Hopf Algebroids in Homological Algebra

Hopf Algebroids in Homological Algebra

Uli Krahmer

joint work with Niels Kowalzig

Shanghai, 16.9.2011

Uli Krahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 1 / 29

A point of departure

The following has featured prominently in many talks this week:

Theorem (Van den Bergh)

Let A be a unital associative algebra over a field k which has afinitely generated projective resolution of finite length as anAe := A⊗k Aop-module, and for which there exists d such thatH i(A,Ae) = 0 for i 6= d. Then one has for all Ae-modules M

H i(A,M) ' Hd−i(A,M ⊗A ω), ω := Hd(A,Ae).

Ginzburg added:

Theorem (Ginzburg)

If in addition ω ' A as right Ae-module (i.e. if A is Calabi-Yau), thenH•(A,A) is a Batalin-Vilkovisky algebra.


A point of departure

The following has featured prominently in many talks this week:

Theorem (Van den Bergh)

Let A be a unital associative algebra over a field k which has afinitely generated projective resolution of finite length as anAe := A⊗k Aop-module, and for which there exists d such thatH i(A,Ae) = 0 for i 6= d. Then one has for all Ae-modules M

H i(A,M) ' Hd−i(A,M ⊗A ω), ω := Hd(A,Ae).

Ginzburg added:

Theorem (Ginzburg)

If in addition ω ' A as right Ae-module (i.e. if A is Calabi-Yau), thenH•(A,A) is a Batalin-Vilkovisky algebra.


Some questions I’d like to talk about

H•(A,M) ' Hd−•(A,M ⊗A ω) is an isomorphism of ...?

We have

H i(A,M) ' ExtAe (A,M), Hj(A,N) ' TorAe

(N ,A),

and most other classical (co)homology theories in algebra worksimilarly, only that (Ae ,A) is replaced by other augmented rings.Do the above theorems generalise to other such theories?

What is a Batalin-Vilkovisky algebra anyway?




We have


(N ,A),






We have


(N ,A),




Hopf algebroids

Hopf algebroids are generalisations of Hopf algebras in which thebase field k is replaced by a possibly noncommutative algebra A.

They are in particular k-algebras U with a k-algebra map

η : Ae → U ,

so every U-module is canonically an A-bimodule.

The Hopf algebroid structure turns U-Mod into a monoidalcategory whose unit object is A and for which the forgetfulfunctor to Ae-Mod is monoidal, just as for Hopf algebras.

For the experts: When I say Hopf algebroid I mean throughoutleft Hopf algebroid or equivalently left ×A-Hopf algebra.


Hopf algebroids



η : Ae → U ,





Hopf algebroids



η : Ae → U ,





Hopf algebroids



η : Ae → U ,





Examples of Hopf algebroids

Example 1: For A = k , Hopf algebroids are just Hopf algebras.

Example 2: For any algebra A, U = Ae is a Hopf algebroid overA. Here η : Ae → U is the identity. Every Hopf algebroid has acounit ε : U → A, and this is here the multiplication map of A.In particular, its in general not a ring map, but only U-linear.

Example 3: U = U(A, L), the universal enveloping algebra of aLie-Rinehart algebra aka Lie algebroid. Here A is a commutativealgebra and L is a Lie algebra and A and L are both acting oneach other. Special cases are Weyl algebras and more generallythe rings of algebraic differential operators on smooth affinevarieties X (here A = k[X ], L = Derk(A)).












Generalising Van den Bergh’s result

The Hopf algebroid structure also deifnes a functor

⊗ : U-Mod× Uop-Mod→ Uop-Mod

which for A = k is ⊗k with right action on M ⊗ N given by

(m ⊗k n)u = S(u(2))m ⊗k nu(1).

Using this one obtains:

TheoremLet U be an A-biprojective Hopf algebroid for which A has a finitelygenerated projective resolution of finite length as a U-module, andfor which there exists d such that ExtiU(A,U) = 0 for i 6= d. Thenone has for all U-modules M

ExtiU(A,M) ' TorUd−i(M ⊗ ω,A), ω := ExtdU(A,U).


Generalising Van den Bergh’s result

The Hopf algebroid structure also deifnes a functor

⊗ : U-Mod× Uop-Mod→ Uop-Mod

which for A = k is ⊗k with right action on M ⊗ N given by

(m ⊗k n)u = S(u(2))m ⊗k nu(1).

Using this one obtains:

TheoremLet U be an A-biprojective Hopf algebroid for which A has a finitelygenerated projective resolution of finite length as a U-module, andfor which there exists d such that ExtiU(A,U) = 0 for i 6= d. Thenone has for all U-modules M

ExtiU(A,M) ' TorUd−i(M ⊗ ω,A), ω := ExtdU(A,U).


Gerstenhaber algebras

DefinitionA Gerstenhaber algebra is a graded commutative k-algebra (V ,`)

V =⊕i∈Z

V i , v ` w = (−1)ijw ` v ∈ V i+j , v ∈ V i ,w ∈ V j

with a graded Lie bracket ·, · : V i+1 ⊗k V j+1 → V i+j+1 on

V [1] :=⊕i∈Z

V i+1

of V for which all operators v , · satisfy the graded Leibniz rule

u, v ` w = u, v ` w + (−1)ijv ` u,w, u ∈ V i+1, v ∈ V j .


Examples

The classical example is the Hochschild cohomology of anyassociative algebra.

In fact, we have:

Theorem (Shoikhet)

If U is a Hopf algebroid which is right A-projective, then ExtU(A,A)is naturally a Gerstenhaber algebra.

Shoikhet’s result deals in fact with only mildly restricted abelianmonoidal categories and mimicks Schwede’s elegant treatmentof the Hochschild case. In our ofrthcoming paper, we will giveexplicit formulas for the structure in terms of the canonicalcochain complex computing ExtU(A,A).


Examples


In fact, we have:

Theorem (Shoikhet)




Examples


In fact, we have:

Theorem (Shoikhet)




Gerstenhaber modules

DefinitionA module over V is a graded (V ,`)-module (Ω,a),

Ω =⊕j∈Z

Ωj , u a ω ∈ Ωj−i , u ∈ V i , ω ∈ Ωj

with a representation of the graded Lie algebra (V [1], ·, ·)

L : V i+1 ⊗k Ωj → Ωj−i , u ⊗k ω 7→ Lu(ω)

which satisfies for u ∈ V i+1, v ∈ V j , ω ∈ Ω the mixed Leibniz rule

Lu(v a ω) = u, v a ω + (−1)ijv a Lu(ω).


Examples

The classical example is the Hochschild homology of anyassociative algebra, and this generalises beautifully as follows:

TheoremIf U is a Hopf algebroid which is right A-projective, and ifM ∈ Uop-Mod has in addition a left U-comodule structure (withcompatible underlying left A-module structure), then TorU(M ,A) isnaturally a Gerstenhaber module over ExtU(A,A).


Batalin-Vilkovisky modules

DefinitionA Gerstenhaber module is BV if there is a k-linear differential

B : Ωj → Ωj+1, B B = 0

such that Lu is for u ∈ V i given by the homotopy formula

Luω = B(u a ω)− (−1)iu a B(ω).

A pair (V ,Ω) of a Gerstenhaber algebra and of a Batalin-Vilkoviskymodule over it is also called a differential calculus.


Cyclic homology

In 1963, Rinehart defined what is nowadays called cyclichomology, and in particular a differential

B : TorAe

n (A,A)→ TorAe

n+1(A,A)

which turns the Gerstenhaber module H•(A,A) over H•(A,A) isin fact Batalin-Vilkovisky.

After 20 years of silence, there were Connes, Tsygan, Loday,Quillen, Goodwillie, Cuntz...

So, how about TorU• (M ,A) for Hopf algebroids?


Cyclic homology


B : TorAe

n (A,A)→ TorAe

n+1(A,A)





Cyclic homology


B : TorAe

n (A,A)→ TorAe

n+1(A,A)





The paracyclic object

In our paper in HHA we showed:

TheoremGiven a Hopf algebroid U and a right module-left comodule M(technical assumps as above), the reduced canonical chain complex

b : C•(U ,M) := M ⊗Aop U⊗Aop• → C•−1(U ,M)

carries a natural structure of a paracyclic k-module.

This means there is a map

B : C•(U ,M)→ C•+1(U ,M),

but it does in general not define a mixed complex,

B B 6= 0, b B + B b 6= 0.


The paracyclic object

In our paper in HHA we showed:

TheoremGiven a Hopf algebroid U and a right module-left comodule M(technical assumps as above), the reduced canonical chain complex

b : C•(U ,M) := M ⊗Aop U⊗Aop• → C•−1(U ,M)

carries a natural structure of a paracyclic k-module.

This means there is a map

B : C•(U ,M)→ C•+1(U ,M),

but it does in general not define a mixed complex,

B B 6= 0, b B + B b 6= 0.


More and less precisely

The paracyclic structure means that in addition to the ordinarysimplicial k-modue structure there are maps

tn : C•(U ,M)→ C•(U ,M)

with certain props. If tn+1n = id one speaks of a cyclic module

and one does get a mixed complex.

For us this happens iff M is a stable anti Yetter-Drinfel’d module.

In general, one can functorially assign to any paracyclick-module a cyclic one, namely the qutient by im (id− tn+1

n ).

DefinitionA paracyclic k-module quasicyclic if this quotient splits.

TheoremIn this case, everything is fine.




tn : C•(U ,M)→ C•(U ,M)





n ).






tn : C•(U ,M)→ C•(U ,M)





n ).






tn : C•(U ,M)→ C•(U ,M)





n ).




Example

After all this abstract b......., finally something that at leastbegins with interesting words:

TheoremLet A be a twisted Calabi-Yau algebra, Hd(A,Ae) ' Aσ, and assumethat σ is semisimple. Then H•(A,A) is a Batalin-Vilkovisk algebra.

This applies to quantum groups, quantised universal envelopingalgebras, quantum planes, quantum spheres...


Short version

Given a Hopf algebroid (U ,A), a right U-module and leftU-comodule M (with compatible induced left A-modulestructures), we will define a paracocyclic k-module structure on

C •(U ,M) := U⊗A• ⊗A M .

When U is a Hopf algebra over A = k , and M = k , then thisreduces to the original definition of Hopf-cyclic cohomology dueto Connes and Moscovici. Present and unpresent people haveextended their setting to stable anti Yetter-Drinfel’d modules(SaYD) M . Kaygun has observed that one can start with anymodule-comodule, Bohm and Stefan have done Hopf algebroids,but only for SaYD and implicitly, so one of our aims was toobtain the most straigthforward generalisation of Connes andMoscovici’s formulas to the general setting.


Short version

Given a Hopf algebroid (U ,A), a right U-module and leftU-comodule M (with compatible induced left A-modulestructures), we will define a paracocyclic k-module structure on

C •(U ,M) := U⊗A• ⊗A M .

When U is a Hopf algebra over A = k , and M = k , then thisreduces to the original definition of Hopf-cyclic cohomology dueto Connes and Moscovici. Present and unpresent people haveextended their setting to stable anti Yetter-Drinfel’d modules(SaYD) M . Kaygun has observed that one can start with anymodule-comodule, Bohm and Stefan have done Hopf algebroids,but only for SaYD and implicitly, so one of our aims was toobtain the most straigthforward generalisation of Connes andMoscovici’s formulas to the general setting.


The cyclic dual

There is a functorial way to associate to any cocyclic object acyclic one. In contrast to common misbelief, this process can beextended to general paracocyclic objects (details later).

If we apply this to C •(U ,M) we obtain a paracyclic objectwhose structure can be seen best by applying a vector spaceisomorphism with

C•(U ,M) := M ⊗Aop (I U )⊗Aop•

Taking here U = Ae ,M = A one obtains on the nose the originaldefinitions of the cyclic homology of an associative algebra.

If we then twist one of the two A-actions on itself by anautomorphism σ ∈ Aut(A), we obtain for M = Aσ the twistedcyclic homology of Kustermans, Murphy, and Tuset.


The cyclic dual







The cyclic dual







The cyclic dual







Constructing C •(U ,M)

Ultimately, this construction goes back to Crainic.

The first step is to define an auxiliary paracocyclic object

B•(U ,M) := U⊗eA•+1 ⊗Ae M .

that is easy to deal with. Here U is considered with the usual(A,A)-bimodule structure given by , . So B•(U ,M) is

U⊗A•+1 ⊗k M modulo the span of elements

u0⊗A· · ·⊗Aun⊗Ae amb−b u0⊗A· · ·⊗Aun a⊗Ae m | a, b ∈ A.

We will use Schauenburg’s shorthand notation

β−1(u ⊗A v) =: u+ ⊗Aop u−v

for the inverse of the Galois map. As mentioned above, whenA = k , that is, U is an ordinary Hopf algebra, then we have

u+ ⊗k u− = u(1) ⊗k S(u(2)).



Ultimately, this construction goes back to Crainic.The first step is to define an auxiliary paracocyclic object

B•(U ,M) := U⊗eA•+1 ⊗Ae M .





β−1(u ⊗A v) =: u+ ⊗Aop u−v


u+ ⊗k u− = u(1) ⊗k S(u(2)).



Ultimately, this construction goes back to Crainic.The first step is to define an auxiliary paracocyclic object

B•(U ,M) := U⊗eA•+1 ⊗Ae M .





β−1(u ⊗A v) =: u+ ⊗Aop u−v


u+ ⊗k u− = u(1) ⊗k S(u(2)).


Paracyclic objects in abelian categories

Paracyclic objects are like cyclic objects, only T := tn+1 is notrequired to be the identity id:

DefinitionA paracyclic object is a simplical object (C•, b•, s•) equipped withmorphisms t : Cn → Cn that satisfy (on Cn)

bi t = −tbi−1, si t = −tsi−1, b0t = (−1)nbn, s0t = (−1)nt2sn, 1 ≤ i ≤ n.

T commutes with all the paracyclic generators t, bi , sj . As aconsequence, a cyclic object can be attached to any paracyclicobject by passing to the coinvariants C/im (id− T ) of T . Inwell-behaved cases, there is no loss of homological information.

Paracocyclic objects are paracyclic objects in the dual category.


Paracyclic objects in abelian categories

Paracyclic objects are like cyclic objects, only T := tn+1 is notrequired to be the identity id:

DefinitionA paracyclic object is a simplical object (C•, b•, s•) equipped withmorphisms t : Cn → Cn that satisfy (on Cn)

bi t = −tbi−1, si t = −tsi−1, b0t = (−1)nbn, s0t = (−1)nt2sn, 1 ≤ i ≤ n.

T commutes with all the paracyclic generators t, bi , sj . As aconsequence, a cyclic object can be attached to any paracyclicobject by passing to the coinvariants C/im (id− T ) of T . Inwell-behaved cases, there is no loss of homological information.

Paracocyclic objects are paracyclic objects in the dual category.


The paracocyclic structure

The paracocyclic structure on B•(U ,M) is given by thefollowing operators, where w := u0 ⊗A · · · ⊗A un:

δ′i(w ⊗Ae m) = u0 ⊗A · · · ⊗A ∆(ui)⊗A · · · ⊗A un ⊗Ae m

where 0 ≤ i ≤ n,

δ′n+1(w ⊗Ae m) = u0(2) ⊗A u1 ⊗A · · · ⊗A m(−1)u

0(1) ⊗Ae m(0)

σ′i(w ⊗Ae m) = u0 ⊗A · · · ⊗A t(ε(ui+1))ui ⊗A · · · ⊗A un ⊗Ae m

where 0 ≤ i ≤ n − 1 and

τ ′n(w ⊗Ae m) = u1 ⊗A · · · ⊗A un ⊗A m(−1)u0 ⊗Ae m(0).

This works for all left comodules M over a bialgebroid U .


Passing to ⊗Uop gives

Theorem

C •(U ,M) := U⊗A• ⊗A M carries for all Hopf algebroids (U ,A) andright modules-left comodules M a canonical paracocyclic k-modulestructure with codegeneracies and cofaces

δi(z ⊗A m) =

1⊗A u1 ⊗A · · · ⊗A un ⊗A mu1 ⊗A · · · ⊗A ∆(ui)⊗A · · · ⊗A un ⊗A mu1 ⊗A · · · ⊗A un ⊗A m(−1) ⊗A m(0)

if i = 0,if 1 ≤ i ≤ n,if i = n + 1,

δj(m) =

1⊗A mm(−1) ⊗A m(0)

if j = 0,if j = 1,

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

and cocyclic operator (z := u1 ⊗A · · · ⊗A un).

τn(z ⊗Am) = u1−(1)u

2⊗A · · · ⊗A u1−(n−1)u

n ⊗A u1−(n)m(−1)⊗Am(0)u

1+.


The dual theory

Secondly, we generalise the concept of cyclic duality to a functorthat assigns paracyclic objects to arbitrary paracocyclic ones andwe compute the cyclic dual (C•(U ,M), d•, s•, t•) of the aboveparacocyclic k-module.

We then provide an isomorphism of this with the paracyclicmodule M ⊗op

A (IU )⊗opA • whose structure maps are given by

di(m ⊗opA x)=

m ⊗op

A u1 ⊗opA · · · ⊗

opA ε(un) I un−1

m ⊗opA · · · ⊗

opA un−iun−i+1 ⊗op

A · · ·mu1 ⊗op

A u2 ⊗opA · · · ⊗

opA un

if i =0,if 1≤ i≤n − 1,if i =n,

si(m ⊗opA x)=

m ⊗op

A u1 ⊗opA · · · ⊗

opA un ⊗op

A 1m ⊗op

A · · · ⊗opA un−i ⊗op

A 1⊗opA un−i+1 ⊗op

A · · ·m ⊗op

A 1⊗opA u1 ⊗op

A · · · ⊗opA un

if i =0,if 1≤ i≤n − 1,if i =n,

tn(m ⊗opA x)=m(0)u

1+ ⊗

opA u2

+ ⊗opA · · · ⊗

opA un

+ ⊗opA un

− · · · u1−m(−1),

where we abbreviate x := u1 ⊗opA · · · ⊗

opA un.


The dual theory

Secondly, we generalise the concept of cyclic duality to a functorthat assigns paracyclic objects to arbitrary paracocyclic ones andwe compute the cyclic dual (C•(U ,M), d•, s•, t•) of the aboveparacocyclic k-module.

We then provide an isomorphism of this with the paracyclicmodule M ⊗op

A (IU )⊗opA • whose structure maps are given by

di(m ⊗opA x)=

m ⊗op

A u1 ⊗opA · · · ⊗

opA ε(un) I un−1

m ⊗opA · · · ⊗

opA un−iun−i+1 ⊗op

A · · ·mu1 ⊗op

A u2 ⊗opA · · · ⊗

opA un

if i =0,if 1≤ i≤n − 1,if i =n,

si(m ⊗opA x)=

m ⊗op

A u1 ⊗opA · · · ⊗

opA un ⊗op

A 1m ⊗op

A · · · ⊗opA un−i ⊗op

A 1⊗opA un−i+1 ⊗op

A · · ·m ⊗op

A 1⊗opA u1 ⊗op

A · · · ⊗opA un

if i =0,if 1≤ i≤n − 1,if i =n,

tn(m ⊗opA x)=m(0)u

1+ ⊗

opA u2

+ ⊗opA · · · ⊗

opA un

+ ⊗opA un

− · · · u1−m(−1),

where we abbreviate x := u1 ⊗opA · · · ⊗

opA un.


How duality works

All one needs is in LNM1289 and Loday. One just has tocompose the usual cyclic duality with a suitable autoequivalenceof the cyclic category to obtain a functor that does lift toparacocyclic objects (just not to a proper duality).

Concretely, we deifne:

DefinitionThe cyclic dual of a paracocyclic k-module C • = (C •, δ•, σ•, τ•) isthe cyclic k-module C• := (C•, d•, s•, t•), where Cn := C n, and

di := σn−(i+1) : Cn → Cn−1, 0 ≤ i < n,

dn := σn−1 τn : Cn → Cn−1,

si := δn−(i+1) : Cn−1 → Cn, 0 ≤ i < n,

tn := τn : Cn → Cn.


How duality works

All one needs is in LNM1289 and Loday. One just has tocompose the usual cyclic duality with a suitable autoequivalenceof the cyclic category to obtain a functor that does lift toparacocyclic objects (just not to a proper duality).

Concretely, we deifne:

DefinitionThe cyclic dual of a paracocyclic k-module C • = (C •, δ•, σ•, τ•) isthe cyclic k-module C• := (C•, d•, s•, t•), where Cn := C n, and

di := σn−(i+1) : Cn → Cn−1, 0 ≤ i < n,

dn := σn−1 τn : Cn → Cn−1,

si := δn−(i+1) : Cn−1 → Cn, 0 ≤ i < n,

tn := τn : Cn → Cn.


Lie-Rinehart algebras (= Lie algebroids) I

Let (A, L) be a Lie-Rinehart algebra over k : L is a k-Lie algebraand a module over the commutative k-algebra A, but it also actsvia derivations ∂X on A such that

[X , aY ] = (∂Xa)Y + a[X ,Y ], ∂aX = a∂X , a ∈ A,X ,Y ∈ L.

Examples: L = Derk(A,A). Foliations are sub Lie-Rinehartalgebras. Poisson algebras A with L = Ω1, [da, db] = da, b.U = U(A, L) is the universal k-algebra equipped with two maps

ιA : AtoU , ιL : LtoU

of k-algebras/k-Lie algebras subject to

ιA(a)ιL(X ) = ιL(aX ), ιL(X )ιA(a)− ιA(a)ιL(X ) = ιA(∂X (a))

for a ∈ A,X ∈ L. The map ιA is injective. If L is A-projective,then ιL is injective as well. We’ll suppress the maps in the sequel.





Examples: L = Derk(A,A). Foliations are sub Lie-Rinehartalgebras.

Poisson algebras A with L = Ω1, [da, db] = da, b.U = U(A, L) is the universal k-algebra equipped with two maps









Examples: L = Derk(A,A). Foliations are sub Lie-Rinehartalgebras. Poisson algebras A with L = Ω1, [da, db] = da, b.

U = U(A, L) is the universal k-algebra equipped with two maps









Examples: L = Derk(A,A). Foliations are sub Lie-Rinehartalgebras. Poisson algebras A with L = Ω1, [da, db] = da, b.U = U(A, L) is the universal k-algebra equipped with two maps






Lie-Rinehart algebras (Lie algebroids) II

Xu: U carries the structure of a bialgebroid withη(−⊗ 1) = η(1⊗−) given by ιA. The prescriptions

∆(X ) = 1⊗A X + X ⊗A 1, ∆(a) = a ⊗A 1

can be extended by the universal property to a coproduct∆ : U → U ⊗A U . The counit is similarly given by the extensionof the anchor ε to U .

Easy observation: The Galois map is bijective, the translationmap is given on generators as

a+ ⊗Aop a− := a ⊗Aop 1, X+ ⊗Aop X− := X ⊗Aop 1− 1⊗Aop X

and is then extended via universalityBy the way: Lie-Rinehart algebras are a good example thatdemonstrates that there is no adjoint action of a Hopf algebroidon itself, ad(u)v := u+vu− is ill-defined. So U is in general nota module algebra over itself.




∆(X ) = 1⊗A X + X ⊗A 1, ∆(a) = a ⊗A 1

can be extended by the universal property to a coproduct∆ : U → U ⊗A U . The counit is similarly given by the extensionof the anchor ε to U .Easy observation: The Galois map is bijective, the translationmap is given on generators as


and is then extended via universality

By the way: Lie-Rinehart algebras are a good example thatdemonstrates that there is no adjoint action of a Hopf algebroidon itself, ad(u)v := u+vu− is ill-defined. So U is in general nota module algebra over itself.




∆(X ) = 1⊗A X + X ⊗A 1, ∆(a) = a ⊗A 1

can be extended by the universal property to a coproduct∆ : U → U ⊗A U . The counit is similarly given by the extensionof the anchor ε to U .Easy observation: The Galois map is bijective, the translationmap is given on generators as


and is then extended via universalityBy the way: Lie-Rinehart algebras are a good example thatdemonstrates that there is no adjoint action of a Hopf algebroidon itself, ad(u)v := u+vu− is ill-defined. So U is in general nota module algebra over itself.


Towards Hopf algebroids: Ae-rings

Bialgebroids generalise bialgebras by replacing the base ring k bya possibly noncommutative base algebra A. For starters, theyare Ae-rings, i.e. algebras U with an algebra map η : Ae → U .

We consider M ∈ U-Mod,N ∈ Uop-Mod as A-bimodules with

a m b := η(a ⊗k b)m, a, b ∈ A,m ∈ M .

a I m J b := nη(b ⊗k a), a, b ∈ A, n ∈ N .

In particular, U itself carries two left and two right A-actions allcommuting with each other. Usually we consider U as anAe-module using a u b, and otherwise we write e.g. I U .





a m b := η(a ⊗k b)m, a, b ∈ A,m ∈ M .







a m b := η(a ⊗k b)m, a, b ∈ A,m ∈ M .




Bialgebroids (×A-bialgebras) I

Now assume U is also a coalgebra in the monoidal category(Ae-Mod,⊗A,A) with coproduct and counit

∆ : U → U ⊗A U , ε : U → A.

Note: There is no natural algebra structure on U ⊗A U!

Takeuchi’s solution: In U ⊗A U consider the embedding

ι : U ×A U → U ⊗A U ,

of the Ae-ring U ×A U which is the centre of the A-bimodule

I U ⊗A UJ :

U×AU :=∑

i

ui⊗Avi ∈ U⊗AU |∑i

a I ui⊗Avi =∑i

ui⊗Avi J a.




∆ : U → U ⊗A U , ε : U → A.



ι : U ×A U → U ⊗A U ,


I U ⊗A UJ :

U×AU :=∑

i


a I ui⊗Avi =∑i

ui⊗Avi J a.




∆ : U → U ⊗A U , ε : U → A.



ι : U ×A U → U ⊗A U ,


I U ⊗A UJ :

U×AU :=∑

i


a I ui⊗Avi =∑i

ui⊗Avi J a.


Bialgebroids (×A-bialgebras) II

Similarly, A is not an Ae-ring unless A is commutative. Tohandle this one needs the canonical map

π : Endk(A)→ A, ϕ 7→ ϕ(1),

and the fact that Endk(A) is an Ae-ring.

Now it makes sense to require ∆, ε to factor through ι and π:

Definition (Takeuchi)

A (left) bialgebroid is an Ae-ring U together with twohomomorphisms ∆ : U → U ×A U, ε : U → Endk(A) of Ae-ringssuch that U is a coalgebra in Ae-Mod via ∆ = ι ∆, ε = π ε.

There is an analogous notion of a right bialgebroid in which forexample A has a canonical right U-module structure.


Bialgebroids (×A-bialgebras) II

Similarly, A is not an Ae-ring unless A is commutative. Tohandle this one needs the canonical map

π : Endk(A)→ A, ϕ 7→ ϕ(1),

and the fact that Endk(A) is an Ae-ring.

Now it makes sense to require ∆, ε to factor through ι and π:

Definition (Takeuchi)

A (left) bialgebroid is an Ae-ring U together with twohomomorphisms ∆ : U → U ×A U, ε : U → Endk(A) of Ae-ringssuch that U is a coalgebra in Ae-Mod via ∆ = ι ∆, ε = π ε.

There is an analogous notion of a right bialgebroid in which forexample A has a canonical right U-module structure.


Hopf algebroids (×A-Hopf algebras)

Let U be a bialgebroid and define the Galois map of U

β : I U ⊗Aop U → U ⊗A U , u ⊗Aop v 7→ u(1) ⊗A u(2)v ,

where I U ⊗Aop U = U ⊗k U/spana I u ⊗k v − u ⊗k v a.

For bialgebras over fields β is bijective if and only if U is a Hopfalgebra with β−1(u ⊗k v) := u(1) ⊗ S(u(2))v , where S is theantipode of U . This motivates:

Definition (Schauenburg)

A (left) bialgebroid U is a (left) Hopf algebroid if β is a bijection.

There is a notion of (full) Hopf algebroid due to Bohm andSzlachanyi which are left and right bialgebroids. The preciseaxioms would barely fit on one slide.Example: U = Ae : η : Ae → Ae is the identity, ε : Ae → A is themultiplication map, ∆(a ⊗k b) = (a ⊗k 1)⊗k (1⊗k b).





where I U ⊗Aop U = U ⊗k U/spana I u ⊗k v − u ⊗k v a.For bialgebras over fields β is bijective if and only if U is a Hopfalgebra with β−1(u ⊗k v) := u(1) ⊗ S(u(2))v , where S is theantipode of U . This motivates:





















Documents

Hopf Algebroids in Homological Algebra