. . . . . .

Lie-Hopf algebras and their Hopf cycliccohomology

Bahram Rangipour

Texas A&M University, College Station

May 1, 2014

. . . . . .

Local Index Formula in NCG

Theorem (Connes-Moscovici, 1995)

For an (odd) spectral triple (A,H,D) such that the residues∫−

ℓT := Ress=0Tr(s

ℓT |D|−2s), T ∈ A, [D, c], |D|−z ; z ∈ Cmake sense, one has:[(φn)n=1,3,...] is a cocycle in the (b,B)-bicomplex of A,

φn(a0, . . . , an) =

∑k,ℓ

cn,k,ℓ

∫−

ℓa0[D, a1](k1) . . . [D, an](kn)|D|−n−2|k|

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

cn,k,ℓ =(−1)|k| Γ(ℓ)

(|k|+ n

2

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

.

. . . . . .

Theorem (Connes-Moscovici 1998)

I For any n ≥ 1 and any oriented flat manifold Mn, there is acanonical Hopf algebra Hn acting on the algebraAn := C∞

c (FM) > Diff(M).

I There is a canonical cyclic cohomology theory, associated toHn, canonically isomorphic to the Gelfand Fuks cohomologyof the Lie algebra of formal vector fields on Rn.

I There is a characteristic map from the mentioned cycliccohomology of Hn to the cyclic cohomology of the algebra An

such that the index cocycle is trapped in its image.

. . . . . .

Modular pair in involution

Let H be a Hopf algebra.

I An algebra map δ : H → C is called a character

I An element σ ∈ H is called a group-like if ∆(σ) = σ ⊗ σ

I The pair (δ, σ) is called modular pair in involution (MPI) ifδ(σ) = 1 and

S2δ (h) = σhσ−1

Here Sδ(h) =∑

δ(h(1))S(h(2))

. . . . . .

Hopf cyclic cohomology, HC (H , σCδ)

We define operators b and B, H⊗qb //

H⊗q+1

Boo

b :=

q+1∑i=0

(−1)idi , B :=

(q∑

i=0

(−1)qi t i

)sq−1(1− t).

where

d0(h1 ⊗ . . .⊗ hq) = 1⊗ h1 ⊗ . . .⊗ hq,

di (h1 ⊗ . . .⊗ hq) =

∑h1 ⊗ . . .⊗∆(hi )⊗ . . .⊗ hq,

dq+1(h1 ⊗ . . .⊗ hq) = h1 ⊗ . . .⊗ hq ⊗ σ,

sj(h1 ⊗ . . .⊗ hq) = h1 ⊗ . . .⊗ ε(hj+1)⊗ . . .⊗ hq,

t(h1 ⊗ . . .⊗ hq) =∑

Sδ(h1(2)) · (h2 ⊗ . . .⊗ hq ⊗ σ),

where H acts on H⊗q diagonally.

. . . . . .

Action of Hopf algebras on algebras

The same way that groups or Lie algebras act on algebras byautomorphisms or derivations respectively, we want Hopf algebrasact on algebras. In this case we say an algebra A is H-modulealgebra if

h (ab) =∑

(h(1) a)(h(2) b), h 1A = ε(h)1A

. . . . . .

Characteristic map

Suppose a Hopf algebra H acts on an algebra A. Provided for aMPI (δ, σ) the algebra A obtains a δ-invariant σ-trace, τ : A → C,i.e

τ(h(a)

)= δ(h)τ(a), τ(ba) = τ

(aσ(b)

)there is a characteristic map

χτ : HC •(H, σCδ) → HC •(A)

χτ (h1 ⊗ . . .⊗ hn)(a0 ⊗ . . .⊗ an)) = τ(a0h

1(a1) · · · hn(an))

. . . . . .

Crossed product algebra

Let M be a manifold and Γ ≤ DiffM.One defines the left action of Γ on C∞(M) by

ϕ f = f ϕ−1

Define the crossed product algebra

C∞(M) > Γ

A typical element is a finite sum of∑

i fi U∗

ϕiwhere fU∗

ϕ stands for

f > ϕ−1.Multiplication reads as

f 1 U∗ϕ1f 2 U∗

ϕ2= f 1(ϕ−1

2 f 2)U∗ϕ2ϕ1

. . . . . .

The Hopf algebra Hn

I Let M = Rn. Vector fields on FM are generated by

Y ij := ykj

∂

∂y ik, Xk = y ik

∂

∂x i

I We lift them on An := C∞c (FM) > Γ by

Xk(fU∗φ) = Xk(f )U

∗φ, Y i

j (fU∗φ) = Y i

j (f )U∗φ

I We have Y ij (ab) = Y i

j (a)b + aY ij (b),

I However, for Xk we have

Xk(ab) = Xk(a)b + aXk(b) + δij ,k(a)Yji (b)

,

I Here δijk(fU∗φ) = γijk(φ)fU

∗φ, and

γijk(φ)(x , y) =[y−1 · φ′(x)−1 · ∂µφ′(x) · y

]ijyµk

. . . . . .

continued ...

One defines the higher order operators δIj ,k|l1,...,lm by

δIj ,k|l1,...,lm = [Xlm , δIj ,k|l1,...,lm−1

]

They satisfy the Bianchi identities δijk|l − δijl |k = δiµlδµjk − δiµkδ

µjl

DefinitionAs an algebra Hn is the subalgebra of L(An) generated by Xk , Y

ij

and all δIj ,k|l1,...,lm . The comultiplication of Hn is obtained by the

Leibniz rule h(ab) = h(1)(a) h(2)(b), h ∈ Hn, a, b ∈ An.So we see that

∆(Xl) = Xl ⊗ 1 + 1⊗ Xl + δij ,l ⊗ Y ji

∆(Y ij ) = Y i

j ⊗ 1 + 1⊗ Y ij

∆(δij ,k) = δij ,k ⊗ 1 + 1⊗ δij ,k

. . . . . .

Actions of Hn on AΓ(FM)

I Hn acts on AΓ(FM)

I AΓ(FM) possesses the trace

τ(fU∗φ) =

∫FM

fϖFM if φ = id

0 otherwise

I The character on Hn defined by

δ(Y ij ) = δij , δ(Xl) = 0, δ(δij0j1|j2,...,jm) = 0

. . . . . .

Structure of Hn

I Xk ,Yij form a representation of g := gℓaffine

n

I δij ,k|ℓ1,...,ℓm generates F := U(a+n )∗

I(U(g),F

)forms a matched pair of Hopf algebras

Hn =(U(g) I F

)cop

. . . . . .

Past progress ([Moscovici-R])

C •(H,Cδ)χτ // C •(AΓ)

C •top(a)

ELH // C •,•c−w(g

∗,F)Θ //

I

OO

C •Bott(Ω)

Φ

OO

. . . . . .

C •,•c−w(g

∗,F)...

......

∧2g∗

∂c−w

OO

bc−w // (∧2g∗ ⊗ ∧2F)F

∂c−w

OO

bc−w // (∧2g∗ ⊗ ∧3F)F

∂c−w

OO

bc−w // . . .

g∗

∂c−w

OO

bc−w // (g∗ ⊗ ∧2F)F

∂c−w

OO

bc−w // (g∗ ⊗ ∧3F)F

∂c−w

OO

bc−w // . . .

C

∂c−w

OO

bc−w // (C⊗ ∧2F)F

∂c−w

OO

bc−w // (C⊗ ∧3F)F

∂c−w

OO

bc−w // . . . ,

bc−w(α⊗ f 0 ∧ · · · ∧ f q) = α⊗ 1 ∧ f 0 ∧ · · · ∧ f q,

∂c−w(α⊗ f 0 ∧ · · · ∧ f q) =

∂α⊗ f 0 ∧ · · · ∧ f q −∑i

θi ∧ α⊗ Xi (f0 ∧ · · · ∧ f q).

. . . . . .

We now define a multiplication on the bicompplex

Cp,qcoinv(g

∗,F) := (∧pg∗ ⊗F⊗q+1)F .

Cp,qcoinv ⊗ C r ,s

coinv → Cp+r ,q+scoinv ,

(ω1 ⊗ f 0 ⊗ . . .⊗ f q) ∗ (ω2 ⊗ g0 ⊗ . . .⊗ g s)

= ω1 ∧ ω2 ⊗ f 0 ⊗ . . .⊗ f q−1 ⊗ f qg0 ⊗ g1 ⊗ . . .⊗ g s

Let us now define the graded multiplication

(ω1 ⊗ f ) · (ω2 ⊗ g) = (−1)qr (ω1 ⊗ f ) ∗ (ω2 ⊗ g)

and eventually we use the canonical projectionπ : F⊗(q+1) → ∧q+1F to define the multiplication onC •,•c−w(g

∗,F), by which it becomes a commutative DG algebra.

. . . . . .

Hopf version of universal connection and curvature

ωij := θij ⊗ 1 + θk ⊗ ηij ,k ∈ C 1,0

c−w(g∗,F)

Ωij := θk ⊗ 1 ∧ ηij ,k ∈ C 1,1

c−w(g∗,F)

. . . . . .

∂T (Ωij) = −θl ∧ θkl ⊗ 1 ∧ ηij ,k − θl ∧ θk ⊗ 1 ∧ ηij ,k|l

− θrj ∧ θk ⊗ 1 ∧ ηir ,k + θrk ∧ θk ⊗ 1 ∧ ηij ,r + θis ∧ θk ⊗ 1 ∧ ηsj ,k .

ωik · Ωk

j

= θik ∧ θp ⊗ 1 ∧ ηkj ,p + θl ∧ θp ⊗ ηik,l ⊗ ηkj ,p − θl ∧ θp ⊗ ηik,lηkj ,p ⊗ 1.

Ωik · ωk

j

= −θl ∧ θkj ⊗ 1 ∧ ηik,l − θl ∧ θp ⊗ 1⊗ ηik,lηkj ,p + θl ∧ θp ⊗ ηik,l ⊗ ηkj ,p.

In other words

∂T (Ωij) = Ωi

k · ωkj − ωi

k · Ωkj

. . . . . .

Similarly we have,

∂T (ωij ) = −θkj ∧ θik ⊗ 1− θℓ ∧ θkℓ ⊗ ηijk − θℓ ∧ θk ⊗ ηij ,k|ℓ

− θpj ∧ θk ⊗ ηip,k + θiq ∧ θk ⊗ ηqj ,k − θpk ∧ θk ⊗ ηij ,p − θk ⊗ 1 ∧ ηij ,k

On the other hand we have

ωik · ωk

j =

θik ∧ θkj ⊗ 1 + θℓ ∧ θkj ⊗ ηik,ℓ + θik ∧ θp ⊗ ηkj ,p + θℓ ∧ θp ⊗ ηik,lηkj ,p

We see that

∂T (ωij ) = −Ωi

j + ωik · ωk

j

. . . . . .

Weil Algebra

The truncated Weil algebra W := ⊕p,q≥0Wp,2n is recalled as

follows.W p,2q = Ap(gln)⊗ Sq

2n(gln)

It is the commutative DG algebra generated by the connectionelements T i

j of degree 1 and the curvature elements R ij of degree 2.

dT ij = −1⊗ R i

j + T ik ∧ T k

j ⊗ 1, dR ij = T k

j ⊗ R ik − T i

k ⊗ Rkj

. . . . . .

Hopf-Weil basis

By the universal property of W (gℓn) we define the following DGalgebra map

L : W (r ,2k)(gℓn) → C r+k,kc−w (g∗,F) (1)

L(T ij ) = ωi

j , L(R ij ) = Ωi

j (2)

TheoremThe map L : W (r ,2k)(gℓn) → C r+k,k

c−w (g∗,F) defined above is aquasi-isomorphism.

. . . . . .

Lie-Hopf Algebras

We first introduce the setting.

I F is a commutative Hopf algebra.

I g is a finite dimensional Lie algebra.

I g acts on F by derivations, : g⊗F → F .

I F coacts on g, H : g → g⊗F .

For a fixed basis X1, · · · ,XN of g, we write the coaction as

H : Xi 7→ Xj ⊗ f ji .

. . . . . .

Lie-Hopf Algebras

For the element f ji ,k := Xk f ji , we say the coaction H : g → g⊗Fsatisfies the structure identity of g if

f kj ,i − f ki ,j =∑s,r

C ks,r f

ri f

sj +

∑l

C li ,j f

kl

Finally we introduce an action of g on F⊗2 as

X • (f ⊗ g) := X<0> f ⊗ X<1>g + f ⊗ X g .

. . . . . .

Lie-Hopf Algebras

DefinitionWe say F is a g−Hopf algebra if

(A) Coaction H : g → g⊗F satisfies the structure identity of g,

(B) Coalgebra structure of F is g−equivariant, i.e.

∆(X f ) = X •∆(f ), and ε(X f ) = 0.

TheoremF is a g−Hopf algebra if and only if (F ,U(g)) is a matched pair ofHopf algebras meach means F I U(g) is a Hopf algebra.

. . . . . .

Coinvariant Lie subalgebra and Hopf algebra

Thanks to (A)

g0 := gF :=X ∈ g | H(X ) = X ⊗ 1

,

is a Lie algebra.And because of (B)

F0 := Fg :=F

⟨g F⟩is a Hopf algebra

. . . . . .

Cartan calculus for Lie-Hopf algebras

For any Y ∈ g0 we define the contraction ιY and the Lie derivativeLY on C •,•(g∗,F).

ιY (ω⊗f ) = ιY (ω)⊗f , LY (ω⊗f ) = LY (ω)⊗f +ω⊗Y f . (3)

(a) The contraction ιY is a derivation of degree −1.

(b) The Lie derivative LY is a derivation of degree 0.

(c) LX = ∂T ιY + ιY ∂T

(d) [∂T ,LY ] = 0

(e) L[Y1,Y2] = [LY1 ,LY2 ]

(f) [ιY1 ,LY2 ] = ι[Y1,Y2]

. . . . . .

Weil homomorphism

Let α : g∗0 → g∗ be a (algebraic) Cartan connection i.e.

α adY = adY α, ιY (α(ω)) = ω(Y ).

We extend α to α : g∗0 → C 1,0c−w(g

∗,F) by

α(ω) = (α(ω))<0> ⊗ (α(ω))<−1> .

The extension of α defines a Cartan connection on C •,•c−w(g

∗,F).As a result we get a map of g0-DG algebras

Cα : ∧•g∗0 ⊗ S•(g∗0)[2q] → C •,•c−w (g

∗,F)

Here q = dim g− dim g0.

. . . . . .

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

. . . . . .

Geometries and the corresponding Hopf algebras

C∞Γ

(M)):= C∞

c (M) > Γ

Geometry Lie alg. Hopf alg. Iso. Lie alg. Algebra

General an Hn gℓn C∞Γ

(F+(Rn)

)Vol pres. san SHn sℓn C∞

Γ

(F+s (Rn)

)Sympl. sp2n SpH2n sp2n C∞

Γ

(F+sp(R2n)

)Contact Fn2n+1 CnH2n+1 cn2n+1 C∞

Γ

(F+cn(R2n+1)

)

. . . . . .

The quantum symmetry of AΓ := C∞c (Rn)o Γ

I Xk = ∂∂x i

, Xk(fU∗ϕ) = Xk(f )U

∗ϕ,

I For Xk we have

Xk(ab) = Xk(a)b +∑

σik(a)Xi (b)

I Here

σik(fU

∗ϕ) =

∂ϕi

∂xkfU∗

ϕ

and higher derivatives

σij1,...,jk

(fU∗ϕ) =

∂kϕi

∂x j1 · · · ∂x jkfU∗

ϕ

. . . . . .

The Hopf algebra Kn

We let the Hopf algebra Kn be the subalgebra of L(AΓ generatedby

Xℓ, σij1,...,jm , σ−1

here σ = det[σij ] is the Jacobi automorphism of AΓ.

Its Hopf algebra structure is

∆(Xl) = Xl ⊗ 1 + σkl ⊗ Xk ,

∆(σij ) = σk

j ⊗ σik ,

∆(σ) = σ ⊗ σ, ∆(σ−1) = σ−1 ⊗ σ−1,

∆(σij1,...,jk

) = [∆(Xjk ),∆(σij1,...,jk−1

)],

ε(σ) = ε(σ−1) = 1, ε(σij ) = δij , ε(Xl) = ε(σi

j1,...,jk) = 0.

. . . . . .

Action of Kn on AΓ

I Kn acts on AΓ

I AΓ possesses the trace

τ(fU∗φ) =

∫Rn

fϖRn if φ = id

0 otherwise

I σ−1C defines a MPI module on Kn

I τ is a σ−1 trace

I

χτ : HC •(Kn,σ−1

C) → HC •(AΓ)

. . . . . .

HC (Kn,σ−1 C)

There is a map of algebras

C •,•c−w (V

∗,FK) → C •,•c−w (g

∗, gℓn,FK)

That induces

HP•(Kn;σ−1

C) ∼= HP•(Hn, gℓn;Cδ)

TheoremHopf cyclic cohomology of Kn with coefficients in σ−1C consists ofall Chern classes.

. . . . . .

Ωij = θk ⊗ 1 ∧ S(γpj )γ

ip,k ∈ C 1,1

c−w(V∗,FK)

HPm(C •,•c−w(V

∗,FK))=

⊕i1+···+ik=m

⟨Tr(Ωi1) · · ·Tr(Ωik )⟩

. . . . . .

For n = 1 we have the following b + B cocycles in HC 1(K1,σ−1 C)

C0 := 1⊗ σ−1X1 ∈ σ−1C⊗K1,

C1 := 1⊗ σ−2σ11,1 ∈ σ−1

C⊗K1

. . . . . .

For n = 2 we have the following b + B cocycles in HC 2(K2,σ−1 C)

C0 := 1⊗ σ−1σi2X1 ⊗ σ−1Xi − σ−1σi

1X2 ⊗ σ−1Xi ,

C1 := 1⊗ σ−1σi2S(σ

js)σ

sj ,1 ⊗ σ−1Xi − σ−1σi

1S(σjs)σ

sj ,2 ⊗ σ−1Xi ,

C2 := 1⊗ σ−1S(σir )σ

t1σ

si ,2 ⊗ σ−1S(σr

k)σks,t⊗

− σ−1S(σir )σ

t2σ

si ,1 ⊗ σ−1S(σr

k)σks,t ,

(C1)2 := 1⊗ σ−1S(σi

s)σt1σ

si ,2 ⊗ σ−1S(σr

k)σkr ,t

− σ−1S(σis)σ

t2σ

si ,1 ⊗ σ−1S(σr

k)σkr ,t .

. . . . . .

A surprise

χτ : HC •(Kn,σ−1

C) → HC •(AΓ)

is not injective.

For n=1 χτ (C1) = 0

For n=2 χτ (C1) = χτ (C21 ) = 0