Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting

Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Ramón González Rodríguez

http://www.dma.uvigo.es/˜rgon/

Departamento de Matemática Aplicada II. Universidade de Vigo

Red Nc-Alg: Escuela de investigación avanzada en Álgebra no Conmutativa

Granada, 9-13 de noviembre de 2015

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups Outline

1 Magmas, comagmas, monoids and comonoids

2 Hopf algebras

3 Weak Hopf algebras

4 Hopf quasigroups

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

1 Magmas, comagmas, monoids and comonoids

2 Hopf algebras

3 Weak Hopf algebras

4 Hopf quasigroups

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

In the following C denotes a strict monoidal category with tensor product ⊗, unit object K.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Deﬁnition.

A monoid is a unital magma A = (A, ηA, µA) where

µA ◦ (A ⊗ µA) = µA ◦ (µA ⊗ A).

Given two monoids A = (A, ηA, µA) and B = (B, ηB , µB ), f : A → B is a monoid morphism if it is a morphism of unital magmas.

Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Deﬁnition.

By a unital magma in C we understand a triple A = (A, ηA, µA) where A is an object in C and ηA : K → A (unit), µA : A ⊗ A → A (product) are morphisms in C such that µA ◦ (A ⊗ ηA) = idA = µA ◦ (ηA ⊗ A). Given two unital magmas A = (A, ηA, µA) and B = (B, ηB , µB ), f : A → B is a morphism of unital magmas if

µB ◦ (f ⊗ f ) = f ◦ µA, f ◦ ηA = ηB .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Deﬁnition.

µB ◦ (f ⊗ f ) = f ◦ µA, f ◦ ηA = ηB .

Deﬁnition.

A monoid is a unital magma A = (A, ηA, µA) where

µA ◦ (A ⊗ µA) = µA ◦ (µA ⊗ A).

Given two monoids A = (A, ηA, µA) and B = (B, ηB , µB ), f : A → B is a monoid morphism if it is a morphism of unital magmas.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Deﬁnition.

A comonoid is a counital comagma D = (D, εD , δD ) where

(δD ⊗ D) ◦ δD = (D ⊗ δD ) ◦ δD .

If D = (D, εD , δD ) and E = (E, εE , δE ) are comonoids, f : D → E is a morphism of comonoids if it is a a morphism of counital comagmas.

Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Deﬁnition.

By duality, a counital comagma in C is a triple D = (D, εD , δD ) where D is an object in C and εD : D → K (counit), δD : D → D ⊗ D (coproduct) are morphisms in C such that (εD ⊗ D) ◦ δD = idD = (D ⊗ εD ) ◦ δD . If D = (D, εD , δD ) and E = (E, εE , δE ) are counital comagmas, f : D → E is a morphism of counital comagmas if

(f ⊗ f ) ◦ δD = δE ◦ f , εE ◦ f = εD .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Deﬁnition.

(f ⊗ f ) ◦ δD = δE ◦ f , εE ◦ f = εD .

Deﬁnition.

A comonoid is a counital comagma D = (D, εD , δD ) where

(δD ⊗ D) ◦ δD = (D ⊗ δD ) ◦ δD .

If D = (D, εD , δD ) and E = (E, εE , δE ) are comonoids, f : D → E is a morphism of comonoids if it is a a morphism of counital comagmas.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting If A, B are unital magmas (monoids) in C, the object A⊗B is a unital magma (monoid) in C where

ηA⊗B = ηA ⊗ ηB , µA⊗B = (µA ⊗ µB ) ◦ (A ⊗ cB,A ⊗ B).

In a dual way, if D, E are counital comagmas (comonoid) in C, D ⊗ E is a counital comagma (comonoid) in C where

εD⊗E = εD ⊗ εE , δD⊗E = (D ⊗ cD,E ⊗ E) ◦ (δD ⊗ δE ).

Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Assume that C is braided with braiding c.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting In a dual way, if D, E are counital comagmas (comonoid) in C, D ⊗ E is a counital comagma (comonoid) in C where

εD⊗E = εD ⊗ εE , δD⊗E = (D ⊗ cD,E ⊗ E) ◦ (δD ⊗ δE ).

Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Assume that C is braided with braiding c.

If A, B are unital magmas (monoids) in C, the object A⊗B is a unital magma (monoid) in C where

ηA⊗B = ηA ⊗ ηB , µA⊗B = (µA ⊗ µB ) ◦ (A ⊗ cB,A ⊗ B).

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Assume that C is braided with braiding c.

If A, B are unital magmas (monoids) in C, the object A⊗B is a unital magma (monoid) in C where

ηA⊗B = ηA ⊗ ηB , µA⊗B = (µA ⊗ µB ) ◦ (A ⊗ cB,A ⊗ B).

In a dual way, if D, E are counital comagmas (comonoid) in C, D ⊗ E is a counital comagma (comonoid) in C where

εD⊗E = εD ⊗ εE , δD⊗E = (D ⊗ cD,E ⊗ E) ◦ (δD ⊗ δE ).

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Deﬁnition

Let A be an monoid. We say that (M, ϕM ) is a left A-module if M is an object in C and ϕM : A ⊗ M → M is a morphism in C (called the action) satisfying

ϕM ◦ (ηA ⊗ M) = idM , ϕM ◦ (A ⊗ ϕM ) = ϕM ◦ (µA ⊗ M).

Given two left A-modules (M, ϕM ) and (N, ϕN ), f : M → N is a morphism of left A-modules if ϕN ◦ (A ⊗ f ) = f ◦ ϕM .

We denote the category of right A-modules by AC. In an analogous way we deﬁne the category of right H-modules and we denote it by CA. In this case the right action will be denoted by φM .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Deﬁnition

Let D be a comonoid. We say that (M,%M ) is a left D-comodule if M is an object in C and %M : M → D ⊗ M is a morphism in C (called the coaction) satisfying

(εD ⊗ M) ◦ %M = idM , (D ⊗ %M ) ◦ %M = (δH ⊗ M) ◦ %M .

Given two left D-comodules (M,%M ) and (N,%N ), f : M → N is a morphism of left D-comodules if %N ◦ f = (D ⊗ f ) ◦ %M . We denote the category of left H-comodules by D C. Analogously, CD denotes the category of right D-comodules. In this case the right coaction will be denoted by ρM .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

1 Magmas, comagmas, monoids and comonoids

2 Hopf algebras

3 Weak Hopf algebras

4 Hopf quasigroups

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

The notion of Hopf algebra emerged slowly from the work of the topologists in the 1940’s dealing with the cohomology of compact Lie groups and their homogeneous spaces......

Pierre Cartier, A primer of Hopf algebras, preprints.ihes.fr/2006/M/M-06-40.pdf

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Deﬁnition

A Hopf algebra H is an object in C with a monoid structure (H, ηH , µH ) and a comonoid structure (H, εH , δH ) such δH and εH are monoid morphisms and there exists a morphism λH : H → H in C (called the antipode of H) veriﬁying:

λH ∗ idH = εH ⊗ ηH = idH ∗ λH .

Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

In the following C denotes a strict braided monoidal category with braiding c.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

In the following C denotes a strict braided monoidal category with braiding c.

Deﬁnition

λH ∗ idH = εH ⊗ ηH = idH ∗ λH .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Note that δH and εH are monoid morphisms, if and only if,

µH ◦ δH = µH⊗H ◦ (δH ⊗ δH ), δH ◦ ηH = ηH ⊗ ηH and εH ◦ µH = εH ⊗ εH , εH ◦ ηH = idK . Then, if H is a Hopf algebra, the assertions

δH and εH are monoid morphisms.

µH and ηH are comonoid morphisms. are equivalent.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Deﬁnition

A Hopf algebra in C is commutative if µH ◦ cH,H = µH and is cocommutative if cH,H ◦ δH = δH .

Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Theorem

If H is a Hopf algebra in C, the antipode λH is unique, antimultiplicative, anticomultiplicative and leaves the unit ηH and the counit εH invariant:

λH ◦ µH = µH ◦ (λH ⊗ λH ) ◦ cH,H , δH ◦ λH = cH,H ◦ (λH ⊗ λH ) ◦ δH ,

λH ◦ ηH = ηH , εH ◦ λH = εH .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Theorem

If H is a Hopf algebra in C, the antipode λH is unique, antimultiplicative, anticomultiplicative and leaves the unit ηH and the counit εH invariant:

λH ◦ µH = µH ◦ (λH ⊗ λH ) ◦ cH,H , δH ◦ λH = cH,H ◦ (λH ⊗ λH ) ◦ δH ,

λH ◦ ηH = ηH , εH ◦ λH = εH .

Deﬁnition

A Hopf algebra in C is commutative if µH ◦ cH,H = µH and is cocommutative if cH,H ◦ δH = δH .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Examples Group algebras are the natural examples of cocommutative Hopf algebras. Let G be a group with unit e, and R a commutative ring. The group algebra is the direct product in R-Mod M RG = Rg g∈G

with the product deﬁned by µRG (g ⊗ h) = g.h. The unit element is e, and RG is a cocommutative Hopf algebra, with coproduct δRG , counit εRG and antipode λRG given by the formulas:

−1 δRG (g) = g ⊗ g, εRG (g) = 1R , λRG (r) = g .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Examples If G is a ﬁnite group with unit element e we can consider (RG)∗ = GR by M GR = Rfg g∈G

with fg : RG → R deﬁned by fg (h) = δg,h. Then, GR is a Hopf algebra, called the dual Hopf algebra of RG. The algebra P structure is given by the formulas fg fh = δg,hfg and 1GR = g∈G fg . The coalgebra structure is X δGR (fg ) = fgh−1 ⊗ fh, εGR (fg ) = δg,e . h∈G

The antipode is given by λGR (fg ) = fg −1 .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Deﬁnition. Let H, B two Hopf algebras. We will say that f : H → B is a morphism of Hopf algebras in f is a monoid-comonoid morphism.

Theorem.

If f : H → B is a morphism of Hopf algebras, λB ◦ f = f ◦ λH .

Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Theorem. 2 If H is a commutative or cocommutative Hopf algebra, then λH = idH .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Theorem.

If f : H → B is a morphism of Hopf algebras, λB ◦ f = f ◦ λH .

Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Theorem. 2 If H is a commutative or cocommutative Hopf algebra, then λH = idH .

Deﬁnition. Let H, B two Hopf algebras. We will say that f : H → B is a morphism of Hopf algebras in f is a monoid-comonoid morphism.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Theorem. 2 If H is a commutative or cocommutative Hopf algebra, then λH = idH .

Deﬁnition. Let H, B two Hopf algebras. We will say that f : H → B is a morphism of Hopf algebras in f is a monoid-comonoid morphism.

Theorem.

If f : H → B is a morphism of Hopf algebras, λB ◦ f = f ◦ λH .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Deﬁnition. H Let H be a Hopf algebra in C with invertible antipode. We shall denote by H YD the category of left Yetter-Dinfeld modules over H. That is, M = (M, ϕM ,%M ) is an object H in H YD if (M, ϕM ) is a left H-module, (M,%M ) is a left H-comodule and (µH ⊗ M) ◦ (H ⊗ cM,H ) ◦ ((rM ◦ ϕM ) ⊗ H) ◦ (H ⊗ cH,M ) ◦ (δH ⊗ M)

= (µH ⊗ ϕM ) ◦ (H ⊗ cH,H ⊗ M) ◦ (δH ⊗ %M ) or, equivalently,

%M ◦ ϕM = (µH ⊗ M) ◦ (H ⊗ cM,H ) ◦ (µH ⊗ ϕM ⊗ H) ◦ (H ⊗ cH,M ⊗ M ⊗ H)◦

(δH ⊗ %M ⊗ λH ) ◦ (H ⊗ cH,M ) ◦ (δH ⊗ M). H The morphisms in H YD are morphisms of left modules and comodules.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

H The category H YD is a strict monoidal category with the usual tensor product in C, H for M, N in H YD, M ⊗ N has the diagonal module and comodule structures given by

ϕM⊗N = (ϕM ⊗ ϕN ) ◦ (H ⊗ cH,M ⊗ N) ◦ (δH ⊗ M ⊗ N),

%M⊗N = (µH ⊗ M ⊗ N) ◦ (H ⊗ cM,H ⊗ N) ◦ (%M ⊗ %N ). It is also a braided category, where the braiding is given by

tM,N : M ⊗ N → N ⊗ N, tM,N = (ϕN ⊗ M) ◦ (H ⊗ cM,N ) ◦ (%M ⊗ N).

It is immediate to see that tM,N is a natural isomorphism with inverse

−1 −1 −1 −1 tM,N = cM,N ◦ (ϕN ⊗ M) ◦ (λH ⊗ N ⊗ M) ◦ (cH,N ⊗ M) ◦ (N ⊗ %M ).

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

1 Magmas, comagmas, monoids and comonoids

2 Hopf algebras

3 Weak Hopf algebras

4 Hopf quasigroups

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Quantum groupoids (weak Hopf algebras) appear naturally in the theory of dynamical deformations of quantum groups......

D. Nikshych, L. Vainerman, Finite quantum groupoids and their applications, New Directions in Hopf Algebras, MSRI Publications, 43 (2002), 211-262.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting By weak Hopf algebras (or quantum groupoids in the terminology of Nikshych and Vainerman ) we understand the objects introduced iby G. Böhm, F. Nill, K. Szlachányi, in a category of vector spaces, as a generalization of Hopf algebras. Here, we recall the deﬁnition of these objects and some relevant results in a braided setting.

Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

In this section we also assume in C that every idempotent morphism splits, i.e., if ∇ : Y → Y is such that ∇ = ∇ ◦ ∇, there exist an object Z and morphisms i : Z → Y and p : Y → Z such that ∇ = i ◦ p, p ◦ i = idZ . Note that, in these conditions, Z, p and i are unique up to isomorphism. The categories satisfying this property constitute a broad class that includes, among others, the categories with equalizers or with coequalizers.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

By weak Hopf algebras (or quantum groupoids in the terminology of Nikshych and Vainerman ) we understand the objects introduced iby G. Böhm, F. Nill, K. Szlachányi, in a category of vector spaces, as a generalization of Hopf algebras. Here, we recall the deﬁnition of these objects and some relevant results in a braided setting.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Deﬁnition.

A weak Hopf algebra H is an object in C with a monoid structure (H, ηH , µH ) and a comonoid structure (H, εH , δH ) such that the following axioms hold:

(i) δH ◦ µH = (µH ⊗ µH ) ◦ δH⊗H ,

(ii) εH ◦ µH ◦ (µH ⊗ H) = (εH ⊗ εH ) ◦ (µH ⊗ µH ) ◦ (H ⊗ δH ⊗ H) −1 = (εH ⊗ εH ) ◦ (µH ⊗ µH ) ◦ (H ⊗ (cH,H ◦ δH ) ⊗ H),

(iii) (δH ⊗ H) ◦ δH ◦ ηH = (H ⊗ µH ⊗ H) ◦ (δH ⊗ δH ) ◦ (ηH ⊗ ηH ) −1 = (H ⊗ (µH ◦ cH,H ) ⊗ H) ◦ (δH ⊗ δH ) ◦ (ηH ⊗ ηH ).

(iv) There exists a morphism λH : H → H in C (called the antipode of H) veriﬁying: (iv-1) idH ∗ λH = ((εH ◦ µH ) ⊗ H) ◦ (H ⊗ cH,H ) ◦ ((δH ◦ ηH ) ⊗ H), (iv-2) λH ∗ idH = (H ⊗ (εH ◦ µH )) ◦ (cH,H ⊗ H) ◦ (H ⊗ (δH ◦ ηH )), (iv-3) λH ∗ idH ∗ λH = λH .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

L R L R If we deﬁne the morphisms ΠH , ΠH , ΠH and ΠH by L ΠH = ((εH ◦ µH ) ⊗ H) ◦ (H ⊗ cH,H ) ◦ ((δH ◦ ηH ) ⊗ H), R ΠH = (H ⊗ (εH ◦ µH )) ◦ (cH,H ⊗ H) ◦ (H ⊗ (δH ◦ ηH )), L ΠH = (H ⊗ (εH ◦ µH )) ◦ ((δH ◦ ηH ) ⊗ H), R ΠH = ((εH ◦ µH ) ⊗ H) ◦ (H ⊗ (δH ◦ ηH )). it is straightforward to show that they are idempotent

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

The antipode is unique and the following equalities hold

λH ◦ ηH = ηH , εH ◦ λH = εH .

L R ΠH = idH ∗ λH , ΠH = λH ∗ idH . Moreover, we have that

L L L L R R R L L R R R ΠH ◦ ΠH = ΠH , ΠH ◦ ΠH = ΠH , ΠH ◦ ΠH = ΠH , ΠH ◦ ΠH = ΠH ,

L L L L R R R L L R R R ΠH ◦ ΠH = ΠH , ΠH ◦ ΠH = ΠH , ΠH ◦ ΠH = ΠH , ΠH ◦ ΠH = ΠH .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Also it is easy to show the formulas

L R L R L R ΠH = ΠH ◦ λH = λH ◦ ΠH , ΠH = ΠH ◦ λH = λH ◦ ΠH ,

L L R R R R L L ΠH ◦ λH = ΠH ◦ ΠH = λH ◦ ΠH , ΠH ◦ λH = ΠH ◦ ΠH = λH ◦ ΠH .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Examples. As group algebras are the natural examples of Hopf algebras, groupoid algebras provide examples of weak Hopf algebras. Let G be a a ﬁnite groupoid (i.e. groupoids with a ﬁnite number of objects), and R a commutative ring. The groupoid algebra is the direct product in R-Mod M RG = Rσ

σ∈G1

with the product of two morphisms being equal to their composition if the latter is deﬁned and 0 in otherwise, i.e. µRG (τ ⊗σ) = τ ◦σ if s(τ) = t(σ) and µRG (τ ⊗σ) = 0 if s(τ) 6= t(σ). The unit element is 1 = P id . The algebra RG is a RG x∈G0 x cocommutative weak Hopf algebra, with coproduct δRG , counit εRG and antipode λRG given by the formulas:

−˙1 δRG (σ) = σ ⊗ σ, εRG (σ) = 1, λRG (σ) = σ .

For the weak Hopf algebra RG the morphisms target and source are respectively,

L R ΠRG (σ) = idt(σ), ΠRG (σ) = ids(σ)

and λRG ◦ λRG = idRG , i.e. the antipode is involutory.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Examples. ∗ If G1 is ﬁnite, then RG is free of a ﬁnite rank as a R-module, hence GR = (RG) = HomR (RG, R) is a commutative weak Hopf algebra with involutory antipode. As R-module M GR = Rfσ σ∈G1

with hfσ, τi = δσ,τ . The algebra structure is given by the formulas fσfτ = δσ,τ fσ and 1 = P f . The coalgebra structure is GR σ∈G1 σ X X δ (f ) = f ⊗ f = f −1 ⊗ f , ε (f ) = δ . GR σ τ ρ σρ ρ GR σ σ,idt(σ) τρ=σ ρ∈G1

The antipode is given by λGR (fσ) = fσ−1 .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Deﬁnition. A morphism between weak Hopf algebras H and B is a morphism f : H → B which is comonoid morphism such that R R f ◦ ΠH = ΠB ◦ f , L L f ◦ ΠH = ΠB ◦ f , R L R L f ◦ ΠH ◦ ΠH = ΠB ◦ ΠB ◦ f , R f ◦ µH = µB ◦ (f ⊗ f ) ◦ ΩH , where R R ΩH = (H ⊗ µH ) ◦ (H ⊗ ΠH ⊗ H) ◦ (δH ⊗ H): H ⊗ H → H ⊗ H is an idempotent morphism.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Examples. If F : G → D is a functor between ﬁnite groupoids then

f = RF : RG → RD, f (σ) = F (σ)

is a morphism of weak Hopf algebras.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Theorem.

If H is a weak Hopf algebra in C, the antipode λH is antimultiplicative, anticomultiplicative, i.e.

λH ◦ µH = µH ◦ (λH ⊗ λH ) ◦ cH,H , δH ◦ λH = cH,H ◦ (λH ⊗ λH ) ◦ δH ,

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Given two left H-comodules (M,%M ) and (N,%N ), we denote by %M⊗N the morphism %M⊗N : M ⊗ N → H ⊗ M ⊗ N deﬁned by

%M⊗N = (µH ⊗ M ⊗ N) ◦ (H ⊗ cM,H ⊗ N) ◦ (%M ⊗ %N ).

Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Let H be a weak Hopf algebra. If (M, ϕM ) and (N, ϕN ) are left H-modules we denote by ϕM⊗N the morphism ϕM⊗N : H ⊗ M ⊗ N → M ⊗ N deﬁned by

ϕM⊗N = (ϕM ⊗ ϕN ) ◦ (H ⊗ cH,M ⊗ N) ◦ (δH ⊗ M ⊗ N).

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Let H be a weak Hopf algebra. If (M, ϕM ) and (N, ϕN ) are left H-modules we denote by ϕM⊗N the morphism ϕM⊗N : H ⊗ M ⊗ N → M ⊗ N deﬁned by

ϕM⊗N = (ϕM ⊗ ϕN ) ◦ (H ⊗ cH,M ⊗ N) ◦ (δH ⊗ M ⊗ N).

Given two left H-comodules (M,%M ) and (N,%N ), we denote by %M⊗N the morphism %M⊗N : M ⊗ N → H ⊗ M ⊗ N deﬁned by

%M⊗N = (µH ⊗ M ⊗ N) ◦ (H ⊗ cM,H ⊗ N) ◦ (%M ⊗ %N ).

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Let (M, ϕM ), (N, ϕN ) be left H-modules. Then the morphism

∇M⊗N = ϕM⊗N ◦ (ηH ⊗ M ⊗ N): M ⊗ N → M ⊗ N is idempotent. In this setting we denote by M × N the image of ∇M⊗N and by pM⊗N : M ⊗ N → M × N, iM⊗N : M × N → M ⊗ N the morphisms such that iM⊗N ◦ pM⊗N = ∇M⊗N and pM⊗N ◦ iM⊗N = idM×N . Using the deﬁnition of × we obtain that the object M × N is a left H-module with action

ϕM×N = pM⊗N ◦ ϕM⊗N ◦ (H ⊗ iM⊗N ): H ⊗ (M × N) → M × N.

Note that, if f : M → M0 and g : N → N0 are morphisms of left H-modules then (f ⊗ g) ◦ ∇M⊗N = ∇M0⊗N0 ◦ (f ⊗ g).

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

In a similar way, if (M,%M ) and (N,%N ) are left H-comodules the morphism

0 ∇M⊗N = (εH ⊗ M ⊗ N) ◦ %M⊗M : M ⊗ N → M ⊗ N

0 0 is idempotent. We denote by M N the image of ∇M⊗N and by pM⊗N : M⊗N → M N, 0 0 0 0 iM⊗N : M N → M ⊗ N the morphisms such that iM⊗N ◦ pM⊗N = ∇M⊗N and 0 0 pM⊗N ◦ iM⊗N = idM N . Using the deﬁnition of we obtain that the object M N is a left H-comodule with coaction

0 0 %M N = (H ⊗ pM⊗N ) ◦ %M⊗N ◦ iM⊗N : M N → H ⊗ (M N). If f : M → M0 and g : N → N0 are morphisms of left H-comodules then (f ⊗ g) ◦ 0 0 ∇M⊗N = ∇M0⊗N0 ◦ (f ⊗ g).

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Deﬁnition. H Let H be a weak Hopf algebra. We shall denote by H YD the category of left-left H Yetter-Drinfeld modules over H. That is, M = (M, ϕM ,%M ) is an object in H YD if (M, ϕM ) is a left H-module, (M,%M ) is a left H-comodule and

(i) (µH ⊗ M) ◦ (H ⊗ cM,H ) ◦ ((%M ◦ ϕM ) ⊗ H) ◦ (H ⊗ cH,M ) ◦ (δH ⊗ M)

= (µH ⊗ ϕM ) ◦ (H ⊗ cH,H ⊗ M) ◦ (δH ⊗ %M ).

(ii) (µH ⊗ ϕM ) ◦ (H ⊗ cH,H ⊗ M) ◦ ((δH ◦ ηH ) ⊗ %M ) = %M . H Let M, N in H YD. The morphism f : M → N is a morphism of left-left Yetter-Drinfeld modules if f ◦ ϕM = ϕN ◦ (H ⊗ f ) and (H ⊗ f ) ◦ %M = %N ◦ f .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Note that if (M, ϕM ,%M ) is a left-left Yetter-Drinfeld module then (ii) is equivalent to

(iii) ((εH ◦ µH ) ⊗ ϕM ) ◦ (H ⊗ cH,H ⊗ M) ◦ (δH ⊗ %M ) = ϕM . The conditions (i) and (ii) of the last deﬁnition can also be restated in the following way: H suppose that (M, ϕM ) ∈ H C and (M,%M ) ∈ C, then M is a left-left Yetter-Drinfeld module if and only if

%M ◦ ϕM = (µH ⊗ M) ◦ (H ⊗ cM,H )◦

(((µH ⊗ ϕM ) ◦ (H ⊗ cH,H ⊗ M) ◦ (δH ⊗ %M )) ⊗ λH ) ◦ (H ⊗ cH,M ) ◦ (δH ⊗ M).

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Theorem. H If the antipode of a weak Hopf algebra H is invertible, H YD is a non-strict braided monoidal category.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

For two left-left Yetter-Drinfeld modules (M, ϕM ,%M ), (N, ϕN ,%N ) the tensor product 0 is deﬁned as object as the image of ∇M⊗N = ∇M⊗N . Using that, M × N = M N

, we obtain that M × N is a left-left Yetter-Drinfeld module with the following action and coaction: ϕM×N = pM⊗N ◦ ϕM⊗N ◦ (H ⊗ iM⊗N )

%M×N = (H ⊗ pM⊗N ) ◦ %M⊗N ◦ iM⊗N .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

L The base object is HL = Im(ΠH ) or, equivalently, the equalizer of δH and

1 L ζH = (H ⊗ ΠH ) ◦ δH ,

2 R or the equalizer of δH and ζH = (H ⊗ΠH )◦ δH . The structure of left-left Yetter-Drinfeld module for HL is the one derived of the following morphisms

ϕHL = pL ◦ µH ◦ (H ⊗ iL),%HL = (H ⊗ pL) ◦ δH ◦ iL.

L where pL : H → HL and iL : HL → H are the morphism such that ΠH = iL ◦ pL and pL ◦ iL = idHL .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

The unit constrains are:

lM = ϕM ◦ (iL ⊗ M) ◦ iHL⊗M : HL × M → M,

L rM = ϕM ◦ cM,H ◦ (M ⊗ (ΠH ◦ iL)) ◦ iM⊗HL : M × HL → M. These morphisms are isomorphisms with inverses:

−1 lM = pHL⊗M ◦ (pL ⊗ ϕM ) ◦ ((δH ◦ ηH ) ⊗ M): M → HL × M,

−1 rM = pM⊗HL ◦ (ϕM ⊗ pL) ◦ (H ⊗ cH,M ) ◦ ((δH ◦ ηH ) ⊗ M): M → M × HL.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

H If M, N, P are objects in the category H YD, the associativity constrains are deﬁned by aM,N,P = pM⊗(N×P)◦(M⊗pN⊗P )◦(iM⊗N ⊗P)◦i(M×N)⊗P :(M×N)×P → M×(N×P). where the inverse is the morphism

−1 aM,N,P = p(M×N)⊗P ◦(pM⊗N ⊗P)◦(M⊗iN⊗P )◦iM⊗(N×P) : M×(N×P) → (M×N)×P

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

If γ : M → M0 and φ : N → N0 are morphisms in the category, then

0 0 γ × φ = pM0×N0 ◦ (γ ⊗ φ) ◦ iM⊗N : M × N → M × N

H is a morphism in H YD and (γ0 × φ0) ◦ (γ × φ) = (γ0 ◦ γ) × (φ0 ◦ φ),

0 0 00 0 0 00 H where γ : M → M and φ : N → N are morphisms in H YD.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Finally, the braiding is

τM,N = pN⊗M ◦ tM,N ◦ iM⊗N : M × N → N × M where tM,N = (ϕN ⊗ M) ◦ (H ⊗ cM,N ) ◦ (%M ⊗ N): M ⊗ N → N ⊗ M

The morphism τM,N is a natural isomorphism with inverse:

−1 0 τM,N = pM⊗N ◦ tM,N ◦ iN⊗M : N × M → M × N where

0 −1 −1 −1 tM,N = cM,N ◦ (ϕN ⊗ M) ◦ (λH ⊗ N ⊗ M) ◦ (cH,N ⊗ M) ◦ (N ⊗ %M ).

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

1 Magmas, comagmas, monoids and comonoids

2 Hopf algebras

3 Weak Hopf algebras

4 Hopf quasigroups

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

.. our axioms unify quasigroups and Mal’tsev algebras just as Hopf algebras historically uniﬁed groups and Lie algebras.

J. Klim, S. Majid, Hopf quasigroups and the algebraic 7-sphere, J. Algebra 323 (2010), 3067-3110.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Deﬁnition.

A Hopf quasigroup H in C is a unital magma (H, ηH , µH ) and a comonoid (H, εH , δH ) such that the following axioms hold:

(i) εH and δH are morphisms of unital magmas.

(ii) There exists λH : H → H in C (called the antipode of H) such that:

(ii-1) µH ◦ (λH ⊗ µH ) ◦ (δH ⊗ H) = εH ⊗ H = µH ◦ (H ⊗ µH ) ◦ (H ⊗ λH ⊗ H) ◦ (δH ⊗ H).

(ii-2) µH ◦ (µH ⊗ H) ◦ (H ⊗ λH ⊗ H) ◦ (H ⊗ δH ) = H ⊗ εH = µH ◦ (µH ⊗ λH ) ◦ (H ⊗ δH ).

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

If H is a Hopf quasigroup, the antipode is unique, antimultiplicative, anticomultiplicative and leaves the unit and the counit invariable:

λH ◦ µH = µH ◦ (λH ⊗ λH ) ◦ cH,H , δH ◦ λH = cH,H ◦ (λH ⊗ λH ) ◦ δH ,

λH ◦ ηH = ηH , εH ◦ λH = εH . Note that,

µH ◦ (λH ⊗ idH ) ◦ δH = µH ◦ (idH ⊗ λH ) ◦ δH = εH ⊗ ηH . holds.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Let H and B be Hopf quasigroups. We say that f : H → B is a morphism of Hopf quasigroups if it is a morphism of unital magmas and comonoids. In this case

λB ◦ f = f ◦ λH

Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

A Hopf quasigroup H is cocommutative if cH,H ◦ δH = δH . In this case, as in the Hopf algebra setting, we have that λH ◦ λH = idH .

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

A Hopf quasigroup H is cocommutative if cH,H ◦ δH = δH . In this case, as in the Hopf algebra setting, we have that λH ◦ λH = idH .

Let H and B be Hopf quasigroups. We say that f : H → B is a morphism of Hopf quasigroups if it is a morphism of unital magmas and comonoids. In this case

λB ◦ f = f ◦ λH

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Examples. A quasigroup is a set Q together with a product such that for any two elements u, v ∈ Q the equations ux = v, xu = v, uv = x have unique solutions in Q. A quasigroup L which contains an element eL such that

ueL = u = eLu for every u ∈ L is called a loop.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting If L is an I.P. loop, it is easy to show that for all u ∈ L the element u−1 is unique and

−1 −1 u u = eL = uu .

Moreover, for all u, v ∈ L, the equality

(uv)−1 = v −1u−1

holds.

Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

A loop L is said to be a loop with the inverse property (for brevity an I.P. loop) if and only if, to every element u ∈ L, there corresponds an element u−1 ∈ L such that the equations u−1(uv) = v = (vu)u−1 hold for every v ∈ L.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

If L is an I.P. loop, it is easy to show that for all u ∈ L the element u−1 is unique and

−1 −1 u u = eL = uu .

Moreover, for all u, v ∈ L, the equality

(uv)−1 = v −1u−1 holds.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

Let R be a commutative ring and L and I.P. loop. Then, M RL = Ru u∈L is a cocommutative Hopf quasigroup with product given by the linear extension of the one deﬁned in L and

−1 δRL(u) = u ⊗ u, εRL(u) = 1R , λRL(u) = u on the basis elements.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups

References: J. N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez, Weak braided Hopf algebras, Indiana Univ. Math. J. 57 (2008), 2423-2458. J.N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez and C. So- neira Calvo, Projections and Yetter-Drinfeld modules over Hopf (co)quasigroups, J. Algebra, 443 (2015), 153-199. G. Böhm, F. Nill, K. Szlachányi, Weak Hopf algebras, I. Integral theory and C ∗- structure, J. Algebra, 221 (1999), 385-438. G. Böhm, J. Gómez Torrecillas, E. López Centella, On the category of weak bial- gebras, J. Algebra, 399 (2014), 801-844. C. Kassel, Quantum Groups, GTM 155, Springer-Verlag, New York, 1995. J. Klim, S. Majid, Hopf quasigroups and the algebraic 7-sphere, J. Algebra 323 (2010), 3067-3110. S. Majid, A quantum groups primer, London Mathematical Society Lecture Note Series 292, Cambridge University Press, 2002. S. Montgomery, Hopf algebras and their actions on rings, Regional Conference Se- ries in Mathematics 82, Providence, Rhode Island: American Mathematical Society, 1993. D.E. Radford, Hopf algebras, World Scientiﬁc 2012. M.E. Sweedler, Hopf algebras, Benjamin, New York 1969.

Ramón González Rodríguez Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting