Magmas, comagmas, monoids and comonoids Hopf algebras Weak Hopf algebras Hopf quasigroups Hopf Algebras, Weak Hopf Algebras and Hopf Quasigroups in a Monoidal Setting 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. 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 : 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;"D ; δD ) where (δD ⊗ D) ◦ δD = (D ⊗ δD ) ◦ δD : If D = (D;"D ; δD ) and E = (E;"E ; 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;"D ; δD ) where D is an object in C and "D : D ! K (counit), δD : D ! D ⊗ D (coproduct) are morphisms in C such that ("D ⊗ D) ◦ δD = idD = (D ⊗ "D ) ◦ δD . If D = (D;"D ; δD ) and E = (E;"E ; δE ) are counital comagmas, f : D ! E is a morphism of counital comagmas if (f ⊗ f ) ◦ δD = δE ◦ f ;"E ◦ f = "D : 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. By duality, a counital comagma in C is a triple D = (D;"D ; δD ) where D is an object in C and "D : D ! K (counit), δD : D ! D ⊗ D (coproduct) are morphisms in C such that ("D ⊗ D) ◦ δD = idD = (D ⊗ "D ) ◦ δD . If D = (D;"D ; δD ) and E = (E;"E ; δE ) are counital comagmas, f : D ! E is a morphism of counital comagmas if (f ⊗ f ) ◦ δD = δE ◦ f ;"E ◦ f = "D : Definition. A comonoid is a counital comagma D = (D;"D ; δD ) where (δD ⊗ D) ◦ δD = (D ⊗ δD ) ◦ δD : If D = (D;"D ; δD ) and E = (E;"E ; 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 "D⊗E = "D ⊗ "E ; 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 "D⊗E = "D ⊗ "E ; 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 "D⊗E = "D ⊗ "E ; 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;'M ) is a left A-module if M is an object in C and 'M : A ⊗ M ! M is a morphism in C (called the action) satisfying 'M ◦ (ηA ⊗ M) = idM ;'M ◦ (A ⊗ 'M ) = 'M ◦ (µA ⊗ M): Given two left A-modules (M;'M ) and (N;'N ), f : M ! N is a morphism of left A-modules if 'N ◦ (A ⊗ f ) = f ◦ 'M : 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 ("D ⊗ 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;"H ; δH ) such δH and "H are monoid morphisms and there exists a morphism λH : H ! H in C (called the antipode of H) verifiying: λH ∗ idH = "H ⊗ 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.