Hopf Algebroids in Homological Algebra Uli Kr ahmer joint work - - PowerPoint PPT Presentation

Hopf Algebroids in Homological Algebra

Uli Kr¨ ahmer

joint work with Niels Kowalzig

Shanghai, 16.9.2011

Uli Kr¨ ahmer (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 a finitely generated projective resolution of finite length as an Ae := A ⊗k Aop-module, and for which there exists d such that Hi(A, Ae) = 0 for i = d. Then one has for all Ae-modules M Hi(A, M) ≃ Hd−i(A, M ⊗A ω), ω := Hd(A, Ae).

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 2 / 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 a finitely generated projective resolution of finite length as an Ae := A ⊗k Aop-module, and for which there exists d such that Hi(A, Ae) = 0 for i = d. Then one has for all Ae-modules M Hi(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), then H•(A, A) is a Batalin-Vilkovisky algebra.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 2 / 29

Some questions I’d like to talk about

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

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 3 / 29

Some questions I’d like to talk about

H•(A, M) ≃ Hd−•(A, M ⊗A ω) is an isomorphism of ...? We have Hi(A, M) ≃ ExtAe(A, M), Hj(A, N) ≃ TorAe(N, A), and most other classical (co)homology theories in algebra work similarly, only that (Ae, A) is replaced by other augmented rings. Do the above theorems generalise to other such theories?

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 3 / 29

Some questions I’d like to talk about

H•(A, M) ≃ Hd−•(A, M ⊗A ω) is an isomorphism of ...? We have Hi(A, M) ≃ ExtAe(A, M), Hj(A, N) ≃ TorAe(N, A), and most other classical (co)homology theories in algebra work similarly, 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?

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 3 / 29

Hopf algebroids

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

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 4 / 29

Hopf algebroids

Hopf algebroids are generalisations of Hopf algebras in which the base 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.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 4 / 29

Hopf algebroids

Hopf algebroids are generalisations of Hopf algebras in which the base 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 monoidal category whose unit object is A and for which the forgetful functor to Ae-Mod is monoidal, just as for Hopf algebras.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 4 / 29

Hopf algebroids

Hopf algebroids are generalisations of Hopf algebras in which the base 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 monoidal category whose unit object is A and for which the forgetful functor to Ae-Mod is monoidal, just as for Hopf algebras. For the experts: When I say Hopf algebroid I mean throughout left Hopf algebroid or equivalently left ×A-Hopf algebra.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 4 / 29

Examples of Hopf algebroids

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

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 5 / 29

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 over

• A. Here η : Ae → U is the identity. Every Hopf algebroid has a

counit 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.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 5 / 29

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 over

• A. Here η : Ae → U is the identity. Every Hopf algebroid has a

counit 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 a Lie-Rinehart algebra aka Lie algebroid. Here A is a commutative algebra and L is a Lie algebra and A and L are both acting on each other. Special cases are Weyl algebras and more generally the rings of algebraic differential operators on smooth affine varieties X (here A = k[X], L = Derk(A)).

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 5 / 29

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).

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 6 / 29

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:

Theorem

Let U be an A-biprojective Hopf algebroid for which A has a finitely generated projective resolution of finite length as a U-module, and for which there exists d such that Exti

U(A, U) = 0 for i = d. Then

• ne has for all U-modules M

Exti

U(A, M) ≃ TorU d−i(M ⊗ ω, A),

ω := Extd

U(A, U).

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 6 / 29

Gerstenhaber algebras

Definition

A 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

• f 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.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 7 / 29

Examples

The classical example is the Hochschild cohomology of any associative algebra.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 8 / 29

Examples

The classical example is the Hochschild cohomology of any associative 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.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 8 / 29

Examples

The classical example is the Hochschild cohomology of any associative 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 abelian monoidal categories and mimicks Schwede’s elegant treatment

• f the Hochschild case. In our ofrthcoming paper, we will give

explicit formulas for the structure in terms of the canonical cochain complex computing ExtU(A, A).

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 8 / 29

Gerstenhaber modules

Definition

A module over V is a graded (V , )-module (Ω, ), Ω =

• j∈Z

Ωj, u 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 → Lu(ω) which satisfies for u ∈ V i+1, v ∈ V j, ω ∈ Ω the mixed Leibniz rule Lu(v ω) = {u, v} ω + (−1)ijv Lu(ω).

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 9 / 29

Examples

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

Theorem

If U is a Hopf algebroid which is right A-projective, and if M ∈ Uop-Mod has in addition a left U-comodule structure (with compatible underlying left A-module structure), then TorU(M, A) is naturally a Gerstenhaber module over ExtU(A, A).

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 10 / 29

Batalin-Vilkovisky modules

Definition

A 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 ω) − (−1)iu B(ω). A pair (V , Ω) of a Gerstenhaber algebra and of a Batalin-Vilkovisky module over it is also called a differential calculus.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 11 / 29

Cyclic homology

In 1963, Rinehart defined what is nowadays called cyclic homology, 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) is in fact Batalin-Vilkovisky.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 12 / 29

Cyclic homology

In 1963, Rinehart defined what is nowadays called cyclic homology, 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) is in fact Batalin-Vilkovisky. After 20 years of silence, there were Connes, Tsygan, Loday, Quillen, Goodwillie, Cuntz...

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 12 / 29

Cyclic homology

In 1963, Rinehart defined what is nowadays called cyclic homology, 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) is in 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?

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 12 / 29

The paracyclic object

In our paper in HHA we showed:

Theorem

Given 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.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 13 / 29

The paracyclic object

In our paper in HHA we showed:

Theorem

Given 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 = 0, b ◦ B + B ◦ b = 0.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 13 / 29

More and less precisely

The paracyclic structure means that in addition to the ordinary simplicial k-modue structure there are maps tn : C•(U, M) → C•(U, M) with certain props. If tn+1

n

= id one speaks of a cyclic module and one does get a mixed complex.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 14 / 29

More and less precisely

The paracyclic structure means that in addition to the ordinary simplicial k-modue structure there are maps tn : C•(U, M) → C•(U, M) with certain props. If tn+1

n

= 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.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 14 / 29

More and less precisely

The paracyclic structure means that in addition to the ordinary simplicial k-modue structure there are maps tn : C•(U, M) → C•(U, M) with certain props. If tn+1

n

= 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 paracyclic k-module a cyclic one, namely the qutient by im (id − tn+1

n

).

Definition

A paracyclic k-module quasicyclic if this quotient splits.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 14 / 29

More and less precisely

The paracyclic structure means that in addition to the ordinary simplicial k-modue structure there are maps tn : C•(U, M) → C•(U, M) with certain props. If tn+1

n

= 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 paracyclic k-module a cyclic one, namely the qutient by im (id − tn+1

n

).

Definition

A paracyclic k-module quasicyclic if this quotient splits.

Theorem

In this case, everything is fine.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 14 / 29

Example

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

Theorem

Let A be a twisted Calabi-Yau algebra, Hd(A, Ae) ≃ Aσ, and assume that σ is semisimple. Then H•(A, A) is a Batalin-Vilkovisk algebra. This applies to quantum groups, quantised universal enveloping algebras, quantum planes, quantum spheres...

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 15 / 29

Short version

Given a Hopf algebroid (U, A), a right U-module and left U-comodule M (with compatible induced left A-module structures), we will define a paracocyclic k-module structure on C •(U, M) := U⊗A• ⊗A M.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 16 / 29

Short version

Given a Hopf algebroid (U, A), a right U-module and left U-comodule M (with compatible induced left A-module structures), 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 this reduces to the original definition of Hopf-cyclic cohomology due to Connes and Moscovici. Present and unpresent people have extended their setting to stable anti Yetter-Drinfel’d modules (SaYD) M. Kaygun has observed that one can start with any module-comodule, B¨

• hm and Stefan have done Hopf algebroids,

but only for SaYD and implicitly, so one of our aims was to

• btain the most straigthforward generalisation of Connes and

Moscovici’s formulas to the general setting.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 16 / 29

The cyclic dual

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

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 17 / 29

The cyclic dual

There is a functorial way to associate to any cocyclic object a cyclic one. In contrast to common misbelief, this process can be extended to general paracocyclic objects (details later). If we apply this to C •(U, M) we obtain a paracyclic object whose structure can be seen best by applying a vector space isomorphism with C•(U, M) := M ⊗Aop ( ◮ U ✁ )⊗Aop•

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 17 / 29

The cyclic dual

There is a functorial way to associate to any cocyclic object a cyclic one. In contrast to common misbelief, this process can be extended to general paracocyclic objects (details later). If we apply this to C •(U, M) we obtain a paracyclic object whose structure can be seen best by applying a vector space isomorphism with C•(U, M) := M ⊗Aop ( ◮ U ✁ )⊗Aop• Taking here U = Ae, M = A one obtains on the nose the original definitions of the cyclic homology of an associative algebra.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 17 / 29

The cyclic dual

There is a functorial way to associate to any cocyclic object a cyclic one. In contrast to common misbelief, this process can be extended to general paracocyclic objects (details later). If we apply this to C •(U, M) we obtain a paracyclic object whose structure can be seen best by applying a vector space isomorphism with C•(U, M) := M ⊗Aop ( ◮ U ✁ )⊗Aop• Taking here U = Ae, M = A one obtains on the nose the original definitions of the cyclic homology of an associative algebra. If we then twist one of the two A-actions on itself by an automorphism σ ∈ Aut(A), we obtain for M = Aσ the twisted cyclic homology of Kustermans, Murphy, and Tuset.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 17 / 29

Constructing C •(U, M)

Ultimately, this construction goes back to Crainic.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 18 / 29

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⊗e

A•+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⊗Aeamb−b ✄ u0⊗A· · ·⊗Aun ✁ a⊗Aem | a, b ∈ A}.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 18 / 29

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⊗e

A•+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⊗Aeamb−b ✄ u0⊗A· · ·⊗Aun ✁ a⊗Aem | 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, when A = k, that is, U is an ordinary Hopf algebra, then we have u+ ⊗k u− = u(1) ⊗k S(u(2)).

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 18 / 29

Paracyclic objects in abelian categories

Paracyclic objects are like cyclic objects, only T := tn+1 is not required to be the identity id:

Definition

A paracyclic object is a simplical object (C•, b•, s•) equipped with morphisms t : Cn → Cn that satisfy (on Cn) bit = −tbi−1, sit = −tsi−1, b0t = (−1)nbn, s0t = (−1)nt2sn, 1 ≤ i ≤ n.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 19 / 29

Paracyclic objects in abelian categories

Paracyclic objects are like cyclic objects, only T := tn+1 is not required to be the identity id:

Definition

A paracyclic object is a simplical object (C•, b•, s•) equipped with morphisms t : Cn → Cn that satisfy (on Cn) bit = −tbi−1, sit = −tsi−1, b0t = (−1)nbn, s0t = (−1)nt2sn, 1 ≤ i ≤ n. T commutes with all the paracyclic generators t, bi, sj. As a consequence, a cyclic object can be attached to any paracyclic

• bject by passing to the coinvariants C/im (id − T) of T. In

well-behaved cases, there is no loss of homological information. Paracocyclic objects are paracyclic objects in the dual category.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 19 / 29

The paracocyclic structure

The paracocyclic structure on B•(U, M) is given by the following 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)u0 (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.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 20 / 29

Passing to ⊗Uop gives

Theorem

C •(U, M) := U⊗A• ⊗A M carries for all Hopf algebroids (U, A) and right modules-left comodules M a canonical paracocyclic k-module structure with codegeneracies and cofaces δi(z ⊗A m) =    1 ⊗A u1 ⊗A · · · ⊗A un ⊗A m u1 ⊗A · · · ⊗A ∆(ui) ⊗A · · · ⊗A un ⊗A m u1 ⊗A · · · ⊗A un ⊗A m(−1) ⊗A m(0) if i = 0, if 1 ≤ i ≤ n, if i = n + 1, δj(m) = 1 ⊗A m m(−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 ⊗A m) = u1

−(1)u2 ⊗A · · · ⊗A u1 −(n−1)un ⊗A u1 −(n)m(−1) ⊗A m(0)u1 +.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 21 / 29

The dual theory

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

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 22 / 29

The dual theory

Secondly, we generalise the concept of cyclic duality to a functor that assigns paracyclic objects to arbitrary paracocyclic ones and we compute the cyclic dual (C•(U, M), d•, s•, t•) of the above paracocyclic k-module. We then provide an isomorphism of this with the paracyclic module M ⊗op

A ( ◮U✁ )⊗op

A • whose structure maps are given by

di(m ⊗op

A x)=

   m ⊗op

A u1 ⊗op A · · · ⊗op A ε(un) ◮ un−1

m ⊗op

A · · · ⊗op A un−iun−i+1 ⊗op A · · ·

mu1 ⊗op

A u2 ⊗op A · · · ⊗op A un

if i =0, if 1≤i ≤n − if i =n, si(m ⊗op

A x)=

   m ⊗op

A u1 ⊗op A · · · ⊗op A un ⊗op A 1

m ⊗op

A · · · ⊗op A un−i ⊗op A 1 ⊗op A un−i+1 ⊗op A · · ·

m ⊗op

A 1 ⊗op A u1 ⊗op A · · · ⊗op A un

if i =0, if 1≤i ≤n − if i =n, tn(m ⊗op

A x)=m(0)u1 + ⊗op A u2 + ⊗op A · · · ⊗op A un + ⊗op A un − · · · u1 −m(−1),

where we abbreviate x := u1 ⊗op

A · · · ⊗op A un.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 22 / 29

How duality works

All one needs is in LNM1289 and Loday. One just has to compose the usual cyclic duality with a suitable autoequivalence

• f the cyclic category to obtain a functor that does lift to

paracocyclic objects (just not to a proper duality).

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 23 / 29

How duality works

All one needs is in LNM1289 and Loday. One just has to compose the usual cyclic duality with a suitable autoequivalence

• f the cyclic category to obtain a functor that does lift to

paracocyclic objects (just not to a proper duality). Concretely, we deifne:

Definition

The cyclic dual of a paracocyclic k-module C • = (C •, δ•, σ•, τ•) is the cyclic k-module ˆ C• := (ˆ C•, d•, s•, t•), where ˆ Cn := C n, and di := σn−(i+1) : ˆ Cn → ˆ Cn−1, 0 ≤ i < n, dn := σn−1 ◦ τn : ˆ Cn → ˆ Cn−1, si := δn−(i+1) : ˆ Cn−1 → ˆ Cn, 0 ≤ i < n, tn := τn : ˆ Cn → ˆ Cn.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 23 / 29

Lie-Rinehart algebras (= Lie algebroids) I

Let (A, L) be a Lie-Rinehart algebra over k: L is a k-Lie algebra and a module over the commutative k-algebra A, but it also acts via derivations ∂X on A such that [X, aY ] = (∂Xa)Y + a[X, Y ], ∂aX = a∂X, a ∈ A, X, Y ∈ L.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 24 / 29

Lie-Rinehart algebras (= Lie algebroids) I

Let (A, L) be a Lie-Rinehart algebra over k: L is a k-Lie algebra and a module over the commutative k-algebra A, but it also acts via 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-Rinehart algebras.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 24 / 29

Lie-Rinehart algebras (= Lie algebroids) I

Let (A, L) be a Lie-Rinehart algebra over k: L is a k-Lie algebra and a module over the commutative k-algebra A, but it also acts via 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-Rinehart

• algebras. Poisson algebras A with L = Ω1, [da, db] = d{a, b}.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 24 / 29

Lie-Rinehart algebras (= Lie algebroids) I

Let (A, L) be a Lie-Rinehart algebra over k: L is a k-Lie algebra and a module over the commutative k-algebra A, but it also acts via 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-Rinehart

• algebras. Poisson algebras A with L = Ω1, [da, db] = d{a, b}.

U = U(A, L) is the universal k-algebra equipped with two maps ιA : AtoU, ιL : LtoU

• f 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.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 24 / 29

Lie-Rinehart algebras (Lie algebroids) II

Xu: U carries the structure of a bialgebroid with η(− ⊗ 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 ˆ ∆ : U → U ⊗A U. The counit is similarly given by the extension

• f the anchor ˆ

ε to U.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 25 / 29

Lie-Rinehart algebras (Lie algebroids) II

Xu: U carries the structure of a bialgebroid with η(− ⊗ 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 ˆ ∆ : U → U ⊗A U. The counit is similarly given by the extension

• f the anchor ˆ

ε to U. Easy observation: The Galois map is bijective, the translation map 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 universality

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 25 / 29

Lie-Rinehart algebras (Lie algebroids) II

Xu: U carries the structure of a bialgebroid with η(− ⊗ 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 ˆ ∆ : U → U ⊗A U. The counit is similarly given by the extension

• f the anchor ˆ

ε to U. Easy observation: The Galois map is bijective, the translation map 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 universality By the way: Lie-Rinehart algebras are a good example that demonstrates that there is no adjoint action of a Hopf algebroid

• n itself, ad(u)v := u+vu− is ill-defined. So U is in general not

a module algebra over itself.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 25 / 29

Towards Hopf algebroids: Ae-rings

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

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 26 / 29

Towards Hopf algebroids: Ae-rings

Bialgebroids generalise bialgebras by replacing the base ring k by a possibly noncommutative base algebra A. For starters, they are 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 ◮ m ◭ b := nη(b ⊗k a), a, b ∈ A, n ∈ N.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 26 / 29

Towards Hopf algebroids: Ae-rings

Bialgebroids generalise bialgebras by replacing the base ring k by a possibly noncommutative base algebra A. For starters, they are 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 ◮ m ◭ b := nη(b ⊗k a), a, b ∈ A, n ∈ N. In particular, U itself carries two left and two right A-actions all commuting with each other. Usually we consider U as an Ae-module using a ✄ u ✁ b, and otherwise we write e.g. ◮ U ✁ .

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 26 / 29

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.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 27 / 29

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!

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 27 / 29

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,

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

◮ U ✁ ⊗A ✄ U ◭ :

U×AU :=

• i

ui⊗Avi ∈ U⊗AU |

• i

a ◮ ui⊗Avi =

• i

ui⊗Avi ◭ a

• .

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 27 / 29

Bialgebroids (×A-bialgebras) II

Similarly, A is not an Ae-ring unless A is commutative. To handle this one needs the canonical map π : Endk(A) → A, ϕ → ϕ(1), and the fact that Endk(A) is an Ae-ring.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 28 / 29

Bialgebroids (×A-bialgebras) II

Similarly, A is not an Ae-ring unless A is commutative. To handle this one needs the canonical map π : Endk(A) → A, ϕ → 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 two homomorphisms ˆ ∆ : U → U ×A U, ˆ ε : U → Endk(A) of Ae-rings such that U is a coalgebra in Ae-Mod via ∆ = ι ◦ ˆ ∆, ε = π ◦ ˆ ε. There is an analogous notion of a right bialgebroid in which for example A has a canonical right U-module structure.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 28 / 29

Hopf algebroids (×A-Hopf algebras)

Let U be a bialgebroid and define the Galois map of U β : ◮ U ⊗Aop U ✁ → U ✁ ⊗A ✄ U, u ⊗Aop v → u(1) ⊗A u(2)v, where ◮ U ⊗Aop U ✁ = U ⊗k U/span{a ◮ u ⊗k v − u ⊗k v ✁ a}.

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 29 / 29

Hopf algebroids (×A-Hopf algebras)

Let U be a bialgebroid and define the Galois map of U β : ◮ U ⊗Aop U ✁ → U ✁ ⊗A ✄ U, u ⊗Aop v → u(1) ⊗A u(2)v, where ◮ U ⊗Aop U ✁ = U ⊗k U/span{a ◮ u ⊗k v − u ⊗k v ✁ a}. For bialgebras over fields β is bijective if and only if U is a Hopf algebra with β−1(u ⊗k v) := u(1) ⊗ S(u(2))v, where S is the antipode of U. This motivates:

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 29 / 29

Hopf algebroids (×A-Hopf algebras)

Let U be a bialgebroid and define the Galois map of U β : ◮ U ⊗Aop U ✁ → U ✁ ⊗A ✄ U, u ⊗Aop v → u(1) ⊗A u(2)v, where ◮ U ⊗Aop U ✁ = U ⊗k U/span{a ◮ u ⊗k v − u ⊗k v ✁ a}. For bialgebras over fields β is bijective if and only if U is a Hopf algebra with β−1(u ⊗k v) := u(1) ⊗ S(u(2))v, where S is the antipode of U. This motivates:

Definition (Schauenburg)

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

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 29 / 29

Hopf algebroids (×A-Hopf algebras)

Let U be a bialgebroid and define the Galois map of U β : ◮ U ⊗Aop U ✁ → U ✁ ⊗A ✄ U, u ⊗Aop v → u(1) ⊗A u(2)v, where ◮ U ⊗Aop U ✁ = U ⊗k U/span{a ◮ u ⊗k v − u ⊗k v ✁ a}. For bialgebras over fields β is bijective if and only if U is a Hopf algebra with β−1(u ⊗k v) := u(1) ⊗ S(u(2))v, where S is the antipode 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 B¨

• hm and

Szlach´ anyi which are left and right bialgebroids. The precise axioms would barely fit on one slide. Example: U = Ae: η : Ae → Ae is the identity, ε : Ae → A is the multiplication map, ∆(a ⊗k b) = (a ⊗k 1) ⊗k (1 ⊗k b).

Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 29 / 29