Morita invariance in Hopf-(co)cyclic (co)homology.kaoutit/Santiago-13.pdf · show that its cyclic...

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Morita invariance in Hopf-(co)cyclic(co)homology.

Laiachi El KaoutitJoint work with Niels Kowalzig

Universidad de Granada. Spain.kaoutit@ugr.es

Congreso de la Real Sociedad Matemática Española.Sesion: Teoría de anillos no conmutativos.

Santiago de Compostela, 21–25 enero 2013.

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Motivations, overviews.

Problem. Give an algebraic formulation of the following geometricresults:

In Poisson geometry

[Ginzburg, Weinstein, Lu, Xu,...]

Two Morita equivalent Poisson man-ifolds, they have isomorphic Poissonhomology. ..

general

In Lie algebroids Theory

[Crainic, Fernandes, Higgins, Mackenzie,...]

Two Morita equivalent Lie alge-broids, they have isomorphic coho-mology.

. . . . . .

In Poisson geometry

general

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In Poisson geometry

Two Morita equivalent Poisson man-ifolds, they have isomorphic Poissonhomology.

general

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In Poisson geometry

general

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In Poisson geometry

general

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A Lie algebroid is a vector bundle E → M over a smooth manifold,together with a map of vector bundles ω : E → TM and Lie structure[−,−] on the vector space Γ(E) of global smooth sections of E , suchthat the induced map Γ(ω) : Γ(E) → Γ(TM) is a Lie algebra mapwhich satisfy: for all X ,Y ∈ Γ(E) and any f ∈ C∞(M) one has

[X , fY ] = f [X ,Y ] + Γ(ω)(X )(f )Y .

Two Lie algebroids (Ei ,Mi), i = 1,2, are said to be Morita equivalentprovided that there exist surjective submersions ϕi : Q → Mi withsimply connected fibers such that:

ϕ∗1E1

!!CCCC

}}{{{{

oo ∼= //_______ ϕ∗2E2

!!CCCC

}}{{{{

π1 !!DDDD

}}{{{{

{{{{ ϕ2

!!CCCC

π2}}zzzz

. . . . . .

Motivations, overviews.A Lie algebroid is a vector bundle E → M over a smooth manifold,together with a map of vector bundles ω : E → TM and Lie structure[−,−] on the vector space Γ(E) of global smooth sections of E , suchthat the induced map Γ(ω) : Γ(E) → Γ(TM) is a Lie algebra mapwhich satisfy: for all X ,Y ∈ Γ(E) and any f ∈ C∞(M) one has

[X , fY ] = f [X ,Y ] + Γ(ω)(X )(f )Y .

ϕ∗1E1

!!CCCC

}}{{{{

oo ∼= //_______ ϕ∗2E2

!!CCCC

}}{{{{

π1 !!DDDD

}}{{{{

{{{{ ϕ2

!!CCCC

π2}}zzzz

. . . . . .

[X , fY ] = f [X ,Y ] + Γ(ω)(X )(f )Y .

ϕ∗1E1

!!CCCC

}}{{{{

oo ∼= //_______ ϕ∗2E2

!!CCCC

}}{{{{

π1 !!DDDD

}}{{{{

{{{{ ϕ2

!!CCCC

π2}}zzzz

. . . . . .

[X , fY ] = f [X ,Y ] + Γ(ω)(X )(f )Y .

ϕ∗1E1

!!CCCC

}}{{{{

oo ∼= //_______ ϕ∗2E2

!!CCCC

}}{{{{

π1 !!DDDD

}}{{{{

{{{{ ϕ2

!!CCCC

π2}}zzzz

. . . . . .

Motivations, overviews.Which algebraic objects then encode the previous geometricinformations?

We know that to each Lie algebroid (E ,M) we can associated a leftHopf algebroid (C∞(M),VΓ(E)). In fact there is a functor

Lie-AlgdM −→ (left)Hopf-AlgdC∞(M)

such that the (co)homology of (E ,M) coincides, up to isomorphism,with the (co)cyclic (co)homolgy of (C∞(M),VΓ(E)).

In this way, the algebraic objects which we are looking for are thenHopf algebroids and theirs cyclic theories!

Questions:

(1) Do Morita equivalent Lie algebroids have Morita equivalentassociated (left) Hopf algebroids?

(2) Are two Morita equivalent (left) Hopf algebroids (R,U)M∼ (S,V ),

have isomorphic Hochschild (co)homology (co)cyclic(co)homology? i.e., Morita invaraiance of HH•,HH•,HC•,HC•?

. . . . . .

Motivations, overviews.Which algebraic objects then encode the previous geometricinformations?We know that to each Lie algebroid (E ,M) we can associated a leftHopf algebroid (C∞(M),VΓ(E)). In fact there is a functor

Questions:

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Questions:

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Questions:

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Questions:

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Questions:

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Motivations, overviews.So far, no answer to these questions is known! We do not even knowwhat Morita equivalent (left) Hopf algebroids (with different baserings) means?

A naive answer to Morita invariance of HH•,HH•,HC•,HC• is givenhere! Explicitly, we assume a Morita base change between two (left)Hopf algebroids of the form (R,U) and (S, U), where R is Moritaequivalent to S and U is constructed from U.

As we will see this approach is not far from some geometricapplication. To this end, we construct a Morita base change left Hopfalgebroid over noncommutative 2-torus (with rational parameter) andshow that its cyclic homology can be computed by means of thehomology of the Lie algebroid of vector fields on the classical 2-torus.

Based on the paper:

L. El Kaoutit and N. Kowalzig, Morita base change in Hopf-cyclic(co)homology. To appear in Lett. Math. Phys.

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Based on the paper:

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Based on the paper:

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Based on the paper:

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Left Hopf algebroids.

Let R be an associative algebra over a commutative ground ring kwith 1, Re = R ⊗k Ro is the enveloping ring of R.Left Hopf algebroids are characterised by their categories of leftmodules. Explicitly, given U an Re-ring, we denote byη∗ : UMod → ReMod the scalers restriction functor.The following assertions are equivalent:

(i) U is a left Hopf R-algebroid;

(ii) The category of left modules UMod is a monoidal category withη∗ is a strict monoidal functor and preserves left inner-homfunctors.

The underlying bimodule ReU admits then a structure of an R-coringsuch that we can define the categories UComod,ComodU of left andright U-comodules.

En general, ModU need not to be monoidal, and there is no suchcharacterisation using comodules.

. . . . . .

Let R be an associative algebra over a commutative ground ring kwith 1, Re = R ⊗k Ro is the enveloping ring of R.

Left Hopf algebroids are characterised by their categories of leftmodules. Explicitly, given U an Re-ring, we denote byη∗ : UMod → ReMod the scalers restriction functor.The following assertions are equivalent:

. . . . . .

Let R be an associative algebra over a commutative ground ring kwith 1, Re = R ⊗k Ro is the enveloping ring of R.Left Hopf algebroids are characterised by their categories of leftmodules. Explicitly, given U an Re-ring, we denote byη∗ : UMod → ReMod the scalers restriction functor.

The following assertions are equivalent:

. . . . . .

The fact that η∗ preserves left inner-hom functors, makes a differencebetween the notions of (left) Hopf algebroids and (left) bialgebroids.As in the case of Hopf algebra this leads to the notion of ”antipode”,although in this case its formulation is not obvious.

The Hopf-Galois map is defined by

β : U1⊗Ro ⊗Ro 1⊗Ro U → 1⊗Ro U ⊗R R⊗1o U, u ⊗Ro v 7→ u(1) ⊗R u(2)v .

The key is that: The functor η∗ preserves left inner-hom functors if,and only if the map β is bijective.We use Sweedler-type notation

u+ ⊗Ro u− := β−1(u ⊗R 1), for all u ∈ U,

for the translation map

β−1(−⊗R 1) : U → U1⊗Ro ⊗Ro 1⊗Ro U,

which plays the rôle of the antipode as in the classical case.

. . . . . .

Left Hopf algebroids.The fact that η∗ preserves left inner-hom functors, makes a differencebetween the notions of (left) Hopf algebroids and (left) bialgebroids.As in the case of Hopf algebra this leads to the notion of ”antipode”,although in this case its formulation is not obvious.

β−1(−⊗R 1) : U → U1⊗Ro ⊗Ro 1⊗Ro U,

. . . . . .

The key is that: The functor η∗ preserves left inner-hom functors if,and only if the map β is bijective.

We use Sweedler-type notation

β−1(−⊗R 1) : U → U1⊗Ro ⊗Ro 1⊗Ro U,

. . . . . .

β−1(−⊗R 1) : U → U1⊗Ro ⊗Ro 1⊗Ro U,

. . . . . .

Cyclic (co)homology for left Hopf algebroids.

Let M be simultaneously a left U-comodule and a right U-module withcompatible left R-action, which we also assume a Stable antiYetter-Drinfel’d module.The associated cyclic object is

C•(U,M) := M ⊗Ro (1⊗Ro U1⊗Ro)⊗Ro •,

whose structure maps are given by:

di(m⊗Rox) =

m ⊗Ro u1 ⊗Ro · · · ⊗Ro

(un−1(1 ⊗ ε(un)o)

), if i = 0,

m ⊗Ro · · · ⊗Ro (un−iun−i+1)⊗Ro · · · ⊗Ro un, if i = 1, ..., n − 1,(mu1)⊗Ro u2 ⊗Ro · · · ⊗Ro un, if i = n

si(m⊗Rox) =

m ⊗Ro u1 ⊗Ro · · · ⊗Ro un ⊗Ro 1, if i = 0,m ⊗Ro · · · ⊗Ro un−i ⊗Ro 1 ⊗Ro un−i+1 ⊗Ro · · · ⊗Ro un, if i = 1, .., n − 1,m ⊗Ro 1 ⊗Ro u1 ⊗Ro u2 ⊗Ro · · · ⊗Ro un, if i = n

tn(m ⊗Ro x) = (m(0)u1+)⊗Ro u2

+ ⊗Ro · · · ⊗Ro un+ ⊗Ro (un

− · · ·u1−m(−1))

where x := u1 ⊗Ro · · · ⊗Ro un.

. . . . . .

Cyclic (co)homology for left Hopf algebroids.Let M be simultaneously a left U-comodule and a right U-module withcompatible left R-action, which we also assume a Stable antiYetter-Drinfel’d module.

The associated cyclic object is

C•(U,M) := M ⊗Ro (1⊗Ro U1⊗Ro)⊗Ro •,

di(m⊗Rox) =

m ⊗Ro u1 ⊗Ro · · · ⊗Ro

(un−1(1 ⊗ ε(un)o)

), if i = 0,

si(m⊗Rox) =

tn(m ⊗Ro x) = (m(0)u1+)⊗Ro u2

+ ⊗Ro · · · ⊗Ro un+ ⊗Ro (un

− · · ·u1−m(−1))

. . . . . .

Cyclic (co)homology for left Hopf algebroids.Let M be simultaneously a left U-comodule and a right U-module withcompatible left R-action, which we also assume a Stable antiYetter-Drinfel’d module.The associated cyclic object is

C•(U,M) := M ⊗Ro (1⊗Ro U1⊗Ro)⊗Ro •,

di(m⊗Rox) =

m ⊗Ro u1 ⊗Ro · · · ⊗Ro

(un−1(1 ⊗ ε(un)o)

), if i = 0,

si(m⊗Rox) =

tn(m ⊗Ro x) = (m(0)u1+)⊗Ro u2

+ ⊗Ro · · · ⊗Ro un+ ⊗Ro (un

− · · ·u1−m(−1))

. . . . . .

C•(U,M) := M ⊗Ro (1⊗Ro U1⊗Ro)⊗Ro •,

di(m⊗Rox) =

m ⊗Ro u1 ⊗Ro · · · ⊗Ro

(un−1(1 ⊗ ε(un)o)

), if i = 0,

si(m⊗Rox) =

tn(m ⊗Ro x) = (m(0)u1+)⊗Ro u2

+ ⊗Ro · · · ⊗Ro un+ ⊗Ro (un

− · · ·u1−m(−1))

. . . . . .

C•(U,M) := M ⊗Ro (1⊗Ro U1⊗Ro)⊗Ro •,

di(m⊗Rox) =

m ⊗Ro u1 ⊗Ro · · · ⊗Ro

(un−1(1 ⊗ ε(un)o)

), if i = 0,

si(m⊗Rox) =

tn(m ⊗Ro x) = (m(0)u1+)⊗Ro u2

+ ⊗Ro · · · ⊗Ro un+ ⊗Ro (un

− · · ·u1−m(−1))

. . . . . .

C•(U,M) := M ⊗Ro (1⊗Ro U1⊗Ro)⊗Ro •,

di(m⊗Rox) =

m ⊗Ro u1 ⊗Ro · · · ⊗Ro

(un−1(1 ⊗ ε(un)o)

), if i = 0,

si(m⊗Rox) =

tn(m ⊗Ro x) = (m(0)u1+)⊗Ro u2

+ ⊗Ro · · · ⊗Ro un+ ⊗Ro (un

− · · ·u1−m(−1))

. . . . . .

Cyclic (co)homology for left Hopf algebroids.

Now its associated cocyclic object is

C•(U,M) := (R⊗1o UR⊗1o)⊗R• ⊗R M,

with structure maps in degree n given by

δi(z ⊗R m) =

1 ⊗R u1 ⊗R · · · ⊗R un ⊗R m, if i = 0u1 ⊗R · · · ⊗R ∆(ui)⊗R · · · ⊗R un ⊗R m, if 1 ≤ i ≤ nu1 ⊗R · · · ⊗R un ⊗R m(−1) ⊗R m(0), if i = n + 1.

δj(m) =

{1 ⊗R m, if j = 0m(−1) ⊗R m(0), if j = 1

σi(z ⊗R m) = u1 ⊗R · · ·⊗R ε(ui+1)⊗R · · ·⊗R un ⊗R m 0 ≤ i ≤ n−1,

τn(z ⊗R m) = u1−(1)u

2 ⊗R · · · ⊗R u1−(n−1)u

n ⊗R u1−(n)m(−1) ⊗R m(0)u1

where we abbreviate z := u1 ⊗R · · · ⊗R un.

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Cyclic (co)homology for left Hopf algebroids.Now its associated cocyclic object is

C•(U,M) := (R⊗1o UR⊗1o)⊗R• ⊗R M,

δi(z ⊗R m) =

δj(m) =

{1 ⊗R m, if j = 0m(−1) ⊗R m(0), if j = 1

τn(z ⊗R m) = u1−(1)u

2 ⊗R · · · ⊗R u1−(n−1)u

n ⊗R u1−(n)m(−1) ⊗R m(0)u1

. . . . . .

C•(U,M) := (R⊗1o UR⊗1o)⊗R• ⊗R M,

δi(z ⊗R m) =

δj(m) =

{1 ⊗R m, if j = 0m(−1) ⊗R m(0), if j = 1

τn(z ⊗R m) = u1−(1)u

2 ⊗R · · · ⊗R u1−(n−1)u

n ⊗R u1−(n)m(−1) ⊗R m(0)u1

. . . . . .

C•(U,M) := (R⊗1o UR⊗1o)⊗R• ⊗R M,

δi(z ⊗R m) =

δj(m) =

{1 ⊗R m, if j = 0m(−1) ⊗R m(0), if j = 1

τn(z ⊗R m) = u1−(1)u

2 ⊗R · · · ⊗R u1−(n−1)u

n ⊗R u1−(n)m(−1) ⊗R m(0)u1

. . . . . .

C•(U,M) := (R⊗1o UR⊗1o)⊗R• ⊗R M,

δi(z ⊗R m) =

δj(m) =

{1 ⊗R m, if j = 0m(−1) ⊗R m(0), if j = 1

τn(z ⊗R m) = u1−(1)u

2 ⊗R · · · ⊗R u1−(n−1)u

n ⊗R u1−(n)m(−1) ⊗R m(0)u1

. . . . . .

√Morita theory.

Let R and S two rings together with two bimodules SPR and RQS withisomorphisms of bimodules:

φ : P ⊗R Q '−→ S, ψ : Q ⊗S P '−→ R.

Thus (R,S,P,Q, φ, ψ) can be considered as a Morita context.This can be extended to a Morita context (Re,Se,Pe,Qe, φe, ψe),where Pe := P ⊗k Qo, Qe := Q ⊗k Po and φe, ψe are obvious.

This is an induced√

Morita equivalence between R and S, in thesense of Takeuchi.By Schauenburg’s result, starting with a left Hopf algebroid (R,U) wecan endow the Se-ring

U := Pe ⊗Re U ⊗Re Qe

with a structure of left S-Hopf algebroid. The pair (S, U) is called theMorita base change (left) Hopf algebroid of (R,U).

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√Morita theory.

φ : P ⊗R Q '−→ S, ψ : Q ⊗S P '−→ R.

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√Morita theory.

φ : P ⊗R Q '−→ S, ψ : Q ⊗S P '−→ R.

Thus (R,S,P,Q, φ, ψ) can be considered as a Morita context.

This can be extended to a Morita context (Re,Se,Pe,Qe, φe, ψe),where Pe := P ⊗k Qo, Qe := Q ⊗k Po and φe, ψe are obvious.

. . . . . .

√Morita theory.

φ : P ⊗R Q '−→ S, ψ : Q ⊗S P '−→ R.

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√Morita theory.

φ : P ⊗R Q '−→ S, ψ : Q ⊗S P '−→ R.

Morita equivalence between R and S, in thesense of Takeuchi.

By Schauenburg’s result, starting with a left Hopf algebroid (R,U) wecan endow the Se-ring

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√Morita theory.

φ : P ⊗R Q '−→ S, ψ : Q ⊗S P '−→ R.

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Morita invariance for Hopf-(co)cyclic (co)homology.

We can construct a quasi-isomorphisms between the chaincomplexes C•(U,M) and C•(U, M) (resp., between the cochaincomplexes C•(U,M) and C•(U, M)). The following is our main result.

TheoremLet (R,U) be a left Hopf algebroid, M a left U-comodule rightU-module which is SaYD, and (R,S,P,Q, φ, ψ) a Morita context.Consider the Morita base change left S-Hopf algebroid and the imageof M

U := Pe ⊗Re U ⊗Re Qe, M := P ⊗R M ⊗R Q.

We then have the following natural k-module isomorphisms:

HH•(U,M) ∼= HH•(U, M), HH•(U,M) ∼= HH•(U, M)

HC•(U,M) ∼= HC•(U, M), HC•(U,M) ∼= HC•(U, M)

between the Hochschild (co)homologies and (co)cyclic(co)homologies of the (co)cyclic objects C•(U,M) and C•(U, M)(resp.,C•(U,M) and C•(U, M)) .

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The Universal Enveloping of Lie Rinehart algebra.

Assume that R is a commutative k-algebra (Q ⊂ k is a ground field)and denote by Derk(R) the Lie algebra of all k-linear derivation of R.Consider L a k-Lie algebra with a structure of R-module, andω : L → Derk(R) a morphism of k-Lie algebras.Following Rinehart, the pair (R,L) is called Lie-Rinehart algebra withanchor map ω, provided that for all X ,Y ∈ L and a, b ∈ R, we have

(aX )(b) = a(X (b)), [X ,aY ] = a[X ,Y ] + X (a)Y

where X (a) stand for ω(X )(a).

ExampleHere are the basic examples which will be handled, and which in factstimulate the above general definition.

(i) The pair (R,Derk(R)) admits trivially a structure of Lie-Rinehart algebra.

(ii) Let (E ,M) be a Lie algebroid. Then the pair (C∞(M), Γ(E)) is obviouslya Lie-Rinehart algebra.

(iii) A smooth manifold M is a Poisson manifold if, and only if its cotangtebundle (T ∗M,M) is a Lie algebroid. Hence (C∞(M), Γ(T ∗M)) is aLie-Rinehart algebra for a Poisson manifold M.

. . . . . .

The Universal Enveloping of Lie Rinehart algebra.Assume that R is a commutative k-algebra (Q ⊂ k is a ground field)and denote by Derk(R) the Lie algebra of all k-linear derivation of R.

Consider L a k-Lie algebra with a structure of R-module, andω : L → Derk(R) a morphism of k-Lie algebras.Following Rinehart, the pair (R,L) is called Lie-Rinehart algebra withanchor map ω, provided that for all X ,Y ∈ L and a, b ∈ R, we have

. . . . . .

The Universal Enveloping of Lie Rinehart algebra.Assume that R is a commutative k-algebra (Q ⊂ k is a ground field)and denote by Derk(R) the Lie algebra of all k-linear derivation of R.Consider L a k-Lie algebra with a structure of R-module, andω : L → Derk(R) a morphism of k-Lie algebras.

Following Rinehart, the pair (R,L) is called Lie-Rinehart algebra withanchor map ω, provided that for all X ,Y ∈ L and a, b ∈ R, we have

. . . . . .

The Universal Enveloping of Lie Rinehart algebra.Assume that R is a commutative k-algebra (Q ⊂ k is a ground field)and denote by Derk(R) the Lie algebra of all k-linear derivation of R.Consider L a k-Lie algebra with a structure of R-module, andω : L → Derk(R) a morphism of k-Lie algebras.Following Rinehart, the pair (R,L) is called Lie-Rinehart algebra withanchor map ω, provided that for all X ,Y ∈ L and a, b ∈ R, we have

. . . . . .

The Universal Enveloping of Lie Rinehart algebra.

Associated to any Lie-Rinehart algebra (R, L), there is a universalobject denoted by (R,VL): Let U(L) be the enveloping Lie algebra ofL. Since L acts on R by derivations, we can consider the smashproduct R#U(L), and so take its factor R-algebra

Π : R#U(L) −→ VL :=R#U(L)

JL, JL := 〈a#X − 1#aX 〉a∈R, X∈L.

The usual k-bialgebra structure of U(L) can be lifted to a structure ofR-bialgebroid on VL, which gives in fact a structure of left HopfR-algebroid. The comultiplication and counit are obvious, thetranslation map is given on generator by

a+#a− := a#1, X+#X− := X#1 − 1#X , modulo JL.

Obviously there are two morphisms ιR : R −→ VL, ιL : L −→ VL ofk-algebras and k-Lie algebras, respectively, which satisfy

ιR(a)ιL(X ) = ιL(aX ), ιL(X )ιR(a)− ιR(a)ιL(X ) = ιR(X (a)), ∀ a ∈ R, X ∈ L.

By construction, the algebra VL and the maps ιR , ιL form anuniversal object subject to these equations.

. . . . . .

The Universal Enveloping of Lie Rinehart algebra.Associated to any Lie-Rinehart algebra (R, L), there is a universalobject denoted by (R,VL):

Let U(L) be the enveloping Lie algebra ofL. Since L acts on R by derivations, we can consider the smashproduct R#U(L), and so take its factor R-algebra

Π : R#U(L) −→ VL :=R#U(L)

JL, JL := 〈a#X − 1#aX 〉a∈R, X∈L.

. . . . . .

The Universal Enveloping of Lie Rinehart algebra.Associated to any Lie-Rinehart algebra (R, L), there is a universalobject denoted by (R,VL): Let U(L) be the enveloping Lie algebra ofL. Since L acts on R by derivations, we can consider the smashproduct R#U(L), and so take its factor R-algebra

Π : R#U(L) −→ VL :=R#U(L)

JL, JL := 〈a#X − 1#aX 〉a∈R, X∈L.

. . . . . .

Π : R#U(L) −→ VL :=R#U(L)

JL, JL := 〈a#X − 1#aX 〉a∈R, X∈L.

. . . . . .

Π : R#U(L) −→ VL :=R#U(L)

JL, JL := 〈a#X − 1#aX 〉a∈R, X∈L.

. . . . . .

Application to noncommutative 2-torus.

Vector bundles versus√Morita theories.

Assume we are given a finitely generated and projective module PRwhich is faithful, in the sense that any equation of the form Pa = 0,for some a ∈ R, implies a = 0. Then it is well known that R is Moritaequivalent to the endomorphisms ring End(PR). This is because R isa commutative ring.Now assume that we are given a (complex) smooth vector bundleπ : P → M of constant rank ≥ 1, then it is well know that the globalsmooth sections Γ(P) form a finitely generated and projectiveC∞(M)-module of constant rank over the complex valued smoothfunctions algebra on M. This module is in fact always faithful (thiscan also follows directly from an argument on maximal idealsassociated to points).In this way C∞(M) is Morita equivalent to the endomorphismsalgebra End(PC∞(M)) ∼= Γ(End(P)), where End(P) is the complexendomorphism algebra bundle.

. . . . . .

Assume we are given a finitely generated and projective module PRwhich is faithful, in the sense that any equation of the form Pa = 0,for some a ∈ R, implies a = 0. Then it is well known that R is Moritaequivalent to the endomorphisms ring End(PR). This is because R isa commutative ring.

Now assume that we are given a (complex) smooth vector bundleπ : P → M of constant rank ≥ 1, then it is well know that the globalsmooth sections Γ(P) form a finitely generated and projectiveC∞(M)-module of constant rank over the complex valued smoothfunctions algebra on M. This module is in fact always faithful (thiscan also follows directly from an argument on maximal idealsassociated to points).In this way C∞(M) is Morita equivalent to the endomorphismsalgebra End(PC∞(M)) ∼= Γ(End(P)), where End(P) is the complexendomorphism algebra bundle.

. . . . . .

Assume we are given a finitely generated and projective module PRwhich is faithful, in the sense that any equation of the form Pa = 0,for some a ∈ R, implies a = 0. Then it is well known that R is Moritaequivalent to the endomorphisms ring End(PR). This is because R isa commutative ring.Now assume that we are given a (complex) smooth vector bundleπ : P → M of constant rank ≥ 1, then it is well know that the globalsmooth sections Γ(P) form a finitely generated and projectiveC∞(M)-module of constant rank over the complex valued smoothfunctions algebra on M. This module is in fact always faithful (thiscan also follows directly from an argument on maximal idealsassociated to points).

In this way C∞(M) is Morita equivalent to the endomorphismsalgebra End(PC∞(M)) ∼= Γ(End(P)), where End(P) is the complexendomorphism algebra bundle.

. . . . . .

Assume we are given a finitely generated and projective module PRwhich is faithful, in the sense that any equation of the form Pa = 0,for some a ∈ R, implies a = 0. Then it is well known that R is Moritaequivalent to the endomorphisms ring End(PR). This is because R isa commutative ring.Now assume that we are given a (complex) smooth vector bundleπ : P → M of constant rank ≥ 1, then it is well know that the globalsmooth sections Γ(P) form a finitely generated and projectiveC∞(M)-module of constant rank over the complex valued smoothfunctions algebra on M. This module is in fact always faithful (thiscan also follows directly from an argument on maximal idealsassociated to points).In this way C∞(M) is Morita equivalent to the endomorphismsalgebra End(PC∞(M)) ∼= Γ(End(P)), where End(P) is the complexendomorphism algebra bundle.

. . . . . .

Application to noncommutative 2-torus.Noncommutative torus revisited.

Consider the Lie group S1 = {z ∈ C \ {0}| |z| = 1} as a real1-dimensional torus by identifying it with the additive quotient R/2πZ.The real d-dimensional torus Td := S1 × · · · × S1 is by the same wayidentified with Rd/2πZd .

Following [Dubois-Violette et al 2001; Khalkhali 2009], fix an elementq ∈ S1 whose argument is rational modulo 2π, and take N ∈ N to bethe smallest natural number such that qN = 1.

Consider the semidirect product group G := Z2N n S1 where

ZN = Z/NZ, and operation

(m, n, θ) (m′, n′, θ′) = (m + m′, n + n′, θθ′qmn′),

for every pair of elements (m, n, θ), (m′, n′, θ′) ∈ G.

Then there is a right action of the group G on the torus T3 given asfollows:

(x, y, z)(m, n, θ) = (qmx, qny, θzym), (x, y, z) ∈ T3, and (m, n, θ) ∈ G.

. . . . . .

Noncommutative torus revisited.

Now, we can show that the following map

T3p // T2

(x, y, z) � // (xN , yN)

satisfies the following properties:

(i) p is a surjective submersion.

(ii) G acts freely on T3 and the orbits of this action coincide with thefibers of p.

As consequence, one can claim by a classical results from differentialgeometry, that (T3,p,T2,G) is a principal fiber bundle.

Now, by extending the G-action on T3 to the trivial bundleT3 × CN → T3, we can construct as follows the associated vectorbundle.

. . . . . .

T3p // T2

(x, y, z) � // (xN , yN)

. . . . . .

T3p // T2

(x, y, z) � // (xN , yN)

. . . . . .

T3p // T2

(x, y, z) � // (xN , yN)

. . . . . .

T3p // T2

(x, y, z) � // (xN , yN)

. . . . . .

T3p // T2

(x, y, z) � // (xN , yN)

. . . . . .

First, this is possible by considering the following left G-action on CN

G // EndC(CN)

(m, n, θ) � //[ω 7→ θUn

0 V−m0 ω

where U0,V0 are the following (N × N)-matrices

0 1 0 0 · · ·0 0 1 0 · · ·...

. . .. . .

...0 · · · 0 11 0 · · · 0 0

, V0 =

1 0 · · · · · · 00 q 0 · · · 00 0 q2 · · · 0...

. . .. . .

...0 0 · · · 0 qN−1

which satisfy the relations

U0V0 = qV0U0, UN0 = V N

0 = IN .

. . . . . .

G // EndC(CN)

(m, n, θ) � //[ω 7→ θUn

0 V−m0 ω

0 1 0 0 · · ·0 0 1 0 · · ·...

. . .. . .

...0 · · · 0 11 0 · · · 0 0

, V0 =

1 0 · · · · · · 00 q 0 · · · 00 0 q2 · · · 0...

. . .. . .

...0 0 · · · 0 qN−1

U0V0 = qV0U0, UN0 = V N

0 = IN .

. . . . . .

G // EndC(CN)

(m, n, θ) � //[ω 7→ θUn

0 V−m0 ω

0 1 0 0 · · ·0 0 1 0 · · ·...

. . .. . .

...0 · · · 0 11 0 · · · 0 0

, V0 =

1 0 · · · · · · 00 q 0 · · · 00 0 q2 · · · 0...

. . .. . .

...0 0 · · · 0 qN−1

U0V0 = qV0U0, UN0 = V N

0 = IN .

. . . . . .

G // EndC(CN)

(m, n, θ) � //[ω 7→ θUn

0 V−m0 ω

0 1 0 0 · · ·0 0 1 0 · · ·...

. . .. . .

...0 · · · 0 11 0 · · · 0 0

, V0 =

1 0 · · · · · · 00 q 0 · · · 00 0 q2 · · · 0...

. . .. . .

...0 0 · · · 0 qN−1

U0V0 = qV0U0, UN0 = V N

0 = IN .

. . . . . .

Therefore, we have a left G-action on T3 × CN defined by

((x, y, z);ω

)(m, n, θ) =

((x, y, z)(m, n, θ); (m, n, θ)−1ω

((qmx, qny, θzym); θ−1U−n

0 V m0 ω

We are now able to associate a non trivial vector bundle to the trivialbundle T3 × CN → T3. That is, we can claim that there is a morphismof vector bundles

T3 × CN //

��

T3 ×G CN := Eq

��

(Eq is the orbits space of T3 × CN)

As a conclusion, the algebra C∞(T2) of all smooth complex valuedfunctions on T2 is Morita equivalent to the endomorphisms algebra ofglobal smooth sections End(Γ(Eq)) ∼= Γ

(End(Eq)

. . . . . .

((x, y, z);ω

)(m, n, θ) =

((x, y, z)(m, n, θ); (m, n, θ)−1ω

0 V m0 ω

T3 × CN //

��

T3 ×G CN := Eq

��

(End(Eq)

. . . . . .

((x, y, z);ω

)(m, n, θ) =

((x, y, z)(m, n, θ); (m, n, θ)−1ω

0 V m0 ω

T3 × CN //

��

T3 ×G CN := Eq

��

(End(Eq)

. . . . . .

((x, y, z);ω

)(m, n, θ) =

((x, y, z)(m, n, θ); (m, n, θ)−1ω

0 V m0 ω

T3 × CN //

��

T3 ×G CN := Eq

��

(End(Eq)

. . . . . .

((x, y, z);ω

)(m, n, θ) =

((x, y, z)(m, n, θ); (m, n, θ)−1ω

0 V m0 ω

T3 × CN //

��

T3 ×G CN := Eq

��

(End(Eq)

. . . . . .

Let u = ei2πt and v = ei2πs be the coordinate functions on the torusT2. There is a C-algebra isomorphism

Γ(End(Eq)) ∼= C∞(T2q) sending (uU0) 7→ U, (vV0) 7→ V .

where the algebra C∞(T2q) is the complex noncommutative 2-torus

whose elements are formal power Laurent series in U,V with rapidlydecreasing sequence of coefficients, subject to the relation

UV = qVU.

In conclusion we have the following Morita context

(C∞(T2), C∞(T2q), Γ(Eq), Γ(Eq)

. . . . . .

UV = qVU.

(C∞(T2), C∞(T2q), Γ(Eq), Γ(Eq)

. . . . . .

UV = qVU.

(C∞(T2), C∞(T2q), Γ(Eq), Γ(Eq)

. . . . . .

CorollaryLet q ∈ S1 be a root of unity (with rational argument mod 2π), and considerthe Morita context

(C∞(T2), C∞(T2q), Γ(Eq), Γ(Eq)

∗, φ, ψ).

Consider the Lie-Rinehart algebra(R = C∞(T2),K = DerC(C∞(T2))

)of the

Lie algebroid of vector fields on the complex torus T2 and its associated leftHopf algebroid (R,VK ). Let M be a right VK -module and left VK -comodule.We then have the following natural C-module isomorphisms

H•(VK ,M) ' H•(VK , M), HC•(VK ,M) ' HC•(VK , M),

where VK is the Morita base change left Hopf algebroid over thenoncommutative torus C∞(T2

. . . . . .

Furthermore, assume that M be R-flat. Then we have that

H•(VK , M) ' H•(K ,M), HC•(VK , M) '⊕

i≥0 H•−2i(K ,M),

H•(VK , M) ' M ⊗R

∧•R K , HP•(VK , M) '

⊕i≡•mod2 Hi(K ,M)

are natural C-module isomorphisms, where H•(K ,M) := TorVK• (M,R), and

where HP• denotes periodic cyclic cohomology.

. . . . . .

Hopf algebroid

G. Böhm Hopf algebroids, Handbook of algebra, Vol. 6, North-Holland,Amsterdam, 2009, pp. 173–236.

D. C. Ravenel, Complex Cobordism and Stable Homotopy Groups ofSpheres. Pure and Applied Mathematics Series, Academic Press, SanDiego, 1986.

P. Schauenburg, Bialgebras over noncommutative rings and a structuretheorem for Hopf bimodules, Appl. Categ. Structures 6 (1998), no. 2,193–222.

P. Schauenburg, Morita base change in quantum groupoids Locallycompact quantum groups and groupoids (Strasbourg, 2002), IRMA Lect.Math. Theor. Phys., 2, de Gruyter, Berlin, 2003, pp. 79–103.

SGA 3: Schémas en groupes. Vol. 1, Propriétés générales des schémasen groupes. Lecture Notes in Mathematics 151. Springer-Verlag, Berlin,Heidelberg, New York, 1970.

M. Sweedler, Groups of simple algebras, Inst. Hautes Études Sci. Publ.Math. (1974), no. 44, 79–189.

M. Takeuchi, Groups of algebras over A ⊗ A, J. Math. Soc. Japan 29(1977), no. 3, 459–492.

. . . . . .

Cyclic (co)Homology.

G. Böhm and D. Stefan, (Co)cyclic (co)homology of bialgebroids: anapproach via (co)monads, Comm. Math. Phys. 282 (2008), no. 1,239–286.

A. Connes, Noncommutative differential geometry, Inst. Hautes ÉtudesSci. Publ. Math. (1985), no. 62, 257–360.

A. Connes, Hopf algebras, cyclic cohomology and the transverse indextheorem, Comm. Math. Phys. 198 (1998), no. 1, 199–246.

M. Crainic, Cyclic cohomology of Hopf algebras, J. Pure Appl. Algebra166 (2002), no. 1-2, 29–66.

R. Dennis and K. Igusa, Hochschild homology and the secondobstruction for pseudoisotopy, Algebraic K -theory, Part I (Oberwolfach,1980), Lecture Notes in Math., vol. 966, Springer, Berlin, 1982, pp. 7–58.

B. Feıgin and B. Tsygan, Additive K -theory, K -theory, arithmetic andgeometry (Moscow, 1984–1986), Lecture Notes in Math., vol. 1289,Springer, Berlin, 1987, pp. 67–209.

P. Hajac, M. Khalkhali, B. Rangipour, and Y. Sommerhäuser, Stableanti-Yetter-Drinfeld modules, C. R. Math. Acad. Sci. Paris 338 (2004),no. 8, 587–590.

. . . . . .

Cyclic (co)Homology.

N. Kowalzig, Hopf algebroids and their cyclic theory, Ph. D. thesis,Universiteit Utrecht and Universiteit van Amsterdam, 2009.

N. Kowalzig and U. Krähmer, Cyclic structures in algebraic (co)homologytheories, Homology, Homotopy and Applications 13 (2011), no. 1,297–318.

N. Kowalzig and H. Posthuma, The cyclic theory of Hopf algebroids, J.Noncomm. Geom. 5 (2011), no. 3, 423–476.

J.-L. Loday, Cyclic homology, second ed., Grundlehren derMathematischen Wissenschaften, vol. 301, Springer-Verlag, Berlin,1998.

J.-L. Loday and D. Quillen, Cyclic homology and the Lie algebrahomology of matrices, Comment. Math. Helv. 59 (1984), no. 4, 569–591.

W. Massey and F. Peterson, The cohomology structure of certain fibrespaces. I, Topology 4 (1965), 47–65.

R. McCarthy, Morita equivalence and cyclic homology, C. R. Acad. Sci.Paris Sér. I Math. 307 (1988), no. 6, 211–215.

. . . . . .

Lie algebroids and Lie groupoids

J. Huebschmann, Poisson cohomology and quantization, J. ReineAngew. Math. 408 (1990), 57–113.

G. Rinehart, Differential forms on general commutative algebras, Trans.Amer. Math. Soc. 108 (1963), 195–222.

K. Mackenzie, Lie Groupoids and Lie Algebroids in DifferentialGeometry. London Math. Soc. Lecture Note Series 124. CambridgeUniversity Press. 1987.

K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids.London Math. Soc. Lecture Note Series 213. Cambridge UniversityPress. 2005.

Morita invariance in Hopf-(co)cyclic (co)homology.kaoutit/Santiago-13.pdf · show that its cyclic...

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