⊗ k, as the weight 1 subcomplex. Here by x ∈ D we mean the element x ∈ H viewed in D .Let d : E i,j −→ E i,j +10 be the vertical, and d : E i,j −→ E i +1 ,j be the horizontal coboundary.Then we first have d ( ⊗ δ ⊗ ) = ⊗ ⊗ δ ⊗ − ⊗ ∆( δ ) ⊗ + ⊗ δ ⊗ ⊗ = 0 . Next, we similarly observe d (cid:0) ⊗ X ⊗ Y ⊗ (cid:1) = ⊗ ⊗ X ⊗ Y ⊗ − ⊗ X ⊗ Y ⊗ ⊗ , and d (cid:16) ⊗ X ⊗ Y ⊗ + ⊗ X ⊗ Y ⊗ ⊗ + 12 ⊗ δ ⊗ Y ⊗ (cid:17) = ⊗ X ⊗ Y ⊗ ⊗ − ⊗ ⊗ X ⊗ Y ⊗ . On the other hand, we have d (cid:0) ⊗ Y ⊗ X ⊗ (cid:1) = ⊗ ⊗ Y ⊗ X ⊗ − ⊗ Y ⊗ X ⊗ ⊗ , and d (cid:16) ⊗ Y ⊗ X ⊗ + ⊗ Y ⊗ X ⊗ − ⊗ Y ⊗ δ ⊗ − ⊗ Y ⊗ δ Y ⊗ (cid:17) = ⊗ Y ⊗ X ⊗ ⊗ − ⊗ ⊗ Y ⊗ X ⊗ . d (cid:0) ⊗ X ∧ Y ⊗ (cid:1) = 0 . Finally we calculate d (cid:0) ⊗ X ⊗ (cid:1) = ⊗ ⊗ X ⊗ + ⊗ X ⊗ ⊗ . We also note that d ( ⊗ X ⊗ ) = − ⊗ X ⊗ ⊗ − ⊗ ⊗ X ⊗ − ⊗ δ ⊗ Y ⊗ , and d ( ⊗ δ Y ⊗ ) = − ⊗ Y ⊗ δ ⊗ − ⊗ δ ⊗ Y ⊗ . Hence, ⊗ Y ⊗ δ ⊗ = d (cid:0) ⊗ X ⊗ (cid:1) + d ( ⊗ X ⊗ − ⊗ δ Y ⊗ ) . As a result, on the E -term we will see E , = (cid:10) ⊗ δ ⊗ (cid:11) , E , = (cid:10) ⊗ X ∧ Y ⊗ (cid:11) . Transgression of these cocycles yields [26, Prop. 4.3] as follows.
Proposition 4.3.
The Hochschild cohomology of the weight 1 subcomplex of H is generated by [ δ ] ∈ HH ( H , k ) , [ X ⊗ Y − Y ⊗ X − δ Y ⊗ Y ] ∈ HH ( H , k ) . Consequently, we recover [26, Thm. 4.4].
Theorem 4.4.
The periodic Hopf-cyclic cohomology of H with coefficients in the SAYD module k ε is given by HP odd ( H , k ) = (cid:10) δ (cid:11) , HP even ( H , k ) = (cid:10) X ⊗ Y − Y ⊗ X − δ Y ⊗ Y (cid:11) . On the Schwarzian quotient we similarly recover [26, Thm. 4.5] as follows.
Theorem 4.5.
The periodic Hopf-cyclic cohomology of H with coefficients in the SAYD module k ε is given by HP odd ( H , k ) = (cid:10) Z (cid:11) , HP even ( H , k ) = (cid:10) X ⊗ Y − Y ⊗ X − ZY ⊗ Y (cid:11) . In this subsection we will compute the (periodic and non-periodic) Hopf-cyclic cohomologygroups of the quantized enveloping algebras U q ( g ). Our strategy will be to realize it as a principalcoextension.Let us first recall Drinfeld-Jimbo quantized enveloping algebras of Lie algebras from [22, Subsect.6.1.2]. 11et g be a finite dimensional complex semi-simple Lie algebra, α , . . . , α ℓ a fixed ordered sequenceof simple roots, and A = [ a ij ] the Cartan matrix. Let also q be a fixed nonzero complex numbersuch that q i = 1, where q i := q d i , 1 ≤ i ≤ ℓ , and d i = ( α i , α i ) / U q ( g ) is the Hopf algebra with 4 ℓ gener-ators E i , F i , K i , K − i , 1 ≤ i ≤ ℓ , and the relations K i K j = K j K i , K i K − i = K − i K i = 1 ,K i E j K − i = q a ij i E j , K i F j K − i = q − a ij i F j ,E i F j − F j E i = δ ij K i − K − i q i − q − i , − a ij X r =0 ( − r (cid:20) − a ij r (cid:21) q i E − a ij − ri E j E ri = 0 , i = j, − a ij X r =0 ( − r (cid:20) − a ij r (cid:21) q i F − a ij − ri F j F ri = 0 , i = j, where (cid:20) nr (cid:21) q = ( n ) q !( r ) q ! ( n − r ) q ! , ( n ) q := q n − q − n q − q − . The rest of the Hopf algebra structure of U q ( g ) is given by∆( K i ) = K i ⊗ K i , ∆( K − i ) = K − i ⊗ K − i ∆( E i ) = E i ⊗ K i + 1 ⊗ E i , ∆( F j ) = F j ⊗ K − j ⊗ F j ε ( K i ) = 1 , ε ( E i ) = ε ( F i ) = 0 S ( K i ) = K − i , S ( E i ) = − E i K − i , S ( F i ) = − K i F i . Let us also recall, from [22], the Hopf-subalgebras U q ( b + ) = Span { E p . . . E p ℓ ℓ K q . . . K q ℓ ℓ | r , . . . r ℓ ≥ , q , . . . , q ℓ ∈ Z } ,U q ( b − ) = Span { K q . . . K q ℓ ℓ F r . . . F r ℓ ℓ | p , . . . p ℓ ≥ , q , . . . , q ℓ ∈ Z } , of U q ( g ).A modular pair in involution for the Hopf algebra U q ( g ) is given by [22, Prop. 6.6]. Let K λ := K n . . . K n ℓ ℓ for any λ = P i n i α i , where n i ∈ Z . Then, ρ ∈ h ∗ being the half-sum of thepositive roots of g , by [22, Prop. 6.6] we have S ( a ) = K ρ aK − ρ , ∀ a ∈ U q ( g ) . Thus, ( ε, K ρ ) is a MPI for the Hopf algebra U q ( g ).For the Hopf subalgebra H := U q ( b + ) ⊆ U q ( g ) =: C , we obtain D = C/CH + = U q ( b − ). Thenby [33, Thm. II] we conclude that C is (faithfully) coflat as left and right D -comodule, andhence the hypothesis of Theorem 4.1 is satisfied.12 emma 4.6. Let C and D be as above, and µ = K p . . . K p ℓ ℓ , p , . . . , p ℓ ≥ . Then we have Cotor nD ( k, µ k ) = ( k ⊕ ( p ... + pℓ ) ! p ... pℓ ! if n = p + p + . . . + p ℓ , if n = p + p + . . . + p ℓ Proof.
We apply Theorem 4.1 to the coextension π : D −→ W := Span { K m . . . K m ℓ ℓ | m , . . . , m ℓ ∈ Z } E r . . . E r ℓ ℓ K m . . . K m ℓ ℓ (cid:26) K m . . . K m ℓ ℓ if r = r = . . . = r ℓ = 0 , otherwise, to have a spectral sequence, converging to Cotor D ( k, µ k ), whose E -term is E i,j = Cotor jW ( k, D ✷ W . . . ✷ W D | {z } i many ✷ W µ k ) . Since D ✷ W . . . ✷ W D | {z } i many ✷ W µ k =Span n E b s a s . . . E β s α s µK − b a . . . K − β α . . . K − b s a s . . . K − β s α s ⊗ · · · ⊗ µK − b a . . . K − β α ⊗ · · · ⊗ µK − b a . . . K − β α | {z } i many ⊗ E b a . . . E β α µK − b a . . . K − β α ⊗ µ ⊗ · · · ⊗ µ | {z } i many ⊗ o ,i ≥ s ≥ ℓ ≥ a , . . . , a s , . . . , α , . . . , α s ≥ i , i , . . . , b , . . . b s , β , . . . , β s ≥
0, the left W -coaction on a typical element is given by ∇ LW ( E a i K − a i . . . K − a µ ⊗ · · · ⊗ E a K − a K − a µ ⊗ E a K − a µ ⊗ ) = K − a i . . . K − a µ ⊗ E a i K − a i . . . K − a µ ⊗ · · · ⊗ E a K − a K − a µ ⊗ E a K − a µ ⊗ . (4.1)Since W consists only of the group-like elements, the result follows from the W -coaction (4.1)to be trivial. Lemma 4.7.
Let C and D be as above, and σ = K ρ . Then we have Cotor nD ( k, C ✷ D . . . ✷ D C | {z } i many ✷ D σ k ) = k ⊕ ℓi if n = ℓ − i, if n = ℓ − i. Proof.
Let us first note that C ✷ D . . . ✷ D C | {z } i many ✷ D σ k =Span n σK − b a . . . K − β α . . . K − b s − a s − . . . K − β s − α s − F b s a s . . . F β s α s ⊗ · · · ⊗ σK − b a . . . K − β α ⊗ · · · ⊗ σK − b a . . . K − β α | {z } i many ⊗ σF b a . . . F β α ⊗ σ ⊗ · · · ⊗ σ | {z } i many ⊗ o , ≥ s ≥ ℓ ≥ a , . . . , a s , . . . , α , . . . , α s ≥ i , i , . . . , b , . . . b s , β , . . . , β s ≥
0. Then, the left D -coaction on a typical element is given by ∇ LD ( σK − a . . . K − a i − F a i ⊗ · · · ⊗ σK − a F a ⊗ σF a ⊗ ) = σK − a . . . K − a i − K − a i ⊗ σK − a . . . K − a i − F a i ⊗ · · · ⊗ σK − a F a ⊗ σF a ⊗ . (4.2)By the proof of Lemma 4.6, there is no repetition on the indexes appearing in (4.2). Accordingly,the result follows from Lemma 4.6. Proposition 4.8.
For σ := K ρ we have Cotor nU q ( g ) ( k, σ k ) = (cid:26) k ⊕ ℓ n = ℓ n = ℓ. Proof.
Letting C and D be as above, it follows from Theorem 4.1 that we have a spectralsequence converging to Cotor C ( k, σ k ) whose E -term is E i,j = Cotor jD ( k, C ✷ D i ✷ D σ k ) . By Lemma 4.7 the cocycles are computed at j = ℓ − i , i.e. i + j = ℓ . The number of cocycles,on the other hand, is ℓ X i =0 (cid:18) ℓi (cid:19) = 2 ℓ . Remark 4.9.
We note that since C = U q ( g ) is a Hopf algebra, we could start with the Hopfsubalgebra H = C ⊆ C to get D = H/H + = W , and yet to arrive at the same result.We are now ready to compute the (periodic) Hopf-cyclic cohomology of U q ( g ). Theorem 4.10.
For σ := K ρ , and ℓ ≡ ǫ (mod 2) , we have HP ǫ ( U q ( g ) , σ k ) = k ⊕ ℓ , HP − ǫ ( U q ( g ) , σ k ) = 0 . Proof.
Let C be as above. As a result of Proposition 4.8 we have HH n ( C, σ k ) = (cid:26) k ⊕ ℓ n = ℓ n = ℓ. Hence, the Connes’ SBI sequence yields HC n ( C, σ k ) = 0 , n < ℓ,HC ℓ +1 ( C, σ k ) ∼ = HC ℓ +3 ( C, σ k ) ∼ = . . . ∼ = 0 k ⊕ ℓ ∼ = HH ℓ ( C, σ k ) ∼ = HC ℓ ( C, σ k ) ∼ = HC ℓ +2 ( C, σ k ) ∼ = . . . where the isomorphisms are given by the periodicity map S : HC p ( C, σ k ) −→ HC p +2 ( C, σ k ) . cknowledgement: S. S¨utl¨u would like to thank Institut des Hautes ´Etudes Scientifiques(IHES) for the hospitality, and the stimulating environment provided during the preparation ofthis work.
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