Hopf Cyclic Cohomology and
Characteristic Classes
Henri Moscovici
Ohio State University
Noncommutative Geometry Festival
TAMU, April 30 - May 3, 2014
Henri Moscovici
Hopf Cyclic Classes
Origins and motivation
The version of cyclic cohomology adapted to Hopf algebras emerged
from the joint work with of A. Connes on the local index formula for
the hypoelliptic signature operator on “spaces of leaves” of foliations.
For foliations of codimension n, we have found a Hopf algebra Hn
which plays the role of a “quantum” structure group for their “spaces
of leaves”. While the characteristic classes of foliations are described
in terms of Gelfand-Fuks Lie algebra cohomology, the appropriate tool
for above problem turned out to be Hopf cyclic cohomology.
The two cohomologies were shown to be canonically isomorphic by an
explicit, but quite intricate, quasi-isomorphism.
The transplantation of the characteristic classes in the Hopf cyclic
cohomological framework broadened the scope of their applicability.
Thus, the issue of finding more explicit constructions of the Hopf
cyclic characteristic classes becomes relevant.
Henri Moscovici
Hopf Cyclic Classes
Chern-Weil construction of characteristic classes
∇= torsion-free connection on M n , with connection form on
ω = (ωji ) ∈ Ω1 (FM) ⊗ gln and curvature Ω = dω + ω ∧ ω,
Ω = (Ωij ) ∈ Ω2 (FM) ⊗ gln ;
∀ P ∈ I (gln ) = S(gl∗n )GLn , the form P(Ω) is closed and basic, i.e.
P(Ω) ∈ Ω∗ (M), hence [P(Ω)] ∈ H ∗ (M, R).
In particular,
n
X
t det Id −
A =
t k ck (A),
2πi
A ∈ gln (C)
k=1
give the classical Chern forms ck (Ω) ∈ Ω2k (M), and the Pontryagin
classes [pk (Ω)] = [c2k (Ω)] ∈ H 4k (M, R).
Henri Moscovici
Hopf Cyclic Classes
Local Index Formula in Noncommutative Geometry
Theorem (A. Connes & HM, 1995)
Assume
Z (A, H, D) = spectral triple, such that ∃ residue
−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,
Z
X
ϕn (a0 , . . . , an ) =
cn,k −a0 [D, a1 ](k1 ) . . . [D, an ](kn ) |D|−n−2|k|
k
2
|k| = k 1 + . . . + k n ,
(−1)|k| Γ |k| + n2
=
.
k1 ! . . . kn !(k1 + 1) . . . (k1 + · · · + kn + n)
∇(T ) = [D , a],
cn,k
2
T (k) = ∇k (T ),
[(ϕn )n=1,3,... ] = ch∗ (H, F ) ∈ HC ∗ (A).
Henri Moscovici
Hopf Cyclic Classes
Relation with Atiyah-Singer Local Index Formula
1
2
3
4
5
The zeta functions associated to the Dirac spectral triple
(C ∞ (M m ), L2 (S), D)
/ are meromorphic with simple poles.
Z
Z
1
σ−n (P).
∀ P ∈ ΨDO(M n );
−P =
(2π)n S ∗ M
(Guillemin-Wodzicki residue)
Z
−f 0 [/
D, f 1 ](k1 ) . . . [/
D, f n ](kn ) |/
D|−(n+2|k|) = 0 , if |k| > 0 ;
Z
−f 0 [/
D, f 1 ] . . . [/
D, f n ]|/
D|−n =
12
Z
∇2 /4πi
= cn
det
∧ f 0 df 1 ∧ . . . ∧ df n ;
sinh ∇2 /4πi
M
under the isomorphism (Connes, HKR)
HP ∗ (C ∞ (M m )) ∼
= H∗dR (M, C),
ch∗ (H,D)
/ ≡ [(ϕn ) ] ∼
D).
= [Â(R)] ≡ Ch∗ (/
Henri Moscovici
Hopf Cyclic Classes
Local index formula for codimension 1
M = S 1 , Γ ⊂ G = Diff(S 1 )δ ,
H = L2 (FS 1 × S 1 , y −1 dx dy dα) ⊗ C2
1
1
Q = 2y ∂y ∂α γ1 + y ∂x γ2 + (y ∂y )2 − ∂α2 −
γ3 ,
i
4
where γ1 , γ2 , γ3 ∈ M2 (C) are the Pauli matrices.
Z
1
1
0 1
• √ ϕ1 (a , a ) = Γ
−a0 [Q, a1 ](Q 2 )−1/2
2
2i
Z
Z
1
3
1
5
− Γ
−a0 ∇[Q, a1 ](Q 2 )−3/2 +
Γ
−a0 ∇2 [Q, a1 ](Q 2 )−5/2
2
2
2·3
2
Z
1
7
−
Γ
−a0 ∇3 [Q, a1 ](Q 2 )−7/2 ≡ 0
2·3·4
2
Henri Moscovici
Hopf Cyclic Classes
Local index cocycle
Z
1
1
3
0 1 2 3
−a0 [Q, a1 ][Q, a2 ][Q, a3 ](Q 2 )−3/2
• √ ϕ3 (a , a , a , a ) = Γ
3i
2
2i
Z
1
5
−
−a0 ∇[Q, a1 ][Q, a2 ][Q, a3 ](Q 2 )−5/2
Γ
2·3·4
2
Z
1
5
−a0 [Q, a1 ]∇[Q, a2 ][Q, a3 ](Q 2 )−5/2
−
Γ
3·4
2
Z
1
5
−
Γ
−a0 [Q, a1 ][Q, a2 ]∇[Q, a3 ](Q 2 )−5/2
2·4
2
= transverse fundamental cocycle + boundary.
0 ,
While the computation is purely symbolical, it requires the symbol σ−4
3
hence about 10 terms!
Henri Moscovici
Hopf Cyclic Classes
Deciphering the transverse local index formula
1
The Chern character cocycle of the transverse signature operator D is
a sum of cochains of the form
XZ
− a0 [Q, a1 ](k1 ) . . . [Q, aq ](kq ) |Q|−q−|2k| , ai = f i Uϕ∗ i ∈ A.
2
By the residue formula, these can be expressed as
X
τ (a0 h1 (a1 ) · · · hq (aq )),
Z
∗
where τ (f Uϕ ) =
f volP , if ϕ = Id and
3
0 otherwise,
hi are transverse differential operators, e.g. X (f Uϕ∗ ) = X (f ) Uϕ∗ .
The operators h generate a Hopf algebra H, which acts on A and
gives rise to a characteristic map
χqΓ : C q (H) ≡ H ⊗ . . . ⊗ H −→ C q (A) .
4
The Chern character of D is in the range of the induced map in cyclic
cohomology χ∗Γ : HC ∗ (H) −→ HC ∗ (AΓ )(1) .
Henri Moscovici
Hopf Cyclic Classes
Hopf algebra H1
Action on A = C ∞ (F + S 1 ) o G,
Y (fUϕ∗ ) = y
Y (f ◦ ϕ̃) = Y (f ) ◦ ϕ̃
∂f ∗
U ,
∂y ϕ
=⇒
ϕ̃(x, y ) = (ϕ(x), ϕ0 (x) y ) ,
X (fUϕ∗ ) = y
∂f ∗
U
∂x ϕ
Y (ab) = Y (a) b + a Y (b).
00
ϕ (x)
(Y (f ) ◦ ϕ̃) (x, y )
ϕ0 (x)
=⇒ X (ab) = X (a) b + a X (b) + δ1 (a) Y (b).
dϕ
dn
δn := [X , δn−1 ] =⇒ δn (fUϕ∗ ) = y n dx
log
fUϕ∗ .
n
dx
X (f ◦ ϕ̃) = (X (f ) ◦ ϕ̃) + y
As algebra= generated by {X , Y , δ1 , δ2 , . . .} subject to relations:
[Y , X ] = X ,
[Y , δk ] = kδk ,
Henri Moscovici
[X , δk ] = δk+1 ,
Hopf Cyclic Classes
[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 + δ12 ⊗ δ1 + 1 ⊗ δ3 , etc.
Counit: (X ) = (Y ) = (δk ) = 0,
(1) = 1.
Antipode: S(1) = 1 , S(X ) = −X + δ1 Y , S(Y ) = −Y ,
S(δ1 ) = −δ1 , S(δ2 ) = δ12 − δ1 , . . .
Character δ ∈ H1∗ :
δ(Y ) = 1,
δ(X ) = 0,
Twisted antipode Sδ (h) = δ(h(1) ) S(h(2) )
is involutive:
Sδ2 = Id .
Henri Moscovici
δ(δn ) = 0.
Hopf Cyclic Classes
Hopf cyclic cohomology
\
Cylic structure: H(δ,1)
= {C n (H; Cδ ) = H⊗n }n≥0
∂0 (h1 ⊗ . . . ⊗ hn−1 ) = 1 ⊗ h1 ⊗ . . . ⊗ hn−1 ,
∂j (h1 ⊗ . . . ⊗ hn−1 ) = h1 ⊗ . . . ⊗ ∆hj ⊗ . . . ⊗ hn−1
∂n (h1 ⊗ . . . ⊗ hn−1 ) = h1 ⊗ . . . ⊗ hn−1 ⊗ 1
σi (h1 ⊗ . . . ⊗ hn+1 ) = h1 ⊗ . . . ⊗ ε(hi+1 ) ⊗ . . . ⊗ hn+1
Bicomplex :
τn (h1 ⊗ . . . ⊗ hn ) = Sδ (h1 ) · h2 ⊗ . . . ⊗ hn ⊗ 1
!
n+1
n
X
X
b=
(−1)i ∂i , B =
(−1)ni τni σn−1 τn .
i=0
i=0
Example (HP ∗ (H1 ; Cδ ))
TF = X ⊗ Y − Y ⊗ X − δ1 Y ⊗ Y
GV = δ1
Henri Moscovici
(fundamental class)
(Godbillon-Vey class)
Hopf Cyclic Classes
Relative Hopf cyclic cohomology
K = Hopf subalgebra of H, C := H ⊗K C, with K acting on H by
right multiplication and on C by the counit.
As left H-module C ' H/HK+ , where K+ = Ker ε|K, via
h + HK+ 7→ ḣ = h ⊗K 1 ∈ H ⊗K C.
Cylic structure:
{C n (H, K; Cσδ ) = C ⊗n }n≥0
∂0 (c 1 ⊗ . . . ⊗ c n−1 ) = 1̇ ⊗ c 1 ⊗ . . . ⊗ . . . ⊗ c n−1 ,
i
i
∂i (c 1 ⊗ . . . ⊗ c n−1 ) = c 1 ⊗ . . . ⊗ c(1)
⊗ c(2)
⊗ . . . ⊗ c n−1 ,
∂n (c 1 ⊗ . . . ⊗ c n−1 ) = c 1 ⊗ . . . ⊗ c n−1 ⊗ 1̇ ;
σi (c 1 ⊗ . . . ⊗ c n+1 ) = c 1 ⊗ . . . ⊗ ε(c i+1 ) ⊗ . . . ⊗ c n+1 ,
τn (ḣ1 ⊗ c 2 ⊗ . . . ⊗ c n ) = Sδ (h1 ) · (c 2 ⊗ . . . ⊗ c n ⊗ 1̇).
Henri Moscovici
Hopf Cyclic Classes
Transverse Index Theorem
Theorem (A. Connes & HM, 1998)
There are canonical constructions for the following entities:
1
2
3
a Hopf algebra Hn associated to Diff(Rn ), with modular character δ,
and modular pair (δ, 1);
an isomorphism κ∗ between the Gelfand-Fuks cohomology
∗ (a , O ) and HP ∗ (H , O ; C );
HGF
n
n
n
n
δ
an action of Hn on AG (F Rn ), and a characteristic map
χ(h1 ⊗ . . . ⊗ hn )(a0 , . . . , an ) = τ (a0 h1 (a1 ) . . . hn (an ))
∗
χ∗ : HP ∗ (Hn , On ; Cδ ) → HP(1)
(AG (PRn )),
4
ch∗ (D)(1) = χ∗ (L),
PRn := F Rn /On
∗ (a , O ).
L ∈ HP ∗ (Hn , On ; Cδ ) ∼
= HGF
n
n
Henri Moscovici
Hopf Cyclic Classes
Diff-invariant de Rham cohomology
de Rham complex of invariant forms on ∞-jet bundle
{Ω∗ (P ∞ M)G , d} for M n = smooth manifold, G = Diff(M)δ
F k M = k − jets at 0
of local diffeos
ρ : Rn → M
F ∞ M := lim F k M
←
−
P k M = F k M/On
M ← P 1 M ← P 2 M ← · · · P ∞ M := lim P k M
←
−
G − action :
φ ∈ G, ρ ∈ F k M =⇒ φ · j0∞ (ρ) := j0∞ (φ ◦ ρ)
M ← F 1M ← F 2M ← · · ·
Gelfand-Fuks Lie algebra cohomology complex
of formal vector fields
d
|t=0 ρt , t 7→ ρt : Rn → Rn }
{C ∗ (an ), d}
an = {v = j0∞ dt
d
∞
ṽ |j0∞ (φ) = j0
|t=0 (φ ◦ ρt ) ; ω̃(ṽ1 , . . . , ṽm ) = ω(v1 , . . . , vm )
dt
DGA-isomorphism
ω ∈ C • (an , On ) 7→ ω̃ ∈ Ω• (P ∞ M)G
Henri Moscovici
Hopf Cyclic Classes
Diff-equivariant de Rham cohomology
Simplicial manifold
4̄G M = {4̄G M[p] = Gp+1 × M}p≥0 ,
∂¯i (ρ0 , . . . , ρp , x) = (ρ0 , . . . , ρ̌i , . . . , ρp ),
0 ≤ i ≤ p,
σ̄i (ρ0 , . . . , ρp , x) = (ρ0 , . . . , ρi , ρi , . . . , ρp , x) , 0 ≤ i ≤ p.
`
Geometric realization |4̄G M| = ∆p × 4̄G M[p]/ ∼
Dupont complex of (covariant) compatible forms {Ω∗ |4̄G M|, d}
ω = {ωp }p≥0 ,
ωp ∈ Ω• (∆p × 4G M[p])
(µ• × Id)∗ ωq = (Id ×µ• )∗ ωp ∈ Ω• (∆p × 4G M[q])
ω(ρ0 ρ, . . . , ρp ρ) = ρ∗ ω(ρ0 , . . . , ρp ),
Bott complex
{C̄ • (G, Ω• (M)) , δ̄, d}
c̄(ρ0 , . . . , ρp ) ∈ Ωq (M)
c̄(ρ0 ρ, . . . , ρp ρ) = ρ∗ c̄(ρ0 , . . . , ρp ),
δ̄c̄(ρ0 , . . . , ρp ) =
p
X
∀ ρ, ρi ∈ G
(−1)i c̄(ρ0 , . . . , ρ̌i , . . . , ρp ).
i=0
Henri Moscovici
∀ ρ, ρi ∈ G
Hopf Cyclic Classes
Differentiable cohomology (à la Haefliger)
Differentiable cochain ω ∈ C̄dp (G, Ωq (M)) if locally,
X ω(ρ0 , . . . , ρp , x) =
fI x, jxk (ρ0 ), . . . , jxk (ρp ) dx I .
Differentiable compatible form ω = {ωp }p≥0 ∈ Ω•d (|4̄G M|) if
X
ωp (t; ρ0 , . . . , ρp , x) =
fI ,J t; x, jxk (ρ0 ), . . . , jxk (ρp ) dt I ∧ dx J .
Theorem (Differentiable analogue of Dupont’s Theorem)
The chain map
Z
Φ
: Ω•d (|4̄G M|) → C̄d• (G, Ω∗ (M)) induces an
∼ H • (M, R).
Hd• (|4G M|, R) =
d,G
ƥ
isomorphism
Henri Moscovici
Hopf Cyclic Classes
Explicit van Est-Haefliger quasi-isomorphism
∇= torsion free connection =⇒ cross-section to π1 : F ∞ M → FM
σ∇ (u) = j0∞ (exp∇
x ◦u);
σ∇φ = φ−1 ◦ σ∇ ◦ φ,
σp (t; ρ0 , . . . , ρp , u) = σPp0
ti ∇ρi (u);
∞
φ ∈ G,
σ̂ = {σp }p≥0 : |4̄G FM| → F M.
Theorem
The chain map C∇ (ω) = σ̂ ∗ (ω̃) ∈ Ω•d (|4̄G FM|) induces quasi-iso of
DG-algebras C∇ : C • (an , On ) → Ω•d (|4̄G (PM, On )|).
Corollary
The composition
Z
D∇ = Φ
is quasi-isomorphism.
ƥ
◦ C∇ : C • (an , On ) → C̄dtot • (G, Ω∗ (PM))
Henri Moscovici
Hopf Cyclic Classes
Hopf cyclic analogue of van Est isomorphism
This involves Connes’ map Φ : C̄ • (G, Ωp (F Rn )) → CC • (Cc∞ (F Rn ) n G).
If λ ∈ Im(D∇ ) where now M = Rn , and ∇= flat connection, then Φ(λ) is
of the form
X
Φ(λ)(a0 , . . . , a` ) =
τ (a0 hα1 (a1 ) . . . hαq (aq )),
hαi ∈ Hn ,
α
q
⊗q
1
with the tensor
uniquely determined by λ.
α hα ⊗ . . . ⊗ hα ∈ Hn
One obtains a chain map Υ : Im(D∇ ) → CC tot• (Hn , Cδ ),
X
Υ(λ) =
hα1 ⊗ . . . ⊗ hαq ∈ Hn⊗ q .
P
α
Theorem (AC & HM 1998, 2001)
The composition Υ ◦ D∇ : C • (an , On ) → CC tot∗ (Hn , On ; Cδ )
quasi-isomorphism.
Henri Moscovici
Hopf Cyclic Classes
is
Characteristic cocycles by simplicial Chern-Weil
P
The universal connection ϑ = (ϑij ), where ϑij ( nk=1 ξ k ∂k ) = ∂j ξ i , and
curvature forms R = (Rji ), where Rji = dϑij + ϑik ∧ ϑkj in C • (an )
generate a DG-subalgebra CW • (an ).
By Gelfand-Fuks Thm. CW • (an ) ,→ C • (an ) is quasi-isomorphism.
CW • (an ) ∼
= Ŵ (gln ) = W (gln )/I2n , where W (gln ) = ∧• gl∗n ⊗ S(gl∗n )
and I2n = ideal generated by elements of S(gl∗n ) of deg > 2n.
Lemma
i
∗ (ϑ
ei
ei ) = ω i and σ ∗ (R
For any torsion-free connection, σ∇
∇ j ) = Ωj .
j
j
Simplicial connectionPand curvature:
ω̂p (t; ρ0 , . . . , ρp ) := pi=0 ti ρ∗i (ω) ∈ Ω1d (|4̄G FM|)
Ω̂ := d ω̂ + ω̂ ∧ ω̂ ∈ Ω2d (|4̄G FM|),
P
P
Ω̂pP
(t; ρ0 , . . . , ρp ) = pi=0 dti ∧ρ∗i (ω)+ pi=0 ti ρ∗i (Ω)−ρ∗i (ω)∧ρ∗i (ω)
+ pi,j=0 ti tj ρ∗i (ω) ∧ ρ∗j (ω).
Henri Moscovici
Hopf Cyclic Classes
Vey basis in differentiable Dupont algebra
The forms ω̂ji and Ω̂ij generate a DG-subalgebra CWd• (|4̄G FM|), and
C∇ gives isomorphism between CW • (an ) ≡ Ŵ (gln ) and
CWd• (|4̄G FM|).
R1 ck (Ω̂) = d(Tck (ω̂)), with Tck (ω̂) = k 0 ck ω̂, Ω̂t , . . . , Ω̂t dt,
Ω̂t = t Ω̂ + (t 2 − t)ω̂ ∧ ω̂.
By restriction to On -basic elements, C∇ induces an isomorphism of
CW • (an , On )) ≡ Ŵ (gln , On ) onto CWd• (|4̄G PM|).
c2k (Ω̂) remain but c2k−1 (Ω̂) = d(Tc
2k−1 (ω̂)), with
R1
Tc2k−1 (ω̂) = (2k − 1) 0 c2k−1 s(ω̂), Ω̂t , . . . , Ω̂t dt ,
Ω̂t = ts(Ω̂) + o(Ω̂) + (t 2 − 1)s(ω̂) ∧ s(ω̂).
{TcI (ω̂) ∧ cJ (Ω̂)}(I ,J)∈Vn , resp. {TcI (ω̂) ∧ cJ (Ω̂)}(I ,J)∈VOn , represents
a basis in cohomology.
Henri Moscovici
Hopf Cyclic Classes
Vey basis in differentiable Bott complex
Corollary
The cocycles obtained by their integration along fibers,
Z
CI ,J (∇) := Φ TcI (ω̂) ∧ cJ (Ω̂) , (I , J) ∈ VOn ,
ƥ
• (PM, R).
form a complete set of representatives for a basis of Hd,G
Z
For example, C∅,{2k} (∇) = C2k (Ω̂) = Φ c2k (Ω̂) is the cocycle
C2k (Ω̂) =
ƥ
(p)
{C2k (Ω̂)}p≥0
with components
Z
(p)
C2k (Ω̂)(φ0 , . . . , φp ) = (−1)p Φ c2k Ω̂(t; φ0 , . . . , φp ) =
∆pZ
X
X
iq
1
= (−1)p
(−1)µ
Ω̂iµ(i
∧ · · · ∧ Ω̂µ(i
(t; φ0 , . . . , φp ).
1)
2k )
1≤i1 <...<i2k ≤n µ∈S2k
Henri Moscovici
∆p
Hopf Cyclic Classes
Generators of the Hopf algebra Hn
Hn acts on A = Cc∞ (F Rn ) n G where G = Diff(Rn )δ by
Xk = ykµ
∂
,
∂ xµ
Yij = yiµ
∂
;
∂ yjµ
Yij (f Uφ∗ ) = Yij (f ) Uφ∗ ,
Uφ∗ Yij Uφ = Yij ;
Xk (f Uφ∗ ) = Xk (f ) Uφ∗ ,
i
Uφ∗ Xk Uφ = Xk − γjk
(φ) Yij ,
i
i
δjk
(f Uφ∗ ) = γjk
(φ) f Uφ∗ ,
i
i
δjk
`1 ...`r := [X`r , . . . [X`1 , δjk ] . . .] ;
∗
i
∗
i
δjk
`1 ...`r (f Uφ ) := γjk `1 ...`r (φ) f Uφ ,
where
i
i
γjk
`1 ...`r (φ) := X`r · · · X`1 γjk (φ)
i
Basic algebra generators : {Xk , Yij , δjk
}
Henri Moscovici
Hopf Cyclic Classes
Transfer to Hopf cyclic cohomology
Let M = Rn , G = Diff(Rn ) and ∇ = trivial connection. Identify
F Rn ∼
= Rn n GLn (R). One has
i
ωji := (y−1 )iµ dyjµ = y−1 dy j ,
i
φ∗ (ωji ) = ωji + γjk
(φ) θk ,
i, j = 1, . . . , n
φ∈G
γji k (φ)(x, y) = y−1 · φ0 (x)−1 · ∂µ φ0 (x) · y
ω̂(t; φ0 , . . . , φp )ij =
Ω̂(t; φ0 , . . . , φp ) =
p
X
r =0
p
X
tr φ∗r (ωji ) =
+
j
ykµ
i
tr γjk
(φr ) θk
r =0
dtr ∧ φ∗r (ω) −
r =0
p
X
p
X
i
p
X
tr φ∗r (ω) ∧ φ∗r (ω)
i=0
tr ts φ∗r (ω) ∧ φ∗s (ω).
r ,s=0
Henri Moscovici
Hopf Cyclic Classes
Vey basis in Hopf cyclic complex
Theorem
(1) The cocycles Υ(CI ,J (∇)), with (I , J) ∈ Vn , form a complete set of
representatives for the periodic Hopf cyclic cohomology HP • (Hn ; Cδ ).
(2) The cocycles Υ(CI ,J (∇)), with (I , J) ∈ VOn , form a complete set of
representatives for HP • (Hn , On ; Cδ ).
(3) Every cohomology class in HP • (Hn ; Cδ ) and HP • (Hn , On ; Cδ ), can
be represented by cocycles manufactured out of the algebra generators
i } of H .
{Xk , Yji , δjk
n
Proof.
i from the previous slide, the map Φ, when applied to
In addition to the δjk
monomials a = f Uφ∗ ∈ A, brings in the operators Xk and Yji , via
P
P
df = nk=1 Xk (f )θk + ni,j=1 Yji (f ) ωij .
Henri Moscovici
Hopf Cyclic Classes
Example: Chern cocycle
(p)
The top component of {Cq (Ω̂)}p≥0 , q = 2k, in the simplified cyclic
model
X X
1
iq
(q)
i1
κq (Ω̂) =
(−1)µ θj0 ∧ · · · ∧ θjq ⊗ ηµ(i
∧ · · · ∧ ηµ(i
.
),j 1
q ),jq
1
1≤is ,jt ≤n µ∈Sq
0
(p)
The lower components κq (Ω̂) are given by similar expressions, with
coefficients of the form
Z
1
k
;
t1k1 · · · tp p dt1 ∧ · · · ∧ dtp =
(k1 + 1) · · · (k1 + . . . + kp + p)
∆p
note the resemblance with the coefficients appearing in the local index
formula.
Henri Moscovici
Hopf Cyclic Classes