cortona moscovici - gwdg

Transverse Index Theory and HopfAlgebras

Henri Moscovici

Cortona, June 2008

Henri Moscovici Transverse Index Theory and Hopf Algebras Cortona, June 2008 1 / 25

Classical geometries

Geometry Automorphism group Structure group

1 General Diff(Rn) GLn

2 Vol preserving DiffVol(Rn) SLn

3 Symplectic DiffSp(R2n) Sp2n

4 Contact DiffCn(R2n+1) Cn2n+1

5 Volume preserving up to constant

6 Symplectic form preserving up to constant

7 Complex analytic version of the above


Geometries modulo primitive Cartan-Lie pseudogroupsand the corresponding Hopf algebras

Geometry modulo Pseudogroup Lie algebra Hopf algebra

1 General Π(Rn) an Hn

2 Vol preserving ΠVol(Rn) san SHn

3 Symplectic ΠSp(R2n) sp2n SpH2n

4 Contact ΠCn(R2n+1) cn2n+1 CnH2n+1

5 Volume preserving up to constant

6 Symplectic form preserving up to constant

7 Complex analytic version of the above


Foliations and characteristic classes

Q [H. Hopf] Is there a completely integrable plane field on S3 ?

A [G. Reeb, 1948] S3 = D2 × S1∐

S1 × D2/ ∼

Q [A. Haefliger] Is any E ⊂ TV homotopic to an integrable one?

A [R. Bott, 1970] Cohomological obstructions:

Pont>2n(TV /E ) = 0, n = rank(TV /E ).


Transverse characteristic classes

The vanishing theorem is obtained by applying the Chern-Weil theoryto E -flat connections.

This vanishing gives rise by transgression to new, R-valued andcontinuously varying, characteristic classes.

The Lie algebra an of formal vector fields on Rn plays the role of theclassifying object, and its Gelfand-Fuks cohomology gives universaltransverse characteristic classes for foliations.

In the dual, K -homological context, the universal object turned out tobe a Hopf algebra Hn , canonically associated to the group Diff Rn,and its Hopf cyclic cohomology delivers the universal transversecharacteristic classes.


The space of leaves

Holonomy gives rise to a pseudogroup on acomplete transversal, or equivalently to a transverse etale groupoid.

Thus, while ordinary, locally compact, topological spaces X can beequivalently described in terms of their function algebras C0(X ), oneneeds to allow noncommuting coordinates to adequately describe‘spaces of leaves’, and replace the notion of isomorphism by that ofMorita equivalence.

For instance, C (S1 o Γ) ≡ C (S1) o Γ, captures the transverse spaceto a codimension 1 foliation (Γ = Z for the Kronecker foliation).

If Γ acts properly, e.g. Γ = Z/NZ, then C (S1 o Γ) and C (S1/Γ) areMorita equivalent; but e.g. for Γ = PSL(2,R) the quotient S1/Γ ispractically meaningless.


Transverse K -homology class [A. Connes & H.M., 1995]

Hypoelliptic signature operator: Mn = oriented smooth manifold,π : PM = F +M/SO(n)→ M, V = Kerπ∗ , N = T (PM)/V

Q := (d∗V dV − dV d∗V )⊕ γV (dH + d∗H), H = connection.

Defines a spectral triple (AΓ,H,D), for any Γ ⊂ Diff+(M), where

AΓ = C∞c (PM) o Γ, H = L2(∧•V∗ ⊗ ∧•N ∗, vol),

and with D determined by funct. eq. Q = D|D|; specifically,

D = D∗, [D, a] := D a − a D bounded ∀ a ∈ AΓ,

f (D + i)−1 ∈ Lp(H), p = v + 2n, ∀ f ∈ C∞c (PM).

Zeta functions: ∀P ∈ ΨDO ′, the function z 7→ Tr(P|D|−z) hasmeromorphic continuation with simple poles and∫

−P := Resz=0 Tr(P|D|−z) =1

(2π)v+n

∫S ′T∗M

σ′−v−2n(P)


Local Index Formula

Theorem (AC & HM, 1995)

Assume (A,H,D) = (odd) spectral cycle, such that ∃ residues∫−T := Ress=0 Tr(T |D|−2s), T ∈ Ψ{A, [D,A], |D|−z ; z ∈ C}.

1 [(ϕn)n=1,3,...] is a cocycle in the (b,B)-bicomplex of A,

ϕn(a0, . . . , an) =∑

k

cn,k

∫−a0[D, a1](k1) . . . [D, an](kn)|D|−n−2|k|

∇(T ) = [D2, a], T (k) = ∇k(T ), |k | = k1 + . . .+ kn ,

cn,k =(−1)|k| Γ

(|k |+ n

2

)k1! . . . kn!(k1 + 1) . . . (k1 + · · ·+ kn + n)

.

2 [(ϕn)n=1,3,...] = ch∗(H,F ) ∈ HC ∗(A).


Relation with Atiyah-Singer Index Theorem

1 The zeta functions associated to the Dirac spectral triple(C∞(Mm), L2(S), /D) are meromorphic with simple poles;

2

∫−P '

∫S∗M

σ−n(P). ∀P ∈ ΨDO∞(Mn);

(Guillemin-Wodzicki residue)

3

∫−f 0[/D, f 1](k1) . . . [/D, f n](kn) |/D|−(n+2|k|) = 0 , if |k | > 0 ;

4

∫−f 0[/D, f 1] . . . [/D, f n] |/D|−n = cn

∫M

A(R) ∧ f 0 df 1 ∧ . . . ∧ df n;

5 under the Connes isomorphism HP∗(C∞(Mm)) ∼= HdR∗ (M,C),

ch∗(H, /D) ≡ [(ϕn) ] ∼= [A(R)] ≡ Chern∗ (/D).


Transverse Index Theorem (mod Diff )

Theorem (AC & HM, 1998)

There are canonical constructions for the following entities:

1 a Hopf algebra Hn associated to Diff(Rn), with modular character δ,and with (δ, 1) modular pair in involution;

2 a co-cyclic structure for any Hopf algebra H endowed with a modularpair in involution (δ, σ);

3 an isomorphism κ∗n between the Gelfand-Fuks cohomology H∗cont(an),resp. H∗cont(an, SOn), and HP∗(Hn; Cδ), resp. HP∗(Hn,SOn; Cδ);

4 an action of Hn on AΓ(FRn), inducing a characteristic mapχ∗Γ : HP∗(Hn, SOn; Cδ)→ HP∗(1)(AΓ(PRn)) ∼= H∗(PRn ×Γ EΓ);

5 a cohomology class Ln ∈ H∗cont(an,SOn), such thatch∗(AΓ(PRn),H(PRn),D)(1) = (χ∗Γ ◦ κ∗n)(Ln).


Example: the Hopf algebra H1

As algebra: generated by {X , Y , δ1, δ2, . . .} subject to

[Y ,X ] = X , [Y , δk ] = kδk , [X , δk ] = δk+1, [δk , δ`] = 0.

As coalgebra: ∆(Y ) = Y ⊗ 1 + 1⊗ Y ,

∆(X ) = X ⊗ 1 + 1⊗ X + δ1 ⊗ Y ,

∆(δ1) = δ1 ⊗ 1 + 1⊗ δ1,

∆(δ2) = δ2 ⊗ 1 + δ1 ⊗ δ1 + 1⊗ δ2,

∆(δ3) = δ3 ⊗ 1 + δ2 ⊗ δ1 + 3δ1 ⊗ δ2 + δ21 ⊗ δ1 + 1⊗ δ3, . . .

Action on C∞(F +S1) n Γ: ϕ(x , y) = (ϕ(x), ϕ′(x) y) ,Y (fU∗ϕ) = y ∂f

∂y U∗ϕ , X (fU∗ϕ) = y ∂f∂x U∗ϕ,

δn(fU∗ϕ) = yn dn

dxn

(log dϕ

dx

)fU∗ϕ.


Hopf cyclic cohomology [AC+HM 1998]

Modular pair = δ ∈ H∗ character, σ ∈ H group-like element,ε(σ) = 1, δ(σ) = 1 , such that S2 = Ad(σ), whereS(h) =

∑(h) δ(h(1)) S(h(2)), ∀ h ∈ H.

Cylic object H\Hopf = {CqHopf (H)}q≥0

∂0(h1 ⊗ . . .⊗ hn) = 1⊗ h1 ⊗ . . .⊗ hn,∂j(h

1 ⊗ . . .⊗ hn) = h1 ⊗ . . .⊗∆hj ⊗ . . .⊗ hn

∂n(h1 ⊗ . . .⊗ hn) = h1 ⊗ . . .⊗ hn ⊗ σσi (h

1 ⊗ . . .⊗ hn) = h1 ⊗ . . .⊗ ε(hi+1)⊗ . . .⊗ hn

τn(h1 ⊗ . . .⊗ hn) = S(h1) · h2 ⊗ . . .⊗ hn ⊗ σ.

Example (HP∗(H1; Cδ))

TF = X ⊗ Y − Y ⊗ X − δ1 Y ⊗ Y (universal fundamental),GV = δ1 (universal Godbillon-Vey).


Hopf cyclic with coefficients [H-K-R-S 2003]

H= Hopf algebra, M= H-module/comodule is an SAYD if:

m<0>

m<−1>

= m,

(mh)<−1>

⊗ (mh)<0>

= S(h(3))m<−1>

h(1) ⊗m<0>

h(2) .

∀H-module coalgebra C , ∃ canonical co-cyclic structure:

Cn := CnH(C ,M) = M ⊗H C⊗n+1, n ≥ 0

∂i : Cn → Cn+1, σj : Cn → Cn−1, τn : Cn → Cn

∂i (m ⊗H c) = m ⊗H c0 ⊗ . . .⊗∆(ci )⊗ . . .⊗ cn

∂n+1(m ⊗H c) = m<0>⊗H c0

(2) ⊗ c1 ⊗ . . .⊗ cn ⊗m<−1>

c0(1)

σj(m ⊗H c) = m ⊗H c0 ⊗ . . .⊗ ε(c i+1)⊗ . . .⊗ cn

τn(m ⊗H c) = m<0>⊗H c1 ⊗ . . .⊗ cn ⊗m

<−1>c0.


Hopf cyclic cohomology of a Hopf algebra:C := H, H acts on itself by multiplication⇒ Cn

H(H,M) ∼= M ⊗H⊗n.

Hopf cyclic cohomology relative to a sub-Hopf algebra K ⊂ H:C := H⊗K C ⇒ Cn

H(H,M) = M ⊗K H⊗n;alternatively,

∂0(m ⊗ h1 ⊗ . . .⊗ hn) = m ⊗ 1⊗ h1 ⊗ . . .⊗ hn

∂i (m ⊗ h1 ⊗ . . .⊗ hn) = m ⊗ h1 ⊗ . . .⊗∆(hi )⊗ . . .⊗ hn

∂n+1(m ⊗ h1 ⊗ . . .⊗ hn) = m<0>⊗ h1 ⊗ . . .⊗ hn ⊗m

<−1>

σj(m ⊗ h1 ⊗ . . .⊗ hn) = m ⊗ h1 ⊗ . . .⊗ 1⊗ . . .⊗ hn

τn(m ⊗ h1 ⊗ . . .⊗ hn) =

m<0>

h1(1)⊗ ⊗ S(h1

(2))(h2 ⊗ . . .⊗ hn ⊗m<−1>

)


Hopf algebras associated to classical groups ofdiffeomorphisms [H.M. & B. Rangipour]

Splitting a la G.I. Kac: Π = primitive Lie pseudogroup of infinite type,DiffΠ = globally defined diffeomorphisms of type Π, then set-theoretically

DiffΠ = GΠ · NΠ, GΠ ∩ NΠ = {e}, where

flat case : GΠ := Rn n G 0Π, with G 0

Π := linear transformations inDiffΠ, NΠ := {φ ∈ DiffΠ; φ(0) = 0, φ′(0) = Id};

non-flat case : identifying R2n+1 with the Heisenberg group Hn,GΠ := Hn n G 0

Π, with G 0Π := linear contact transformations,

NΠ := all contact diffeomorphisms preserving the origin to order 1 inthe sense of Heisenberg tangency, i.e. φ′(0) = IdH= the identity mapof the Heisenberg tangent bundle, THR2n+1 := (TR2n+1/D)⊕D,where D = contact distribution.


Mutual actions :

The factorization DiffΠ = GΠ · NΠ allows to represent uniquely anyφ ∈ DiffΠ as a product φ = ϕ · ψ, with ϕ ∈ GΠ and ψ ∈ NΠ.

Factorizing the product of any two elements ϕ ∈ GΠ and ψ ∈ NΠ inthe reverse order, ψ · ϕ = (ψ . ϕ) · (ψ / ϕ), one obtains a leftaction ψ 7→ ψ(ϕ) := ψ . ϕ of NΠ on GΠ, and a right action / ofGΠ on NΠ.

These actions are restrictions of the natural actions of DiffΠ on thecoset spaces DiffΠ /NΠ

∼= GΠ and GΠ\DiffΠ∼= NΠ.


Dynamical definition of the Hopf algebra HΠ

Fix basis {Xi}1≤i≤m for the Lie algebra gΠ of GΠ. Each X ∈ gΠ gives riseto a left-invariant vector field on GΠ, which is then extended to a linearoperator on AΠ := C∞(GΠ) n DiffΠ ,

X (f Uφ−1) = X (f ) Uφ−1 , f ∈ C∞(GΠ), φ ∈ DiffΠ .

One has

Uφ−1 Xi Uφ =m∑

j=1

Γji (φ) Xj , with Γj

i (φ) ∈ C∞(Gcn);

define corresponding multiplication operators on AΠ by

∆ji (f Uφ−1) = (Γ(φ)−1)j

i f U∗φ, where Γ(φ) =(Γj

i (φ))

1≤i ,j≤m.


As an algebra, HΠ is generated by the operators Xk ’s and ∆ji ’s. In

particular, HΠ contains all iterated commutators

∆ji ,k1...kr

:= [Xkr , . . . , [Xk1 ,∆ji ] . . .],

which are multiplication operators by the functions

Γji ,k1...kr

(φ) := Xkr . . .Xk1(Γji (φ)), φ ∈ DiffΠ .

For any a, b ∈ A(Πcn), one has

Xk(ab) = Xk(a) b +∑

j

∆jk(a) Xj(b), ∆j

i (ab) =∑k

∆ki (a) ∆j

k(b),

Every h ∈ H(Πcn) satisfies a Leibniz rule of the form

h(ab) =∑

h(1)(a) h(2)b) , ∀ a, b ∈ AΠ.

The operators ∆••···• satisfy the following Bianchi-type identities:

∆ki ,j − ∆k

j ,i =∑r ,s

ckrs ∆r

i ∆sj −

∑`

c`ij ∆k` ,

where c ijk are the structure constants of the Lie algebra gΠ.


Theorem (Realization as Hopf algebra)

Let HΠ be the abstract Lie algebra generated by the operators{Xk , ∆j

i ,k1...kr} and their commutation relations.

1 The algebra HΠ is isomorphic to the quotient of the universalenveloping algebra U(HΠ) by the ideal BΠ generated by the Bianchiidentities.

2 The Leibniz rule determines uniquely a coproduct, with respect towhich HΠ a Hopf algebra and AΠ an HΠ-module algebra.

Example (Projective pseoudogroup)

G := SL2(R) = G · N, with G = {(

a b0 a−1

)| a > 0, b ∈ R}, and

N = {(

1 0c 1

)| c ∈ R}. Then H(G) ∼= H1/(δ′2).


Bicrossed product definition of HΠ

FΠ:= the algebra of functions on N generated by the jet ‘coordinates’

ηji ,k1...kr

(ψ) := Γji ,k1...kr

(ψ)(e), ψ ∈ N;

FΠ is a Hopf algebra with coproduct

∆f (ψ1, ψ2) := f (ψ1 ◦ ψ2), ∀ψ1, ψ2 ∈ N,

with antipodeSf (ψ) := f (ψ−1), ψ ∈ N,

and with counit evaluation at e, ψ ∈ N 7→ ψ(e).

Clearly, the definition is independent of the choice of basis for gΠ.


FΠ coacts on UΠ = U(gΠ) and makes it a comodule coalgebra,

H : UΠ → UΠ ⊗FΠ, as follows:

with {XI = X i11 · · ·X im

m ; i1, . . . , im ∈ Z+} PBW basis of UΠ,

Uψ−1 XI Uψ =∑J

βJI (ψ) XJ ,

with βJI (ψ) in the algebra of functions on GΠ generated by Γj

i ,K (ψ).Define H : UΠ → UΠ ⊗FΠ by

H(XI ) =∑J

XJ ⊗ βJI (·)(e).

The right action / of GΠ on NΠ induces an action of GΠ on FΠ, hencea left action of UΠ on FΠ, that makes FΠ a left UΠ-module algebra.


Theorem (Realization as bicrossed product )

With the above operations, UΠ and FΠ form a matched pair of Hopfalgebras, and their bicrossed product FΠ IC UΠ is canonically isomorphicto the Hopf algebra Hcop

Π .

ε(u . f ) = ε(u)ε(f ),

∆(u . f ) = u(1)<0> . f (1) ⊗ u(1)<1>(u(2) . f (2)),

H(1) = 1⊗ 1,

H(uv) = u(1)<0>v<0> ⊗ u(1)<1>(u(2) . v<1>),

u(2)<0> ⊗ (u(1) . f )u(2)<1> = u(1)<0> ⊗ u(1)<1>(u(2) . f ).


The bicrossed product structure allows to replace the original Hopfco-cyclic module by a simpler bicomplex, which combines theChevalley-Eilenberg cohomology complex of the Lie algebra g = gΠ

with coefficients in Cδ ⊗F⊗• and the coalgebra cohomology complexof F = FΠ with coefficients in ∧•g:

...

∂g

��

...

∂g

��

...

∂g

��Cδ ⊗ ∧2g

βF //

∂g

��

Cδ ⊗F ⊗ ∧2gβF //

∂g

��

Cδ ⊗F⊗2 ⊗ ∧2gβF //

∂g

��

. . .

Cδ ⊗ gβF //

∂g

��

Cδ ⊗F ⊗ gβF //

∂g

��

Cδ ⊗F⊗2 ⊗ g

∂g

��

βF // . . .

Cδ ⊗ CβF // Cδ ⊗F ⊗ C

βF // Cδ ⊗F⊗2 ⊗ CβF // . . . .


Theorem (Hopf cyclic bicomplex)

1 The above bicomplex computes the periodic Hopf cyclic cohomologyHP∗(HΠ; Cδ).

2 There is a relative version of the above bicomplex that computes therelative periodic Hopf cyclic cohomology HP∗(HΠ,U(h); Cδ), for anyreductive subalgebra h of the linear isotropy Lie algebra g0

Π.

Corollary

Explicit cocycle representatives for basis of HP∗(Hn, gln; Cδ), i.e.universal Hopf cyclic Chern classes.

For each partition λ = (λ1 ≥ . . . ≥ λk) of {1, . . . , p}, where 1 ≤ p ≤ n,let λ ∈ Sp also denotes a permutation whose cycles have lengthsλ1 ≥ . . . ≥ λk . Define

Cp,λ :=∑

(−1)µ1⊗ ηj1µ(1),jλ(1)

∧ · · · ∧ ηjpµ(p),jλ(p)

⊗ Xµ(p+1) ∧ · · · ∧ Xµ(n),

where the summation is over all µ ∈ Sn and all 1 ≤ j1, j2, . . . , jp ≤ n.Henri Moscovici Transverse Index Theory and Hopf Algebras Cortona, June 2008 24 / 25

Theorem (Hopf cyclic Chern classes)

The cochains {Cp,λ ; 1 ≤ p ≤ n, [λ] ∈ [Sp]} are cocycles and their classesform a basis of the group HPε(Hn,U(gln); Cδ), where ε ≡ n mod 2;HP1−ε(Hn,U(gln); Cδ) = 0.

Correspondence with the universal Chern classes:

Pn[c1, . . . cn] = C[c1, . . . cn]/In

where deg(cj) = 2j , and In is the ideal generated by the monomials ofdegree > 2n; to each partition λ as above, one associates the degree 2pmonomial

cp,λ := cλ1 · · · cλk, λ1 + . . .+ λk = p;

the corresponding classes {cp,λ ; 1 ≤ p ≤ n, λ ∈ [Sp]} form a basis of thevector space Pn[c1, . . . cn].


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