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Symmetric monoidal functors Typical examples of symmetric monoidal categories: • ( Set , × , ⊤ ) ; • ( S, ⊗ , S ) for a commutative ring S ; • Any entropic variety ( V , ⊗ , 1 ) ; • Any category ( C , + , ⊥ ) with coproduct + and initial object ⊥ . A symmetric monoidal functor is a Monoid homomorphism. Examples:

Symmetric monoidal functors Typical examples of symmetric monoidal categories: • ( Set , × , ⊤ ) ; • ( S, ⊗ , S ) for a commutative ring S ; • Any entropic variety ( V , ⊗ , 1 ) ; • Any category ( C , + , ⊥ ) with coproduct + and initial object ⊥ . A symmetric monoidal functor is a Monoid homomorphism. Examples: • Free algebra functor F : ( Set , × , ⊤ ) → ( V , ⊗ , 1 ) for an entropic variety V ;

Symmetric monoidal functors Typical examples of symmetric monoidal categories: • ( Set , × , ⊤ ) ; • ( S, ⊗ , S ) for a commutative ring S ; • Any entropic variety ( V , ⊗ , 1 ) ; • Any category ( C , + , ⊥ ) with coproduct + and initial object ⊥ . A symmetric monoidal functor is a Monoid homomorphism. Examples: • Free algebra functor F : ( Set , × , ⊤ ) → ( V , ⊗ , 1 ) for an entropic variety V ; • Underlying set functor U : ( S, ⊕ , { 0 } ) → ( Set , × , ⊤ ) .

Monoid and comonoid diagrams

� � � � � � Monoid and comonoid diagrams 1 A ⊗∇ � 1 A ⊗ η A ⊗ A ⊗ A A ⊗ A A ⊗ A A ⊗ 1 monoid ❏ ❏ ❏ ❏ ∇ ❏ ρ A η ⊗ 1 A ❏ ∇⊗ 1 A ∇ ❏ ❏ ❏ ❏ � A � A A ⊗ A 1 ⊗ A ∇ λ A

� � � � � � Monoid and comonoid diagrams 1 A ⊗∇ � 1 A ⊗ η A ⊗ A ⊗ A A ⊗ A A ⊗ A A ⊗ 1 monoid ❏ ❏ ❏ ❏ ∇ ❏ ρ A η ⊗ 1 A ❏ ∇⊗ 1 A ∇ ❏ ❏ ❏ ❏ � A � A A ⊗ A 1 ⊗ A ∇ λ A unit η , multiplication ∇

� � � � � � � � � � � � � Monoid and comonoid diagrams 1 A ⊗∇ � 1 A ⊗ η A ⊗ A ⊗ A A ⊗ A A ⊗ A A ⊗ 1 monoid ❏ ❏ ❏ ❏ ∇ ❏ ρ A η ⊗ 1 A ❏ ∇⊗ 1 A ∇ ❏ ❏ ❏ ❏ � A � A A ⊗ A 1 ⊗ A ∇ λ A unit η , multiplication ∇ 1 A ⊗ ∆ 1 A ⊗ ε � A ⊗ A ⊗ A A ⊗ A A ⊗ A A ⊗ 1 comonoid � ❏❏❏❏❏❏❏❏❏❏ ∆ ρ − 1 ε ⊗ 1 A ∆ ⊗ 1 A ∆ A A ⊗ A A 1 ⊗ A A ∆ λ − 1 A

� � � � � � � � � � � � � Monoid and comonoid diagrams 1 A ⊗∇ � 1 A ⊗ η A ⊗ A ⊗ A A ⊗ A A ⊗ A A ⊗ 1 monoid ❏ ❏ ❏ ❏ ∇ ❏ ρ A η ⊗ 1 A ❏ ∇⊗ 1 A ∇ ❏ ❏ ❏ ❏ � A � A A ⊗ A 1 ⊗ A ∇ λ A unit η , multiplication ∇ 1 A ⊗ ∆ 1 A ⊗ ε � A ⊗ A ⊗ A A ⊗ A A ⊗ A A ⊗ 1 comonoid � ❏❏❏❏❏❏❏❏❏❏ ∆ ρ − 1 ε ⊗ 1 A ∆ ⊗ 1 A ∆ A A ⊗ A A 1 ⊗ A A ∆ λ − 1 A counit ε , comultiplication in Sweedler notation ∆: a �→ a L ⊗ a R or ( a L 1 ⊗ a R 1 ) . . . ( a L na ⊗ a R na ) ( ) ∆: A → A ⊗ A ; a �→ w a

Bi-algebra diagram

� � � � � � � � � � � Bi-algebra diagram ∇ 1 ∆ � 1 1 ⊗ 1 1 ⊗ 1 1 � ❃❃❃❃❃❃❃ � ������� ε ⊗ ε η ⊗ η ε η ∇ ∆ A ⊗ A A A ⊗ A ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ∆ ⊗ ∆ ❡ ∇⊗∇ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ A ⊗ A ⊗ A ⊗ A A ⊗ A ⊗ A ⊗ A 1 A ⊗ τ ⊗ 1 A

� � � � � � � � � � � Bi-algebra diagram ∇ 1 ∆ � 1 1 ⊗ 1 1 ⊗ 1 1 � ❃❃❃❃❃❃❃ � ������� ε ⊗ ε η ⊗ η ε η ∇ ∆ A ⊗ A A A ⊗ A ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ∆ ⊗ ∆ ❡ ∇⊗∇ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ ❨ ❡ A ⊗ A ⊗ A ⊗ A A ⊗ A ⊗ A ⊗ A 1 A ⊗ τ ⊗ 1 A     ∆ monoid     means  is a  homomorphism. ∇ comonoid  

Antipode diagram

� � � � � Antipode diagram S ⊗ 1 A � A ⊗ A A ⊗ A ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✻ ✟ ✻ ✟ ✻ ✟ ∆ ∇ ✻ ✟ ✟ ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✻ ✟ ✻ ✟ ✟ ε A � A A 1 ✻ η A ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✟ ∆ ∇ ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✟ ✟ � A ⊗ A A ⊗ A 1 A ⊗ S

� � � � � Antipode diagram S ⊗ 1 A � A ⊗ A A ⊗ A ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✻ ✟ ✻ ✟ ✻ ✟ ∆ ∇ ✻ ✟ ✟ ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✻ ✟ ✻ ✟ ✟ ε A � A A 1 ✻ η A ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✟ ∆ ∇ ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✟ ✻ ✟ ✟ � A ⊗ A A ⊗ A 1 A ⊗ S Bi-algebra with an antipode S is a Hopf algebra or quantum group .

Examples of Hopf algebras

Examples of Hopf algebras  ∇ : g ⊗ h �→ gh ;      ∆: g �→ g ⊗ g ;  • In ( Set , × , ⊤ ) , a group ( G, · , 1) with: η : ⊤ → { 1 } ;      S : g �→ g − 1 . 

Examples of Hopf algebras  ∇ : g ⊗ h �→ gh ;      ∆: g �→ g ⊗ g ;  • In ( Set , × , ⊤ ) , a group ( G, · , 1) with: η : ⊤ → { 1 } ;      S : g �→ g − 1 .  • Applying the free algebra functor F : Set → V yields a group algebra .

Examples of Hopf algebras  ∇ : g ⊗ h �→ gh ;      ∆: g �→ g ⊗ g ;  • In ( Set , × , ⊤ ) , a group ( G, · , 1) with: η : ⊤ → { 1 } ;      S : g �→ g − 1 .  • Applying the free algebra functor F : Set → V yields a group algebra . • In S , dualizing (for G finite) yields a dual group algebra .

Examples of Hopf algebras  ∇ : g ⊗ h �→ gh ;      ∆: g �→ g ⊗ g ;  • In ( Set , × , ⊤ ) , a group ( G, · , 1) with: η : ⊤ → { 1 } ;      S : g �→ g − 1 .  • Applying the free algebra functor F : Set → V yields a group algebra . • In S , dualizing (for G finite) yields a dual group algebra . • Combining these constructions gives the quantum double of a finite group.

Examples of Hopf algebras  ∇ : g ⊗ h �→ gh ;      ∆: g �→ g ⊗ g ;  • In ( Set , × , ⊤ ) , a group ( G, · , 1) with: η : ⊤ → { 1 } ;      S : g �→ g − 1 .  • Applying the free algebra functor F : Set → V yields a group algebra . • In S , dualizing (for G finite) yields a dual group algebra . • Combining these constructions gives the quantum double of a finite group. • In S , the universal enveloping algebra U ( L ) of a Lie algebra L ,

Examples of Hopf algebras  ∇ : g ⊗ h �→ gh ;      ∆: g �→ g ⊗ g ;  • In ( Set , × , ⊤ ) , a group ( G, · , 1) with: η : ⊤ → { 1 } ;      S : g �→ g − 1 .  • Applying the free algebra functor F : Set → V yields a group algebra . • In S , dualizing (for G finite) yields a dual group algebra . • Combining these constructions gives the quantum double of a finite group. • In S , the universal enveloping algebra U ( L ) of a Lie algebra L , with ∆: x → x ⊗ 1 + 1 ⊗ x for x ∈ L ,

Examples of Hopf algebras  ∇ : g ⊗ h �→ gh ;      ∆: g �→ g ⊗ g ;  • In ( Set , × , ⊤ ) , a group ( G, · , 1) with: η : ⊤ → { 1 } ;      S : g �→ g − 1 .  • Applying the free algebra functor F : Set → V yields a group algebra . • In S , dualizing (for G finite) yields a dual group algebra . • Combining these constructions gives the quantum double of a finite group. • In S , the universal enveloping algebra U ( L ) of a Lie algebra L , with ∆: x → x ⊗ 1 + 1 ⊗ x for x ∈ L , and ∇ as the linearized algebra multiplication.

The big picture GROUP THEORY ❅ ❅ ❅ ❅ ❅ ❘ / QUANTUM GROUPS HOPF ALGEBRAS

The big picture QUASIGROUPS, LOOPS � ✒ � � � � GROUP THEORY ❅ ❅ ❅ ❅ ❅ ❘ / QUANTUM GROUPS HOPF ALGEBRAS

The big picture QUASIGROUPS, LOOPS � ✒ ❅ � ❅ � ❅ � ❅ � ❘ ❅ GROUP THEORY ???????? ❅ � ✒ � ❅ � ❅ � ❅ ❘ ❅ � / QUANTUM GROUPS HOPF ALGEBRAS

References J.D.H. Smith, An Introduction to Quasigroups and Their Representations , Chapman and Hall/CRC, Boca Raton, FL, 2007. D.E. Radford, Hopf Algebras , World Scientific, Singapore, 2012. J.M. P´ erez-Izquierdo, “Algebras, hyperalgebras, nonassociative bialgebras and loops”, Adv. Math. 208 (2007), 834–876. G. Benkart, S. Madaraga, and J.M. P´ erez-Izquierdo, “Hopf algebras with triality”, Trans. Amer. Math. Soc. 365 (2012), 1001–1023.

The big picture QUASIGROUPS, LOOPS � ✒ ❅ � ❅ � ❅ � ❅ � ❘ ❅ GROUP THEORY QUANTUM QUASIGROUPS ❅ � ✒ � ❅ � ❅ � ❅ ❘ ❅ � / QUANTUM GROUPS HOPF ALGEBRAS

Magmas and comagmas

Magmas and comagmas Magma ( A, ∇ : A ⊗ A → A )

Magmas and comagmas Magma ( A, ∇ : A ⊗ A → A ) Unital magma ( A, ∇ : A ⊗ A → A, η : 1 → A )

� � � � Magmas and comagmas Magma ( A, ∇ : A ⊗ A → A ) 1 A ⊗ η Unital magma ( A, ∇ : A ⊗ A → A, η : 1 → A ) A ⊗ A A ⊗ 1 with ❏ ❏ ❏ ❏ ∇ ❏ ρ A η ⊗ 1 A ❏ ❏ ❏ ❏ ❏ � A 1 ⊗ A λ A

� � � � Magmas and comagmas Magma ( A, ∇ : A ⊗ A → A ) 1 A ⊗ η Unital magma ( A, ∇ : A ⊗ A → A, η : 1 → A ) A ⊗ A A ⊗ 1 with ❏ ❏ ❏ ❏ ∇ ❏ ρ A η ⊗ 1 A ❏ ❏ ❏ ❏ ❏ � A 1 ⊗ A λ A Comagma ( A, ∆: A → A ⊗ A )

� � � � Magmas and comagmas Magma ( A, ∇ : A ⊗ A → A ) 1 A ⊗ η Unital magma ( A, ∇ : A ⊗ A → A, η : 1 → A ) A ⊗ A A ⊗ 1 with ❏ ❏ ❏ ❏ ∇ ❏ ρ A η ⊗ 1 A ❏ ❏ ❏ ❏ ❏ � A 1 ⊗ A λ A Comagma ( A, ∆: A → A ⊗ A ) Counital magma ( A, ∆: A → A ⊗ A, ε : A → 1 )

� � � � � � � Magmas and comagmas Magma ( A, ∇ : A ⊗ A → A ) 1 A ⊗ η Unital magma ( A, ∇ : A ⊗ A → A, η : 1 → A ) A ⊗ A A ⊗ 1 with ❏ ❏ ❏ ❏ ∇ ❏ ρ A η ⊗ 1 A ❏ ❏ ❏ ❏ ❏ � A 1 ⊗ A λ A Comagma ( A, ∆: A → A ⊗ A ) 1 A ⊗ ε � Counital magma ( A, ∆: A → A ⊗ A, ε : A → 1 ) A ⊗ A A ⊗ 1 with � ❏❏❏❏❏❏❏❏❏❏ ∆ ρ − 1 ε ⊗ 1 A A 1 ⊗ A A λ − 1 A

Comagmas in ( Set , × , ⊤ )

Comagmas in ( Set , × , ⊤ ) General comagma ( A, ∆) in ( Set , × , ⊤ ) is ∆: A → A ⊗ A ; a �→ a L ⊗ a R

Comagmas in ( Set , × , ⊤ ) General comagma ( A, ∆) in ( Set , × , ⊤ ) is ∆: A → A ⊗ A ; a �→ a L ⊗ a R with functions L : A → A ; a �→ a L and R : A → A ; a �→ a R .

Comagmas in ( Set , × , ⊤ ) General comagma ( A, ∆) in ( Set , × , ⊤ ) is ∆: A → A ⊗ A ; a �→ a L ⊗ a R with functions L : A → A ; a �→ a L and R : A → A ; a �→ a R . Proposition: If ( A, ∆) is counital, then ∆: a �→ a ⊗ a .

Comagmas in ( Set , × , ⊤ ) General comagma ( A, ∆) in ( Set , × , ⊤ ) is ∆: A → A ⊗ A ; a �→ a L ⊗ a R with functions L : A → A ; a �→ a L and R : A → A ; a �→ a R . Proposition: If ( A, ∆) is counital, then ∆: a �→ a ⊗ a . (Each element a is setlike .)

� � � � Comagmas in ( Set , × , ⊤ ) General comagma ( A, ∆) in ( Set , × , ⊤ ) is ∆: A → A ⊗ A ; a �→ a L ⊗ a R with functions L : A → A ; a �→ a L and R : A → A ; a �→ a R . Proposition: If ( A, ∆) is counital, then ∆: a �→ a ⊗ a . (Each element a is setlike .) 1 A ⊗ ε � A ⊗ A A ⊗ 1 Proof: The counital diagram � ❏❏❏❏❏❏❏❏❏❏ ∆ ρ − 1 ε ⊗ 1 A A 1 ⊗ A A λ − 1 A

� � � � � � � Comagmas in ( Set , × , ⊤ ) General comagma ( A, ∆) in ( Set , × , ⊤ ) is ∆: A → A ⊗ A ; a �→ a L ⊗ a R with functions L : A → A ; a �→ a L and R : A → A ; a �→ a R . Proposition: If ( A, ∆) is counital, then ∆: a �→ a ⊗ a . (Each element a is setlike .) 1 A ⊗ ε � A ⊗ A A ⊗ 1 Proof: The counital diagram � ❏❏❏❏❏❏❏❏❏❏ ∆ ρ − 1 ε ⊗ 1 A A 1 ⊗ A A λ − 1 A a L ⊗ a R ✤ 1 A ⊗ ε � a ⊗ x yields , ❴ � ❑❑❑❑❑❑❑❑❑❑❑✤ ∆ ρ − 1 ε ⊗ 1 A A ❴ ☛ x ⊗ a a λ − 1 A

� � � � � � � Comagmas in ( Set , × , ⊤ ) General comagma ( A, ∆) in ( Set , × , ⊤ ) is ∆: A → A ⊗ A ; a �→ a L ⊗ a R with functions L : A → A ; a �→ a L and R : A → A ; a �→ a R . Proposition: If ( A, ∆) is counital, then ∆: a �→ a ⊗ a . (Each element a is setlike .) 1 A ⊗ ε � A ⊗ A A ⊗ 1 Proof: The counital diagram � ❏❏❏❏❏❏❏❏❏❏ ∆ ρ − 1 ε ⊗ 1 A A 1 ⊗ A A λ − 1 A a L ⊗ a R ✤ 1 A ⊗ ε � so a L = a = a R . a ⊗ x � yields , ❴ � ❑❑❑❑❑❑❑❑❑❑❑✤ ∆ ρ − 1 ε ⊗ 1 A A ❴ ☛ x ⊗ a a λ − 1 A

Bimagmas

� � � � � � � Bimagmas Bimagma ( A, ∇ , ∆) with A ❙ ❙ ❦ ❦ ❙ ❦ ❙ ❦ ❙ ❦ ❙ ∇ ❦ ❙ ∆ ❦ ❙ ❦ ❙ ❦ ❙ ❦ ❙ ❦ ❙ ❦ ❙ ❦ ❙ ❦ ❙ ❦ ❙ ❦ A ⊗ A A ⊗ A ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ∆ ⊗ ∆ ❚ ∇⊗∇ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ A ⊗ A ⊗ A ⊗ A A ⊗ A ⊗ A ⊗ A 1 A ⊗ τ ⊗ 1 A

� � � � � � � Bimagmas Bimagma ( A, ∇ , ∆) with A ❙ ❙ ❦ ❦ ❙ ❦ ❙ ❦ ❙ ❦ ❙ ∇ ❦ ❙ ∆ ❦ ❙ ❦ ❙ ❦ ❙ ❦ ❙ ❦ ❙ ❦ ❙ ❦ ❙ ❦ ❙ ❦ ❙ ❦ A ⊗ A A ⊗ A ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ∆ ⊗ ∆ ❚ ∇⊗∇ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ ❚ ❥ A ⊗ A ⊗ A ⊗ A A ⊗ A ⊗ A ⊗ A 1 A ⊗ τ ⊗ 1 A     ∆ magma     So  is a  homomorphism. ∇ comagma  

Biunital bimagmas

Biunital bimagmas A biunital bimagma is a unital and counital bimagma ( A, ∇ , ∆ , η, ε )

� � � � � � Biunital bimagmas A biunital bimagma is a unital and counital bimagma ( A, ∇ , ∆ , η, ε ) with commuting biunitality diagram ∇ 1 ∆ � 1 1 ⊗ 1 1 ⊗ 1 1 � ❃❃❃❃❃❃❃ � ������� η ⊗ η ε ⊗ ε ε η ∇ ∆ A ⊗ A A A ⊗ A

� � � � � � Biunital bimagmas A biunital bimagma is a unital and counital bimagma ( A, ∇ , ∆ , η, ε ) with commuting biunitality diagram ∇ 1 ∆ � 1 1 ⊗ 1 1 ⊗ 1 1 � ❃❃❃❃❃❃❃ � ������� η ⊗ η ε ⊗ ε ε η ∇ ∆ A ⊗ A A A ⊗ A     ∆ unital magma     So  is a  homomorphism. ∇ counital comagma  

Quantum quasigroups and loops

Quantum quasigroups and loops A quantum quasigroup is a bimagma ( A, ∇ , ∆) with invertible

Quantum quasigroups and loops A quantum quasigroup is a bimagma ( A, ∇ , ∆) with invertible ∆ ⊗ 1 A � A ⊗ A ⊗ A 1 A ⊗∇ � A ⊗ A left composite A ⊗ A

Quantum quasigroups and loops A quantum quasigroup is a bimagma ( A, ∇ , ∆) with invertible ∆ ⊗ 1 A � A ⊗ A ⊗ A 1 A ⊗∇ � A ⊗ A left composite A ⊗ A and 1 A ⊗ ∆ � A ⊗ A ⊗ A ∇⊗ 1 A � A ⊗ A . right composite A ⊗ A

Quantum quasigroups and loops A quantum quasigroup is a bimagma ( A, ∇ , ∆) with invertible ∆ ⊗ 1 A � A ⊗ A ⊗ A 1 A ⊗∇ � A ⊗ A left composite A ⊗ A and 1 A ⊗ ∆ � A ⊗ A ⊗ A ∇⊗ 1 A � A ⊗ A . right composite A ⊗ A A quantum loop is a biunital bimagma ( A, ∇ , ∆ , η, ε ) in which the reduct ( A, ∇ , ∆) is a quantum quasigroup.

Quantum quasigroups and loops A quantum quasigroup is a bimagma ( A, ∇ , ∆) with invertible ∆ ⊗ 1 A � A ⊗ A ⊗ A 1 A ⊗∇ � A ⊗ A left composite A ⊗ A and 1 A ⊗ ∆ � A ⊗ A ⊗ A ∇⊗ 1 A � A ⊗ A . right composite A ⊗ A A quantum loop is a biunital bimagma ( A, ∇ , ∆ , η, ε ) in which the reduct ( A, ∇ , ∆) is a quantum quasigroup. Remark: These definitions are self-dual,