Hopf cyclic cohomology and transverse characteristic classes

Transcription

1 Hopf cyclic cohomoloy and transverse characteristic classes Lecture 2: Hopf cyclic complexes Bahram Ranipour, UNB Based on joint works with Henri Moscovici and with Serkan Sütlü Vanderbilt University May 2011

2 Contents Section 1: Hopf cyclic cohomoloy with coefficients in SAYD modules Section 2: Hopf cyclic Bicomplexes of a Lie-Hopf alebra and van Est isomorphism Section 4: Hopf cyclic cohomoloy of H n Section 5: Hopf cyclic cohomoloy of K n

3 Hopf cyclic cohomoloy and SAYD modules MPI and SAYD and their examples Hopf cyclic cohomoloy Lie alebra cohomoloy Examples

4 Hopf Cyclic Cohomoloy Hopf cyclic cohomoloy invented by Connes and Moscovici as a computational tool for computin the index cocycle appears in the local index formula Rouhly speakin, they used the Hopf alebra of eneral transverse symmetry to compute their desire part of the cyclic cohomoloy, differential cyclic cohomoloy, of the crossed product alebra representin the transverse manifold of a foliation The inredient for Hopf cyclic cohomoloy is a Hopf alebra endowed with a modular pair in involution Based on different needs we eneralize Hopf cyclic cohomoloy in different directions: Allowin noncommutative symmetries, ıe, Hopf (co)module (co)alebra Allowin hiher dimensional coefficients, ıe, SAYD modules Allowin Hopf alebras and (co)alebras with several objects, ıe, -Hopf alebras

5 SAYD modules Let H be a Hopf alebra By definition, a character δ : H C is an alebra map A roup-like σ H is the dual object of the character, ie, (σ) = σ σ The pair (δ, σ) are called modular pair in involution if δ(σ) = 1, and S 2 δ = Ad σ Here Ad σ (h) = σhσ 1 and S δ is defined by S δ (h) = δ(h(1))s(h(2)) A riht-left stable-anti-yetter-drinfeld module over a Hopf alebra H is a riht H-module M, which is also a left H-comodule and satisfyin (m h) = S(h(3))m < 1> h(1) m <0> h(2), m <0> m < 1> = m, Lemma Any MPI defines a one dimensional SAYD module and all one dimensional SAYD modules come this way

6 Canonical MPI associated to Lie-Hopf alebras Let (, F) be a Lie Hopf alebra Define the derivation δ : C, δ (X ) = Tr(Ad X ) Extend δ to an alebra map on F U δ := ε δ : F U() C by δ(f u) = ε(f )δ (u) The canonical rep functions fj i F defined for a fixed basis X 1,, X m of, It is clear that f i j (X j ) = m=dim i=1 X i f i j are independent of {X 1,, X m } Set σ F := ( 1) π f π(1) 1 f π(m) π S m m Theorem The pair (δ, σ) is a modular pair involution for the Hopf alebra F U

7 Induced module Let M be a left -module and a riht F-comodule via M : M M F We say that M is an induced module if M (X m) = X <0> m <0> X <1> m <1> + m <0> X m <1> H act on M from left via (f u) m = ε(f )u m H coact on M via, M (m) = m <0> m <1> 1 Lemma Any induced module is a YD-module over F U Let σ M δ := M as a vector space Then the followin make σ M δ a SAYD module over F U m (f u) := ε(f )δ(u(2))s(u(2))m (m) := σs(m <1> ) 1 m <0>

8 Induced Hopf cyclic coefficients in eometric cases Let M be a left := 1 2 -module such that the restriction of the action results a locally finite 2 -module Then, via the 1 action on M, by restriction, and the coaction defined by (m) = m <0> m <1>, iff v m = m <1> (v)m <0> M becomes an induced ( 1, R( 2 ))-module Conversely, every induced ( 1, R( 2 ))-module comes this way So for any representation M of = 1 2 we have a σ M δ as a SAYD module on R( 2 ) U( 1 ) Similarly for a matched pair of Lie roups G = G 1 G 2, and any representation M so that the action of G 2 is locally finite we have a SAYD module σ M δ on R(G 2 ) U( 1 ) For a complete characterization of SAYD modules on eometric bicrossed product Hopf alebars one needs more advanced technoloy which is not suitable to be discussed in this lecture series

9 Cyclic module of a Hopf alebra with a SAYD module C q (H, M) := M H q, q 0 We recall the followin operators on C (H, M) face operators i : C q (H, M) C q+1 (H, M), 0 i q + 1 deeneracy operators σ j : C q (H, M) C q 1 (H, M), 0 j q 1 cyclic operators τ : C q (H, M) C q (H, M), by 0 (m h 1 h q ) = m 1 h 1 h q, i (m h 1 h q ) = m h 1 h i (1) h i (2) h q, q+1 (m h 1 h q ) = m <0> h 1 h q m < 1>, σ j (m h 1 h q ) = m h 1 ε(h j+1 ) h q, τ(m h 1 h q ) = m <0> h 1 (1) S(h 1 (2)) (h 2 h q m < 1> where H acts on H q diaonally

10 cyclic module The raded module C (Hc, M) endowed with the above operators is then a cocyclic module This which means that i, σ j and τ satisfy the followin identities j i = i j 1, if i < j, σ j σ i = σ i σ j+1, if i j, i σ j 1, if i < j σ j i = Id if i = j or i = j + 1 i 1 σ j if i > j + 1, τ i = i 1 τ, τ 0 = q+1, τσ i = σ i 1 τ, τσ 0 = σ n τ 2, τ q+1 = Id 1 i q 1 i q

11 Hopf cyclic cohomoloy(definition) One uses the face operators to define the Hochschild coboundary b : C q (H, M) C q+1 (H, M), by q+1 b := ( 1) i i i=0 One uses the rest of the operators to define the Connes boundary operator, B : C q (H, M) C q 1 (H, M), by ( q ) B := ( 1) qi τ i σ q 1 (1 τ) i=0 It is shown for any cocyclic module that b 2 = B 2 = (b + B) 2 = 0

12 Continue As a result, one defines the cyclic cohomoloy of H with coefficients in SAYD module M, which is denoted by HC (H, M), as the total cohomoloy of the bicomplex M H q, if 0 p q, C p,q (H, M) = 0, otherwise One also defines the periodic cyclic cohomoloy of H with coefficients in M, which is denoted by HP (H, M), as the total cohomoloy of direct sum total of the followin bicomplex (1) M H q, if p q, C p,q (H, M) = 0, otherwise It is seen that the periodic cyclic complex and hence cohomoloy is Z 2 raded (2)

13 Lie alebra cohomoloy Let be a finite dimensional Lie alebra and V is a riht -module The Chevalley-Eilenber complex of the (, V ) is defined by C n (, V ) = Hom( q, V ) q 0 with V 0 C 1 (, V ) 1 C 2 (, V ) 2, (ω)(y 0,, Y q ) = i<j ( 1) i+j ω([y i, Y j ], Y 0,, Ŷi,, Ŷj,, Y q ) + i ( 1) i ω(y 0,, Ŷi, Y q )Y i Alternatively, for {θ i, X i } as a dual basis pair for and 0 (v) = vx i θ i, q (v ω) = vx i θ i ω + v dr (ω) Here dr : q q+1 is the de Rham coboundary which is a derivation of deree 1 and recalled here by dr (θ k ) = 1 2 C k i,jθ i θ j

14 Hopf cyclic cohomoloy of U() Let V be a representation of We easily see that V is a SAYD module on U() provided the coaction defined trivially The antisymmetrization map α : V q V U() q defined by α(v Y 1 Y q ) = 1 v Y q! σ(1) Y σ(q) σ S q We see that bα = 0, Bα = α Usin some standard homotopy aruments we see HP (U(), V ) H i (, V ) i= mod 2

15 Hopf cyclic cohomoloy of R(), and R(G) Let l be the solvable radical of, ie, l is the unique maximal solvable ideal of The Levi decomposition of Lie alebras implies that = s l, where s is a semisimple subalebra of called a Levi subalebra D Gr : V F q C q (, h, V ), D Gr (v f 1 f q )(X 1,, X q ) = ( 1) µ d d dt 1 dt µ S q t 1 =0 q f 1 (exp(t 1 X µ(1) )) f q (exp(t q X µ(q) )v tq=0 Theorem Let G be a Lie roup, V a representative G-module, and = s > l be a Levi decomposition Then D Gr induces the isomorphism HP (R(G), V ) = H i (, s, V ) =i mod 2

16 Continue D Al : V R() q (V q l ) s, D Al (v f 1 f q )(X 1,, X q ) = σ S q ( 1) σ f 1 (X σ(1) ) f q (X σ(q) )v Theorem Let be a finite dimensional Lie alebra with a Levi decomposition = s l Then for any finite dimensional -module V, the map D Al induces the isomorphism HP (R(), V ) = =i mod 2 H i (, s, V )

17 Hopf cyclic cohomoloy of Lie-Hopf alebras Bicomplexes associated to Lie Hopf alebras van Est isomorphism

18 Bullet action Let F a -Hopf alebra We denote the bicrossed product Hopf alebra F U() by H Let the character δ and the roup-like σ be the canonical modular pair in involution In addition, let M be an induced (, F)-module and σ M δ be the associated SAYD module over Hc The Hopf alebra U := U() admits the followin riht action on σ M δ F q, which plays a key role in the definition of the next bicocyclic module: (m f )u = δ (u(1))s(u(2)) m S(u(3)) f, u (f 1 f n ) := u(1) <0> f 1 u(1) <1> u(2) <0> f 2 u(1) <n 1> u(n 1) <1> u(n) f n

19 Bicocyclic module associated to Lie Hopf alebras One then defines a bicocyclic module C, (U, F, M), where C p,q (U, F, s M δ ) := s M δ U p F q, p, q 0, (3) whose horizontal part is the cocyclic module associated to U with coefficients in σ M δ F q The vertical part is the cocyclic module associated to F with coefficients in σ M δ U p One notes that, by definition, a bicocyclic module is a biraded module whose rows and columns form cocyclic modules and any horizontal arrow commute with any vertical one

20 Bicomplex associated to Lie Hopf alebra So each row and column have their own Hochschild coboundary and Connes boundary maps These boundaries and coboundaries are denoted by B, B, b, and b, which are demonstrated in the followin diaram b B σ M δ U 2 b B b B σ M δ U 2 F b B b B σ M δ U 2 F 2 b B b B σ M δ U b B b B σ M δ U F b B b B σ M δ U F 2 b B b B B b σ M σ δ M δ F B b b B b B b σ M δ F 2 B

21 Diaonal complex In the next move, we identify the standard Hopf cocyclic module C (H, σ M δ ) with the diaonal subcomplex D of C, This is achieved by means of the map Ψ : D C (H, σ M δ ) toether with its inverse Ψ 1 : C (H, σ M δ ) D They are explicitly defined as follows: Ψ (m u 1 u n f 1 f n ) = m f 1 u 1 <0> f 2 u 1 <1> u2 <0> respectively Ψ 1 (m f 1 u 1 f n u n ) f n u 1 <n 1> un 1 <1> un, = m u 1 <0> un 1 <0> un f 1 f 2 S(u 1 <n 1> ) f 3 S(u 1 <n 2> u2 <n 2> ) f n S(u 1 <1> un 1 <1> )

22 Simplification of the bicomplex The bicocyclic module C, (U, F, σ M δ ) can be further reduced to the bicomplex C, (, F, σ M δ ) := σ M δ F, via the antisymmetrization map α : σ M δ q F p σ M δ U q F p, α = α Id, the pullback of the vertical b-coboundary in vanishes, while the vertical B-coboundary becomes On the other hand, since F is commutative, the coaction : F, extends from to a unique coaction : p p F After tensorin with the riht coaction of σ M δ, (m X 1 X q ) = m <0> X 1 <0> X q <0> σ 1 m <1> X 1 <1> X q <1>

23 We arrive at the bicomplex C, (, F, σ M δ ), described by the diaram σ M δ 2 b F σ M δ 2 F b F σ M δ 2 F 2 b F σ M δ b F σ M δ F b F σ M δ F 2 b F σ M δ b F σ M δ F b F σ M δ F 2 b F

24 Application of Poincaré isomorphism Let C p,q (, F, M) = M p F q We apply the Poincaré isomorphism D P : C p,q (, F, σ M δ ) C m p,q (, F, M) The result is the followin bicomplex whose columns are Lie alebra cohomoloy complex and the row are Hopf cyclic cohomoloy M 2 b F M 2 F b F M 2 F 2 b F M b F M F b F M F 2 b F M b F M F b F M F 2 b F

25 H( 1 2, h, M) One uses the facts that a = 1 2, and 2 = h < l is a Levi decomposition to see that the relative Lie alebra cohomoloy H(a, h, M) is computed by the total complex of the followin bicomplex Here the vertical and horizontal arrows are induced by the Lie alebra cohomoloy coboundaries of 1 and 2 with values in M 2 and M 1 (M 2 1 )h (M 2 1 l ) h (M l ) h (M 1 )h (M 1 l ) h (M 1 2 l ) h M h (M l ) h (M 2 l ) h

26 Van Est isomorphism Let (, 2 ) be a matched pair of Lie alebras and M be a finite dimensional representation of 1 2 Let F be ( 1, 2 )-related Hopf alebra, that is F is a 1 -Hopf alebra with a 1 respected Hopf duality from U( 2 ) One then defines the followin Van Est Isomorphism V : M p 1 F q M p 1 q 2 V(m ω f 1 f q )(Y 1,, Y p, ζ 1,, ζ q ) = σ S q ( 1) σ ω(y 1,, Y p )f 1 (ζ σ(1) ) f q (ζ σ(q) )

27 HP(F U, σ M δ ) Theorem Let ( 1, 2 ) be a matched pair of Lie alebras and F a ( 1, 2 )-related Hopf alebra such as R( 2 ), R(G 2 ), or P(G 2 ) Assume that 2 = h l is a Levi decomposition such that h is 1 -invariant and the natural action of h on 1 is iven by derivations Then for any F-comodule and 1 -module M, the Van Est map V, is a map of bicomplexes and induces an isomorphism between Hopf cyclic cohomoloy of F U( 1 ) with coefficients in σ M δ and the Lie alebra cohomoloy of a := 1 2 relative to h with coefficients in the a-module induced by M In other words, HP (F U( 1 ), σ M δ ) = H i ( 1 2, h, M) i= mod 2

28 Hopf cyclic cohomoloy of H n Hopf-Koszul bicomplex Group cohomoloy bicomplex Equivariant bicomplex Wede-Equivariant bicmplex Hopf cyclic cohomoloy of H n relative to l n Hopf cyclic cohomoloy of H n

29 Passae to Lie alebra cohomoloy We recall that the Koszul resolution associated to the Lie alebra is the complex V ( ) : U K U K 2 U K, acts on U from the riht via u X = Xu K : V ( ) F V ( ) defined as follows: K (ω u) = S(u <0> (2)) S(u <1> ω <1> ) ω <0> u <0> (1) V ( ) U C b F,K B F,K V ( ) U F b F,K B F,K V ( ) U F 2 b F,K B F,K Theorem C U (F, V ( )) is a -equivariant Hopf cyclic complex of F with coefficients in the differential raded SAYD V ( ), and the map κ : C U (F, V ( )) C (F, ) induces an isomorphism of total complexes κ(ω u U f 1 f q ) = ω u (f 1 f q ),

30 Linkae with roup cohomoloy We define C p,q coinv (, F) := ( p F q+1 ) F by α f ( p F q+1 ) F if α <0> f S(α <1> ) = α f <0> f <1> ; We define I : p F q ( p F q+1 ) F by I(α f ) = = a <0> f 1 (1) S(f 1 (2))f 2 (1) S(f q 1 (2))f q (1) S(α <1> f q (2)) induces an isomorphism The isomorphism I turns the action into the diaonal action X (f 0 f q ) = q f 0 X f i f q i=0

31 One obtains the biraded module coinv ( 2 F) F coinv ( F) F coinv (C F) F b coinv b coinv b coinv B coinv B coinv B coinv coinv ( 2 F 2 ) F coinv ( F 2 ) F coinv (C F 2 ) F b coinv B coinv b coinv B coinv b coinv B coinv coinv ( 2 F 3 ) F coinv ( F 3 ) F coinv (C F 3 ) F B coinv b coinv b coinv B coinv b coinv B coinv (4)

32 We set N := lim N k, with N k := Jet (k) 0 (N) Recall that the k alebra F is precisely the polynomial alebra enerated by the components of such jets By ψ N we mean j 0 (ψ) N with ψ N Let C q pol (N, p ) be the set of polynomial functions c : N } {{ N } p q+1 times satisfyin the covariance condition c(ψ 0 ψ,, ψ q ψ) = ψ 1 c(ψ 0,, ψ q ), ψ N, q+1 b pol c(ψ 0,, ψ q+1 ) = ( 1) i c(ψ 0,, ˆψ i,, ψ q+1 ) i=0 τ pol (c)(ψ 0,, ψ q ) = c(ψ 1,, ψ q, ψ 0 )

33 We thus arrive at a bicomplex which mixes roup cohomoloy of N with coefficients in and Lie alebra cohomoloy of with coefficients in F, described by the diaram C 0 pol pol ( N, 2 ) b pol pol Cpol 0 pol C 0 pol B pol ( N, ) b pol B pol ( N, C ) b pol B pol pol Cpol 1 ( N, 2 ) pol Cpol 1 pol C 1 pol ( N, ) ( N, C ) b pol B pol b pol B pol b pol B pol pol Cpol 2 pol Cpol 2 pol C 2 pol ( N, ) ( N, ) b pol B pol b pol B pol ( ) N, C b pol B pol

34 In turn, this can be related to the bicomplex (4) via the obvious map J : C, coinv (, F) C pol (N, ) defined, with the self-explanatory notation, by the formula J ( α f ) (ψ0,, ψ q ) = f 0 (ψ 0 ) f q (ψ q )α (5) Proposition The map J : C, coinv (, F) C pol (N, ) is an isomorphism of bicomplexes

35 Wede-Coinvariant Bicomplex Since F is commutative we may consider the subcomplex C p,q c w (, F) C, coinv (, F) c w c w c w 2 c w b c w ( 2 2 F) F c w b c w ( 2 3 F) F c w b c w c w b c w ( 2 F) F c w b c w ( 3 F) F c w b c w C b c w (C 2 F) F b c w (C 3 F) F b c w, Proposition The antisym map α F : C p,q c w (, F) C p,q coinv (, F) is a quasi-isomorphism

36 van Est Isomorphism To construct the desired homomorphism, we associate to any ω CGF r (a) and any pair of inteers (p, q), p + m = q + r, 0 q m = dim G, a p-cochain C p,q (ω)(ψ 0,, ψ p ) on the roup N with values in q-currents on G as follows For each q-form with compact support ζ Ω q c (G), one sets C p,q (ω)(ψ 0,, ψ p ), ζ = Jπ1 (ζ) ω ; Σ(ψ 0,,ψ p) the interation is taken over the finite dimensional cycle in J 0 Theorem The map C : C top(a) C top(n, Ω (G)) is a map of complexes It lands in C, c w (, F) and induces an isomorphism in the level of cohomoloy