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Quantum quasigroups

Jonathan D.H. Smith Iowa State University email: jdhsmith@iastate.edu

http://orion.math.iastate.edu/jdhsmith/homepage.html

The big picture

The big picture

GROUP THEORY

The big picture

GROUP THEORY

• ✒

QUASIGROUPS, LOOPS

References

J.D.H. Smith, An Introduction to Quasigroups and Their Representations, Chapman and Hall/CRC, Boca Raton, FL, 2007.

Quasigroups and loops

Quasigroups and loops

Quasigroup: (Q, ·, /, \) with

Quasigroups and loops

Quasigroup: (Q, ·, /, \) with

y\(y · x) = x = (x · y)/y

and

y · (y\x) = x = (x/y) · y.

Quasigroups and loops

Quasigroup: (Q, ·, /, \) with

y\(y · x) = x = (x · y)/y

and

y · (y\x) = x = (x/y) · y.

In a magma (M, ◦), with element y, define

Quasigroups and loops

Quasigroup: (Q, ·, /, \) with

y\(y · x) = x = (x · y)/y

and

y · (y\x) = x = (x/y) · y.

In a magma (M, ◦), with element y, define right multiplication R◦(y): M → M; x → x ◦ y and

Quasigroups and loops

Quasigroup: (Q, ·, /, \) with

y\(y · x) = x = (x · y)/y

and

y · (y\x) = x = (x/y) · y.

In a magma (M, ◦), with element y, define right multiplication R◦(y): M → M; x → x ◦ y and left multiplication L◦(y): M → M; x → y ◦ x.

Quasigroups and loops

Quasigroup: (Q, ·, /, \) with

y\(y · x) = x = (x · y)/y

and

y · (y\x) = x = (x/y) · y.

In a magma (M, ◦), with element y, define right multiplication R◦(y): M → M; x → x ◦ y and left multiplication L◦(y): M → M; x → y ◦ x. Quasigroup identities say L(y) = L·(y) and R(y) = R·(y) bijective.

Quasigroups and loops

Quasigroup: (Q, ·, /, \) with

y\(y · x) = x = (x · y)/y

and

y · (y\x) = x = (x/y) · y.

In a magma (M, ◦), with element y, define right multiplication R◦(y): M → M; x → x ◦ y and left multiplication L◦(y): M → M; x → y ◦ x. Quasigroup identities say L(y) = L·(y) and R(y) = R·(y) bijective. Loop: Quasigroup Q with identity element e satisfying x · e = x = e · x.

The big picture

GROUP THEORY

• ✒

QUASIGROUPS, LOOPS

The big picture

GROUP THEORY

❅ ❅ ❅ ❅ ❅ ❘

QUANTUM GROUPS

/

HOPF ALGEBRAS

References

D.E. Radford, Hopf Algebras, World Scientific, Singapore, 2012.

Entropic algebras

Entropic algebras

Algebra (A, Ω) is entropic if each operation

ω: Aωτ → A; (a1, . . . , aωτ) → a1, . . . , aωτω

is a homomorphism.

Entropic algebras

Algebra (A, Ω) is entropic if each operation

ω: Aωτ → A; (a1, . . . , aωτ) → a1, . . . , aωτω

is a homomorphism. Examples:

Entropic algebras

Algebra (A, Ω) is entropic if each operation

ω: Aωτ → A; (a1, . . . , aωτ) → a1, . . . , aωτω

is a homomorphism. Examples:

• Modules over a commutative ring;

Entropic algebras

Algebra (A, Ω) is entropic if each operation

ω: Aωτ → A; (a1, . . . , aωτ) → a1, . . . , aωτω

is a homomorphism. Examples:

• Modules over a commutative ring;
• Commutative semigroups, e.g., semilattices;

Entropic algebras

Algebra (A, Ω) is entropic if each operation

ω: Aωτ → A; (a1, . . . , aωτ) → a1, . . . , aωτω

is a homomorphism. Examples:

• Modules over a commutative ring;
• Commutative semigroups, e.g., semilattices;
• Barycentric algebras (with convex combinations as operations);

Entropic algebras

Algebra (A, Ω) is entropic if each operation

ω: Aωτ → A; (a1, . . . , aωτ) → a1, . . . , aωτω

is a homomorphism. Examples:

• Modules over a commutative ring;
• Commutative semigroups, e.g., semilattices;
• Barycentric algebras (with convex combinations as operations);
• Sets.

Tensor products

Tensor products

Variety/category V of entropic algebras (homomorphisms as morphisms).

Tensor products

Variety/category V of entropic algebras (homomorphisms as morphisms). Note V(Y, X) is a subalgebra of the power XY .

Tensor products

Variety/category V of entropic algebras (homomorphisms as morphisms). Note V(Y, X) is a subalgebra of the power XY . Write V(Z, Y ; X) for the set of bihomomorphisms from Z × Y to X.

Tensor products

Variety/category V of entropic algebras (homomorphisms as morphisms). Note V(Y, X) is a subalgebra of the power XY . Write V(Z, Y ; X) for the set of bihomomorphisms from Z × Y to X. The tensor product Z ⊗ Y features in the adjointness

V(Z ⊗ Y, X) ∼ = V ( Z, V(Y, X) ) ∼ = V(Z, Y ; X)

Tensor products

Variety/category V of entropic algebras (homomorphisms as morphisms). Note V(Y, X) is a subalgebra of the power XY . Write V(Z, Y ; X) for the set of bihomomorphisms from Z × Y to X. The tensor product Z ⊗ Y features in the adjointness

V(Z ⊗ Y, X) ∼ = V ( Z, V(Y, X) ) ∼ = V(Z, Y ; X)

Setting X = Z ⊗ Y , taking idZ⊗Y on left, obtain a bihomomorphism

⊗: Z × Y → Z ⊗ Y ; (z, y) → z ⊗ y

Tensor products

Variety/category V of entropic algebras (homomorphisms as morphisms). Note V(Y, X) is a subalgebra of the power XY . Write V(Z, Y ; X) for the set of bihomomorphisms from Z × Y to X. The tensor product Z ⊗ Y features in the adjointness

V(Z ⊗ Y, X) ∼ = V ( Z, V(Y, X) ) ∼ = V(Z, Y ; X)

Setting X = Z ⊗ Y , taking idZ⊗Y on left, obtain a bihomomorphism

⊗: Z × Y → Z ⊗ Y ; (z, y) → z ⊗ y

Lemma: Z ⊗ Y is generated by {z ⊗ y | z ∈ Z, y ∈ Y }

V as a tensor category

V as a tensor category

Define isomorphism τ : Z ⊗ Y → Y ⊗ Z; z ⊗ y → y ⊗ z.

V as a tensor category

Define isomorphism τ : Z ⊗ Y → Y ⊗ Z; z ⊗ y → y ⊗ z. Free algebra 1 on one generator x in V.

V as a tensor category

Define isomorphism τ : Z ⊗ Y → Y ⊗ Z; z ⊗ y → y ⊗ z. Free algebra 1 on one generator x in V. Define isomorphisms 1 ⊗ A

λA

− − → A

ρA

← − − A ⊗ 1

by x ⊗ a ✤

λA a

a ⊗ x ✤

ρA

• for a ∈ A.

V as a tensor category

Define isomorphism τ : Z ⊗ Y → Y ⊗ Z; z ⊗ y → y ⊗ z. Free algebra 1 on one generator x in V. Define isomorphisms 1 ⊗ A

λA

− − → A

ρA

← − − A ⊗ 1

by x ⊗ a ✤

λA a

a ⊗ x ✤

ρA

• for a ∈ A.

Define isomorphism αC,B,A : (C ⊗ B) ⊗ A → C ⊗ (B ⊗ A) by (c ⊗ b) ⊗ a → c ⊗ (b ⊗ a).

V as a tensor category

Define isomorphism τ : Z ⊗ Y → Y ⊗ Z; z ⊗ y → y ⊗ z. Free algebra 1 on one generator x in V. Define isomorphisms 1 ⊗ A

λA

− − → A

ρA

← − − A ⊗ 1

by x ⊗ a ✤

λA a

a ⊗ x ✤

ρA

• for a ∈ A.

Define isomorphism αC,B,A : (C ⊗ B) ⊗ A → C ⊗ (B ⊗ A) by (c ⊗ b) ⊗ a → c ⊗ (b ⊗ a). (Write c ⊗ b ⊗ a for identified image.)

V as a tensor category

Define isomorphism τ : Z ⊗ Y → Y ⊗ Z; z ⊗ y → y ⊗ z. Free algebra 1 on one generator x in V. Define isomorphisms 1 ⊗ A

λA

− − → A

ρA

← − − A ⊗ 1

by x ⊗ a ✤

λA a

a ⊗ x ✤

ρA

• for a ∈ A.

Define isomorphism αC,B,A : (C ⊗ B) ⊗ A → C ⊗ (B ⊗ A) by (c ⊗ b) ⊗ a → c ⊗ (b ⊗ a). (Write c ⊗ b ⊗ a for identified image.) Proposition:

( V, ⊗, 1 )

with τ is a symmetric monoidal (or tensor) category

V as a tensor category

Define isomorphism τ : Z ⊗ Y → Y ⊗ Z; z ⊗ y → y ⊗ z. Free algebra 1 on one generator x in V. Define isomorphisms 1 ⊗ A

λA

− − → A

ρA

← − − A ⊗ 1

by x ⊗ a ✤

λA a

a ⊗ x ✤

ρA

• for a ∈ A.

Define isomorphism αC,B,A : (C ⊗ B) ⊗ A → C ⊗ (B ⊗ A) by (c ⊗ b) ⊗ a → c ⊗ (b ⊗ a). (Write c ⊗ b ⊗ a for identified image.) Proposition:

( V, ⊗, 1 )

with τ is a symmetric monoidal (or tensor) category (“commutative Monoid”).

Symmetric monoidal functors

Symmetric monoidal functors

Typical examples of symmetric monoidal categories:

Symmetric monoidal functors

Typical examples of symmetric monoidal categories:

• (Set, ×, ⊤);

Symmetric monoidal functors

Typical examples of symmetric monoidal categories:

• (Set, ×, ⊤);
• (S, ⊗, S) for a commutative ring S;

Symmetric monoidal functors

Typical examples of symmetric monoidal categories:

• (Set, ×, ⊤);
• (S, ⊗, S) for a commutative ring S;
• Any entropic variety (V, ⊗, 1);

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 ⊥.

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.

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 A ⊗ A ⊗ A

1A⊗∇ ∇⊗1A

• A ⊗ A

∇

• A ⊗ A

∇

A

monoid

A ⊗ A

∇

• ❏

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ A ⊗ 1

1A⊗η

• ρA
• 1 ⊗ A

η⊗1A

• λA

A

Monoid and comonoid diagrams A ⊗ A ⊗ A

1A⊗∇ ∇⊗1A

• A ⊗ A

∇

• A ⊗ A

∇

A

monoid

A ⊗ A

∇

• ❏

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ A ⊗ 1

1A⊗η

• ρA
• 1 ⊗ A

η⊗1A

• λA

A

unit η, multiplication ∇

Monoid and comonoid diagrams A ⊗ A ⊗ A

1A⊗∇ ∇⊗1A

• A ⊗ A

∇

• A ⊗ A

∇

A

monoid

A ⊗ A

∇

• ❏

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ A ⊗ 1

1A⊗η

• ρA
• 1 ⊗ A

η⊗1A

• λA

A

unit η, multiplication ∇

A ⊗ A ⊗ A A ⊗ A

1A⊗∆

• A ⊗ A

∆⊗1A

• A

∆

• ∆
• comonoid

A ⊗ A

1A⊗ε ε⊗1A

• A ⊗ 1

1 ⊗ A A

∆

❏❏❏❏❏❏❏❏❏❏

λ−1

A

• ρ−1

A

Monoid and comonoid diagrams A ⊗ A ⊗ A

1A⊗∇ ∇⊗1A

• A ⊗ A

∇

• A ⊗ A

∇

A

monoid

A ⊗ A

∇

• ❏

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ A ⊗ 1

1A⊗η

• ρA
• 1 ⊗ A

η⊗1A

• λA

A

unit η, multiplication ∇

A ⊗ A ⊗ A A ⊗ A

1A⊗∆

• A ⊗ A

∆⊗1A

• A

∆

• ∆
• comonoid

A ⊗ A

1A⊗ε ε⊗1A

• A ⊗ 1

1 ⊗ A A

∆

❏❏❏❏❏❏❏❏❏❏

λ−1

A

• ρ−1

A

• counit ε, comultiplication in Sweedler notation ∆: a → aL ⊗ aR or

∆: A → A ⊗ A; a → ( (aL1 ⊗ aR1) . . . (aLna ⊗ aRna ) ) wa

Bi-algebra diagram

Bi-algebra diagram 1 ⊗ 1

∇

1 1

η

• ∆
• 1
• 1 ⊗ 1

η⊗η

• A ⊗ A

∆⊗∆

• ∇
• ε⊗ε
• ❨

❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ A

∆

• ε

❃❃❃❃❃❃❃ A ⊗ A A ⊗ A ⊗ A ⊗ A

1A⊗τ⊗1A

• ❡

❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ A ⊗ A ⊗ A ⊗ A

∇⊗∇

Bi-algebra diagram 1 ⊗ 1

∇

1 1

η

• ∆
• 1
• 1 ⊗ 1

η⊗η

• A ⊗ A

∆⊗∆

• ∇
• ε⊗ε
• ❨

❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❨ A

∆

• ε

❃❃❃❃❃❃❃ A ⊗ A A ⊗ A ⊗ A ⊗ A

1A⊗τ⊗1A

• ❡

❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡ A ⊗ A ⊗ A ⊗ A

∇⊗∇

• means

   ∆ ∇    is a   

monoid comonoid

   homomorphism.

Antipode diagram

Antipode diagram A ⊗ A

S⊗1A

A ⊗ A

∇

• ✻

✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ A

εA

• ∆
• ✟

✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟

∆

• ✻

✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ 1

ηA

A A ⊗ A

1A⊗S

A ⊗ A

∇

• ✟

✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟

Antipode diagram A ⊗ A

S⊗1A

A ⊗ A

∇

• ✻

✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ A

εA

• ∆
• ✟

✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟

∆

• ✻

✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ 1

ηA

A A ⊗ A

1A⊗S

A ⊗ A

∇

• ✟

✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟

Bi-algebra with an antipode S is a Hopf algebra or quantum group.

Examples of Hopf algebras

Examples of Hopf algebras

• In (Set, ×, ⊤), a group (G, ·, 1) with:

             ∇: g ⊗ h → gh ; ∆: g → g ⊗ g ; η: ⊤ → {1} ; S : g → g−1.

Examples of Hopf algebras

• In (Set, ×, ⊤), a group (G, ·, 1) with:

             ∇: g ⊗ h → gh ; ∆: g → g ⊗ g ; η: ⊤ → {1} ; S : g → g−1.

• Applying the free algebra functor F : Set → V yields a group algebra.

Examples of Hopf algebras

• In (Set, ×, ⊤), a group (G, ·, 1) with:

             ∇: g ⊗ h → gh ; ∆: g → g ⊗ g ; 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

• In (Set, ×, ⊤), a group (G, ·, 1) with:

             ∇: g ⊗ h → gh ; ∆: g → g ⊗ g ; 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

• In (Set, ×, ⊤), a group (G, ·, 1) with:

             ∇: g ⊗ h → gh ; ∆: g → g ⊗ g ; 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

• In (Set, ×, ⊤), a group (G, ·, 1) with:

             ∇: g ⊗ h → gh ; ∆: g → g ⊗ g ; 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

• In (Set, ×, ⊤), a group (G, ·, 1) with:

             ∇: g ⊗ h → gh ; ∆: g → g ⊗ g ; 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

GROUP THEORY

• ✒

❅ ❅ ❅ ❅ ❅ ❘

QUANTUM GROUPS

/

HOPF ALGEBRAS QUASIGROUPS, LOOPS

The big picture

GROUP THEORY

• ✒

❅ ❅ ❅ ❅ ❅ ❘

QUANTUM GROUPS

/

HOPF ALGEBRAS QUASIGROUPS, LOOPS

❅ ❅ ❅ ❅ ❅ ❘

• ✒

????????

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

GROUP THEORY

• ✒

❅ ❅ ❅ ❅ ❅ ❘

QUANTUM GROUPS

/

HOPF ALGEBRAS QUASIGROUPS, LOOPS

❅ ❅ ❅ ❅ ❅ ❘

• ✒

QUANTUM QUASIGROUPS

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) Unital magma (A, ∇: A ⊗ A → A, η: 1 → A) with

A ⊗ A

∇

• ❏

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ A ⊗ 1

1A⊗η

• ρA
• 1 ⊗ A

η⊗1A

• λA

A

Magmas and comagmas

Magma (A, ∇: A ⊗ A → A) Unital magma (A, ∇: A ⊗ A → A, η: 1 → A) with

A ⊗ A

∇

• ❏

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ A ⊗ 1

1A⊗η

• ρA
• 1 ⊗ A

η⊗1A

• λA

A

Comagma (A, ∆: A → A ⊗ A)

Magmas and comagmas

Magma (A, ∇: A ⊗ A → A) Unital magma (A, ∇: A ⊗ A → A, η: 1 → A) with

A ⊗ A

∇

• ❏

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ A ⊗ 1

1A⊗η

• ρA
• 1 ⊗ A

η⊗1A

• λA

A

Comagma (A, ∆: A → A ⊗ A) Counital magma (A, ∆: A → A ⊗ A, ε: A → 1)

Magmas and comagmas

Magma (A, ∇: A ⊗ A → A) Unital magma (A, ∇: A ⊗ A → A, η: 1 → A) with

A ⊗ A

∇

• ❏

❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ A ⊗ 1

1A⊗η

• ρA
• 1 ⊗ A

η⊗1A

• λA

A

Comagma (A, ∆: A → A ⊗ A) Counital magma (A, ∆: A → A ⊗ A, ε: A → 1) with

A ⊗ A

1A⊗ε ε⊗1A

• A ⊗ 1

1 ⊗ A A

∆

❏❏❏❏❏❏❏❏❏❏

λ−1

A

• ρ−1

A

Comagmas in (Set, ×, ⊤)

Comagmas in (Set, ×, ⊤)

General comagma (A, ∆) in (Set, ×, ⊤) is ∆: A → A ⊗ A; a → aL ⊗ aR

Comagmas in (Set, ×, ⊤)

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

Comagmas in (Set, ×, ⊤)

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

Comagmas in (Set, ×, ⊤)

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

A ⊗ A

1A⊗ε ε⊗1A

• A ⊗ 1

1 ⊗ A A

∆

❏❏❏❏❏❏❏❏❏❏

λ−1

A

• ρ−1

A

Comagmas in (Set, ×, ⊤)

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

A ⊗ A

1A⊗ε ε⊗1A

• A ⊗ 1

1 ⊗ A A

∆

❏❏❏❏❏❏❏❏❏❏

λ−1

A

• ρ−1

A

• yields

aL ⊗ aR ✤ 1A⊗ε ❴

ε⊗1A

• a ⊗ x

x ⊗ a a ☛

∆

❑❑❑❑❑❑❑❑❑❑❑✤

λ−1

A

• ❴

ρ−1

A

• ,

Comagmas in (Set, ×, ⊤)

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

A ⊗ A

1A⊗ε ε⊗1A

• A ⊗ 1

1 ⊗ A A

∆

❏❏❏❏❏❏❏❏❏❏

λ−1

A

• ρ−1

A

• yields

aL ⊗ aR ✤ 1A⊗ε ❴

ε⊗1A

• a ⊗ x

x ⊗ a a ☛

∆

❑❑❑❑❑❑❑❑❑❑❑✤

λ−1

A

• ❴

ρ−1

A

• ,

so aL = a = aR.

Bimagmas

Bimagmas

Bimagma (A, ∇, ∆) with

A

∆

• ❙

❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ A ⊗ A

∆⊗∆

• ∇
• ❦

❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦

• ❚

❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ A ⊗ A A ⊗ A ⊗ A ⊗ A

1A⊗τ⊗1A

• ❥

❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ A ⊗ A ⊗ A ⊗ A

∇⊗∇

Bimagmas

Bimagma (A, ∇, ∆) with

A

∆

• ❙

❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ A ⊗ A

∆⊗∆

• ∇
• ❦

❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦

• ❚

❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ A ⊗ A A ⊗ A ⊗ A ⊗ A

1A⊗τ⊗1A

• ❥

❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ ❥ A ⊗ A ⊗ A ⊗ A

∇⊗∇

• So

   ∆ ∇    is a   

magma comagma

   homomorphism.

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

η

• ∆
• 1
• 1 ⊗ 1

η⊗η

• A ⊗ A

∇

• ε⊗ε
• A

∆

• ε

❃❃❃❃❃❃❃ A ⊗ A

Biunital bimagmas

A biunital bimagma is a unital and counital bimagma (A, ∇, ∆, η, ε) with commuting biunitality diagram

1 ⊗ 1

∇

1 1

η

• ∆
• 1
• 1 ⊗ 1

η⊗η

• A ⊗ A

∇

• ε⊗ε
• A

∆

• ε

❃❃❃❃❃❃❃ A ⊗ A

So

   ∆ ∇    is a   

unital magma counital comagma

   homomorphism.

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 left composite A ⊗ A

∆⊗1A A ⊗ A ⊗ A 1A⊗∇ A ⊗ A

Quantum quasigroups and loops

A quantum quasigroup is a bimagma (A, ∇, ∆) with invertible left composite A ⊗ A

∆⊗1A A ⊗ A ⊗ A 1A⊗∇ A ⊗ A

and right composite A ⊗ A

1A⊗∆ A ⊗ A ⊗ A ∇⊗1A A ⊗ A .

Quantum quasigroups and loops

A quantum quasigroup is a bimagma (A, ∇, ∆) with invertible left composite A ⊗ A

∆⊗1A A ⊗ A ⊗ A 1A⊗∇ A ⊗ A

and right composite A ⊗ A

1A⊗∆ A ⊗ A ⊗ A ∇⊗1A 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 left composite A ⊗ A

∆⊗1A A ⊗ A ⊗ A 1A⊗∇ A ⊗ A

and right composite A ⊗ A

1A⊗∆ A ⊗ A ⊗ A ∇⊗1A 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,

Quantum quasigroups and loops

A quantum quasigroup is a bimagma (A, ∇, ∆) with invertible left composite A ⊗ A

∆⊗1A A ⊗ A ⊗ A 1A⊗∇ A ⊗ A

and right composite A ⊗ A

1A⊗∆ A ⊗ A ⊗ A ∇⊗1A 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, and invariant under symmetric monoidal functors.

Combinatorial quantum loops

Combinatorial quantum loops

Proposition: Counital quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups.

Combinatorial quantum loops

Proposition: Counital quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups. Proof: Left composite is a ⊗ b ✤∆⊗1A a ⊗ a ⊗ b ✤1A⊗∇ a ⊗ (a · b) .

Combinatorial quantum loops

Proposition: Counital quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups. Proof: Left composite is a ⊗ b ✤∆⊗1A a ⊗ a ⊗ b ✤1A⊗∇ a ⊗ (a · b) . Inverse is c ⊗ (c\d)

c ⊗ d ✤

Combinatorial quantum loops

Proposition: Counital quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups. Proof: Left composite is a ⊗ b ✤∆⊗1A a ⊗ a ⊗ b ✤1A⊗∇ a ⊗ (a · b) . Inverse is c ⊗ (c\d)

c ⊗ d ✤

• for binary operation c\d

Combinatorial quantum loops

Proposition: Counital quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups. Proof: Left composite is a ⊗ b ✤∆⊗1A a ⊗ a ⊗ b ✤1A⊗∇ a ⊗ (a · b) . Inverse is a ⊗ (a\(a · b))

a ⊗ (a · b) ✤

• for binary operation c\d

with

b = a\(a · b)

Combinatorial quantum loops

Proposition: Counital quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups. Proof: Left composite is c ⊗ (c\d) ✤∆⊗1A c ⊗ c ⊗ (c\d) ✤1A⊗∇ c ⊗ (c · (c\d)) . Inverse is c ⊗ (c\d)

c ⊗ d ✤

• for binary operation c\d

with

b = a\(a · b)

and

d = c · (c\d).

Combinatorial quantum loops

Proposition: Counital quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups. Proof: Left composite is a ⊗ b ✤∆⊗1A a ⊗ a ⊗ b ✤1A⊗∇ a ⊗ (a · b) . Inverse is c ⊗ (c\d)

c ⊗ d ✤

• for binary operation c\d

with

b = a\(a · b)

and

d = c · (c\d).

Dually, have (b · a)/a = b and (d/c) · c = d, so a quasigroup.

Combinatorial quantum loops

Proposition: Counital quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups. Proof: Left composite is a ⊗ b ✤∆⊗1A a ⊗ a ⊗ b ✤1A⊗∇ a ⊗ (a · b) . Inverse is c ⊗ (c\d)

c ⊗ d ✤

• for binary operation c\d

with

b = a\(a · b)

and

d = c · (c\d).

Dually, have (b · a)/a = b and (d/c) · c = d, so a quasigroup. Conversely, a quasigroup (Q, ·, /, \) provides a counital quantum quasigroup.

Combinatorial quantum loops

Proposition: Counital quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups. Proof: Left composite is a ⊗ b ✤∆⊗1A a ⊗ a ⊗ b ✤1A⊗∇ a ⊗ (a · b) . Inverse is c ⊗ (c\d)

c ⊗ d ✤

• for binary operation c\d

with

b = a\(a · b)

and

d = c · (c\d).

Dually, have (b · a)/a = b and (d/c) · c = d, so a quasigroup. Conversely, a quasigroup (Q, ·, /, \) provides a counital quantum quasigroup.

• Corollary: Quantum loops in (Set, ×, ⊤) are equivalent to loops.

Combinatorial quantum quasigroups

Combinatorial quantum quasigroups

Proposition: Finite quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups with an ordered pair (L, R) of automorphisms.

Combinatorial quantum quasigroups

Proposition: Finite quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups with an ordered pair (L, R) of automorphisms. Proof: Finite quantum quasigroup (A, ∇, ∆) in Set, comagma ∆: a → aL ⊗ aR.

Combinatorial quantum quasigroups

Proposition: Finite quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups with an ordered pair (L, R) of automorphisms. Proof: Finite quantum quasigroup (A, ∇, ∆) in Set, comagma ∆: a → aL ⊗ aR. Bimagma diagram gives L: A → A and R: A → A as endomorphisms of (A, ∇).

Combinatorial quantum quasigroups

Proposition: Finite quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups with an ordered pair (L, R) of automorphisms. Proof: Finite quantum quasigroup (A, ∇, ∆) in Set, comagma ∆: a → aL ⊗ aR. Bimagma diagram gives L: A → A and R: A → A as endomorphisms of (A, ∇). Left composite is

a ⊗ b ✤∆⊗1A aL ⊗ aR ⊗ b ✤1A⊗∇ aL ⊗ (aR · b)

Combinatorial quantum quasigroups

Proposition: Finite quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups with an ordered pair (L, R) of automorphisms. Proof: Finite quantum quasigroup (A, ∇, ∆) in Set, comagma ∆: a → aL ⊗ aR. Bimagma diagram gives L: A → A and R: A → A as endomorphisms of (A, ∇). Left composite is

a ⊗ b ✤∆⊗1A aL ⊗ aR ⊗ b ✤1A⊗∇ aL ⊗ (aR · b)

Invertibility implies L surjective; dually, R.

Combinatorial quantum quasigroups

Proposition: Finite quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups with an ordered pair (L, R) of automorphisms. Proof: Finite quantum quasigroup (A, ∇, ∆) in Set, comagma ∆: a → aL ⊗ aR. Bimagma diagram gives L: A → A and R: A → A as endomorphisms of (A, ∇). Left composite is

a ⊗ b ✤∆⊗1A aL ⊗ aR ⊗ b ✤1A⊗∇ aL ⊗ (aR · b)

Invertibility implies L surjective; dually, R. Then |A| < ∞

⇒ L and R are automorphisms.

Combinatorial quantum quasigroups

Proposition: Finite quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups with an ordered pair (L, R) of automorphisms. Proof: Finite quantum quasigroup (A, ∇, ∆) in Set, comagma ∆: a → aL ⊗ aR. Bimagma diagram gives L: A → A and R: A → A as endomorphisms of (A, ∇). Left composite is

a ⊗ b ✤∆⊗1A aL ⊗ aR ⊗ b ✤1A⊗∇ aL ⊗ (aR · b)

Invertibility implies L surjective; dually, R. Then |A| < ∞

⇒ L and R are automorphisms.

Now write inverse as cL−1 ⊗ (cL−1R\d)

c ⊗ d ✤

Combinatorial quantum quasigroups

Proposition: Finite quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups with an ordered pair (L, R) of automorphisms. Proof: Finite quantum quasigroup (A, ∇, ∆) in Set, comagma ∆: a → aL ⊗ aR. Bimagma diagram gives L: A → A and R: A → A as endomorphisms of (A, ∇). Left composite is

a ⊗ b ✤∆⊗1A aL ⊗ aR ⊗ b ✤1A⊗∇ aL ⊗ (aR · b)

Invertibility implies L surjective; dually, R. Then |A| < ∞

⇒ L and R are automorphisms.

Now write inverse as cL−1 ⊗ (cL−1R\d)

c ⊗ d ✤

• Derive quasigroup identities.

Combinatorial quantum quasigroups

Proposition: Finite quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups with an ordered pair (L, R) of automorphisms. Proof: Finite quantum quasigroup (A, ∇, ∆) in Set, comagma ∆: a → aL ⊗ aR. Bimagma diagram gives L: A → A and R: A → A as endomorphisms of (A, ∇). Left composite is

a ⊗ b ✤∆⊗1A aL ⊗ aR ⊗ b ✤1A⊗∇ aL ⊗ (aR · b)

Invertibility implies L surjective; dually, R. Then |A| < ∞

⇒ L and R are automorphisms.

Now write inverse as cL−1 ⊗ (cL−1R\d)

c ⊗ d ✤

• Derive quasigroup identities.

Converse clear.

Combinatorial quantum quasigroups

Proposition: Finite quantum quasigroups in (Set, ×, ⊤) are equivalent to quasigroups with an ordered pair (L, R) of automorphisms. Proof: Finite quantum quasigroup (A, ∇, ∆) in Set, comagma ∆: a → aL ⊗ aR. Bimagma diagram gives L: A → A and R: A → A as endomorphisms of (A, ∇). Left composite is

a ⊗ b ✤∆⊗1A aL ⊗ aR ⊗ b ✤1A⊗∇ aL ⊗ (aR · b)

Invertibility implies L surjective; dually, R. Then |A| < ∞

⇒ L and R are automorphisms.

Now write inverse as cL−1 ⊗ (cL−1R\d)

c ⊗ d ✤

• Derive quasigroup identities.

Converse clear.

• Problem: Classify arbitrary quantum quasigroups in (Set, ×, ⊤).

The quantum couple

The quantum couple

Theorem: Group G with automorphic right action on finite quasigroup Q.

The quantum couple

Theorem: Group G with automorphic right action on finite quasigroup Q. Commutative ring S, tensor product GQ of free modules GS and QS.

The quantum couple

Theorem: Group G with automorphic right action on finite quasigroup Q. Commutative ring S, tensor product GQ of free modules GS and QS. For g ∈ G and q ∈ Q, write g|q for g ⊗ q.

The quantum couple

Theorem: Group G with automorphic right action on finite quasigroup Q. Commutative ring S, tensor product GQ of free modules GS and QS. For g ∈ G and q ∈ Q, write g|q for g ⊗ q. Define (f|p ⊗ g|q)∇ =

   fg|q

if pg = q;

• therwise

The quantum couple

Theorem: Group G with automorphic right action on finite quasigroup Q. Commutative ring S, tensor product GQ of free modules GS and QS. For g ∈ G and q ∈ Q, write g|q for g ⊗ q. Define (f|p ⊗ g|q)∇ =

   fg|q

if pg = q;

• therwise

and ∆: g|q → ∑

qLqR=q g|qL ⊗ g|qR.

The quantum couple

Theorem: Group G with automorphic right action on finite quasigroup Q. Commutative ring S, tensor product GQ of free modules GS and QS. For g ∈ G and q ∈ Q, write g|q for g ⊗ q. Define (f|p ⊗ g|q)∇ =

   fg|q

if pg = q;

• therwise

and ∆: g|q → ∑

qLqR=q g|qL ⊗ g|qR.

Extend these maps by linearity.

The quantum couple

Theorem: Group G with automorphic right action on finite quasigroup Q. Commutative ring S, tensor product GQ of free modules GS and QS. For g ∈ G and q ∈ Q, write g|q for g ⊗ q. Define (f|p ⊗ g|q)∇ =

   fg|q

if pg = q;

• therwise

and ∆: g|q → ∑

qLqR=q g|qL ⊗ g|qR.

Extend these maps by linearity. Then the quantum couple (GQ, ∇, ∆) is an associative quantum quasigroup in S.

The quantum couple

Theorem: Group G with automorphic right action on finite quasigroup Q. Commutative ring S, tensor product GQ of free modules GS and QS. For g ∈ G and q ∈ Q, write g|q for g ⊗ q. Define (f|p ⊗ g|q)∇ =

   fg|q

if pg = q;

• therwise

and ∆: g|q → ∑

qLqR=q g|qL ⊗ g|qR.

Extend these maps by linearity. Then the quantum couple (GQ, ∇, ∆) is an associative quantum quasigroup in S.

• If G = ⊤, then ⊤Q is a dual quasigroup algebra.

The quantum couple

Theorem: Group G with automorphic right action on finite quasigroup Q. Commutative ring S, tensor product GQ of free modules GS and QS. For g ∈ G and q ∈ Q, write g|q for g ⊗ q. Define (f|p ⊗ g|q)∇ =

   fg|q

if pg = q;

• therwise

and ∆: g|q → ∑

qLqR=q g|qL ⊗ g|qR.

Extend these maps by linearity. Then the quantum couple (GQ, ∇, ∆) is an associative quantum quasigroup in S.

• If G = ⊤, then ⊤Q is a dual quasigroup algebra.
• If Q = ⊤, then G⊤ is the (quasi-)group algebra GS of G.

The quantum couple

Theorem: Group G with automorphic right action on finite quasigroup Q. Commutative ring S, tensor product GQ of free modules GS and QS. For g ∈ G and q ∈ Q, write g|q for g ⊗ q. Define (f|p ⊗ g|q)∇ =

   fg|q

if pg = q;

• therwise

and ∆: g|q → ∑

qLqR=q g|qL ⊗ g|qR.

Extend these maps by linearity. Then the quantum couple (GQ, ∇, ∆) is an associative quantum quasigroup in S.

• If G = ⊤, then ⊤Q is a dual quasigroup algebra.
• If Q = ⊤, then G⊤ is the (quasi-)group algebra GS of G.
• If finite G acts on G by conjugation, then GG is the group quantum double.

Hopf algebras as quantum loops

Hopf algebras as quantum loops

Theorem: If (A, ∇, ∆, η, ε, S) is a Hopf algebra, the reduct (A, ∇, ∆, η, ε) is a quantum loop.

Hopf algebras as quantum loops

Theorem: If (A, ∇, ∆, η, ε, S) is a Hopf algebra, the reduct (A, ∇, ∆, η, ε) is a quantum loop. Proof: Coassociativity gives xLL ⊗ xLR ⊗ xR = xL ⊗ xRL ⊗ xRR.

Hopf algebras as quantum loops

Theorem: If (A, ∇, ∆, η, ε, S) is a Hopf algebra, the reduct (A, ∇, ∆, η, ε) is a quantum loop. Proof: Coassociativity gives xLL ⊗ xLR ⊗ xR = xL ⊗ xRL ⊗ xRR. Applying ⊗y and (1A ⊗ 1A ⊗ S ⊗ 1A)(1A ⊗ 1A ⊗ ∇)(1A ⊗ ∇) gives

xLL ⊗ xLRxRSy = xL ⊗ xRLxRRSy

Hopf algebras as quantum loops

Theorem: If (A, ∇, ∆, η, ε, S) is a Hopf algebra, the reduct (A, ∇, ∆, η, ε) is a quantum loop. Proof: Coassociativity gives xLL ⊗ xLR ⊗ xR = xL ⊗ xRL ⊗ xRR. Applying ⊗y and (1A ⊗ 1A ⊗ S ⊗ 1A)(1A ⊗ 1A ⊗ ∇)(1A ⊗ ∇) gives

xLL ⊗ xLRxRSy = xL ⊗ xRLxRRSy = xL ⊗ xRεηy

Hopf algebras as quantum loops

Theorem: If (A, ∇, ∆, η, ε, S) is a Hopf algebra, the reduct (A, ∇, ∆, η, ε) is a quantum loop. Proof: Coassociativity gives xLL ⊗ xLR ⊗ xR = xL ⊗ xRL ⊗ xRR. Applying ⊗y and (1A ⊗ 1A ⊗ S ⊗ 1A)(1A ⊗ 1A ⊗ ∇)(1A ⊗ ∇) gives

xLL ⊗ xLRxRSy = xL ⊗ xRLxRRSy = xL ⊗ xRεηy

[by the antipode property aLaRS = aεη]

Hopf algebras as quantum loops

Theorem: If (A, ∇, ∆, η, ε, S) is a Hopf algebra, the reduct (A, ∇, ∆, η, ε) is a quantum loop. Proof: Coassociativity gives xLL ⊗ xLR ⊗ xR = xL ⊗ xRL ⊗ xRR. Applying ⊗y and (1A ⊗ 1A ⊗ S ⊗ 1A)(1A ⊗ 1A ⊗ ∇)(1A ⊗ ∇) gives

xLL ⊗ xLRxRSy = xL ⊗ xRLxRRSy = xL ⊗ xRεηy

[by the antipode property aLaRS = aεη]

= xLxRεη ⊗ y

Hopf algebras as quantum loops

Theorem: If (A, ∇, ∆, η, ε, S) is a Hopf algebra, the reduct (A, ∇, ∆, η, ε) is a quantum loop. Proof: Coassociativity gives xLL ⊗ xLR ⊗ xR = xL ⊗ xRL ⊗ xRR. Applying ⊗y and (1A ⊗ 1A ⊗ S ⊗ 1A)(1A ⊗ 1A ⊗ ∇)(1A ⊗ ∇) gives

xLL ⊗ xLRxRSy = xL ⊗ xRLxRRSy = xL ⊗ xRεηy

[by the antipode property aLaRS = aεη]

= xLxRεη ⊗ y

[moving “scalars” round the tensor product]

Hopf algebras as quantum loops

Theorem: If (A, ∇, ∆, η, ε, S) is a Hopf algebra, the reduct (A, ∇, ∆, η, ε) is a quantum loop. Proof: Coassociativity gives xLL ⊗ xLR ⊗ xR = xL ⊗ xRL ⊗ xRR. Applying ⊗y and (1A ⊗ 1A ⊗ S ⊗ 1A)(1A ⊗ 1A ⊗ ∇)(1A ⊗ ∇) gives

xLL ⊗ xLRxRSy = xL ⊗ xRLxRRSy = xL ⊗ xRεηy

[by the antipode property aLaRS = aεη]

= xLxRεη ⊗ y

[moving “scalars” round the tensor product]

= x ⊗ y

Hopf algebras as quantum loops

Theorem: If (A, ∇, ∆, η, ε, S) is a Hopf algebra, the reduct (A, ∇, ∆, η, ε) is a quantum loop. Proof: Coassociativity gives xLL ⊗ xLR ⊗ xR = xL ⊗ xRL ⊗ xRR. Applying ⊗y and (1A ⊗ 1A ⊗ S ⊗ 1A)(1A ⊗ 1A ⊗ ∇)(1A ⊗ ∇) gives

xLL ⊗ xLRxRSy = xL ⊗ xRLxRRSy = xL ⊗ xRεηy

[by the antipode property aLaRS = aεη]

= xLxRεη ⊗ y

[moving “scalars” round the tensor product]

= x ⊗ y

[by the counitality property aLaRεη = a]

Hopf algebras as quantum loops

Theorem: If (A, ∇, ∆, η, ε, S) is a Hopf algebra, the reduct (A, ∇, ∆, η, ε) is a quantum loop. Proof: Coassociativity gives xLL ⊗ xLR ⊗ xR = xL ⊗ xRL ⊗ xRR. Applying ⊗y and (1A ⊗ 1A ⊗ S ⊗ 1A)(1A ⊗ 1A ⊗ ∇)(1A ⊗ ∇) gives

xLL ⊗ xLRxRSy = xL ⊗ xRLxRRSy = xL ⊗ xRεηy

[by the antipode property aLaRS = aεη]

= xLxRεη ⊗ y

[moving “scalars” round the tensor product]

= x ⊗ y

[by the counitality property aLaRεη = a] so

( (∆ ⊗ 1A)(1A ⊗ S ⊗ 1A)(1A ⊗ ∇) )( (∆ ⊗ 1A)(1A ⊗ ∇) ) = 1A⊗A.

Hopf algebras as quantum loops

Theorem: If (A, ∇, ∆, η, ε, S) is a Hopf algebra, the reduct (A, ∇, ∆, η, ε) is a quantum loop. Proof: Coassociativity gives xLL ⊗ xLR ⊗ xR = xL ⊗ xRL ⊗ xRR. Applying ⊗y and (1A ⊗ 1A ⊗ S ⊗ 1A)(1A ⊗ 1A ⊗ ∇)(1A ⊗ ∇) gives

xLL ⊗ xLRxRSy = xL ⊗ xRLxRRSy = xL ⊗ xRεηy

[by the antipode property aLaRS = aεη]

= xLxRεη ⊗ y

[moving “scalars” round the tensor product]

= x ⊗ y

[by the counitality property aLaRεη = a] so

( (∆ ⊗ 1A)(1A ⊗ S ⊗ 1A)(1A ⊗ ∇) )( (∆ ⊗ 1A)(1A ⊗ ∇) ) = 1A⊗A.

Other invertibility verifications similar.

Pointed comonoids

Pointed comonoids

Category K of vector spaces over field K.

Pointed comonoids

Category K of vector spaces over field K. A comonoid is simple if it has exactly two subcomonoids (one trivial, the other improper).

Pointed comonoids

Category K of vector spaces over field K. A comonoid is simple if it has exactly two subcomonoids (one trivial, the other improper). Comonoid is pointed if each simple subcomonoid is 1-dimensional.

Pointed comonoids

Category K of vector spaces over field K. A comonoid is simple if it has exactly two subcomonoids (one trivial, the other improper). Comonoid is pointed if each simple subcomonoid is 1-dimensional. Bimonoid is pointed if its comonoid reduct is.

Quantum loops as Hopf algebras

Quantum loops as Hopf algebras

Theorem: If (A, ∇, ∆, η, ε) is an associative, coassociative finite-dimensional quantum loop in K, and (A, ∆, ε) is pointed, then (A, ∇, ∆, η, ε) is a Hopf algebra.

Quantum loops as Hopf algebras

Theorem: If (A, ∇, ∆, η, ε) is an associative, coassociative finite-dimensional quantum loop in K, and (A, ∆, ε) is pointed, then (A, ∇, ∆, η, ε) is a Hopf algebra. Proof: Finite set A1 = {x ∈ A | a∆ = a ⊗ a , aε = 1} of setlike elements of A forms a monoid under multiplication, identity 1η = e.

Quantum loops as Hopf algebras

Theorem: If (A, ∇, ∆, η, ε) is an associative, coassociative finite-dimensional quantum loop in K, and (A, ∆, ε) is pointed, then (A, ∇, ∆, η, ε) is a Hopf algebra. Proof: Finite set A1 = {x ∈ A | a∆ = a ⊗ a , aε = 1} of setlike elements of A forms a monoid under multiplication, identity 1η = e. For x, y ∈ A1, injective left composite

x ⊗ y ✤∆⊗1A x ⊗ x ⊗ y ✤1A⊗∇ x ⊗ xy gives bijective j : A1 ⊗ A1 → A1 ⊗ A1.

Quantum loops as Hopf algebras

Theorem: If (A, ∇, ∆, η, ε) is an associative, coassociative finite-dimensional quantum loop in K, and (A, ∆, ε) is pointed, then (A, ∇, ∆, η, ε) is a Hopf algebra. Proof: Finite set A1 = {x ∈ A | a∆ = a ⊗ a , aε = 1} of setlike elements of A forms a monoid under multiplication, identity 1η = e. For x, y ∈ A1, injective left composite

x ⊗ y ✤∆⊗1A x ⊗ x ⊗ y ✤1A⊗∇ x ⊗ xy gives bijective j : A1 ⊗ A1 → A1 ⊗ A1.

So

∀ u ∈ A1 , ∃ v, w ∈ A1 . w ⊗ wv = (w ⊗ v)j = u ⊗ e, whence w = u and uv = e.

Quantum loops as Hopf algebras

Theorem: If (A, ∇, ∆, η, ε) is an associative, coassociative finite-dimensional quantum loop in K, and (A, ∆, ε) is pointed, then (A, ∇, ∆, η, ε) is a Hopf algebra. Proof: Finite set A1 = {x ∈ A | a∆ = a ⊗ a , aε = 1} of setlike elements of A forms a monoid under multiplication, identity 1η = e. For x, y ∈ A1, injective left composite

x ⊗ y ✤∆⊗1A x ⊗ x ⊗ y ✤1A⊗∇ x ⊗ xy gives bijective j : A1 ⊗ A1 → A1 ⊗ A1.

So

∀ u ∈ A1 , ∃ v, w ∈ A1 . w ⊗ wv = (w ⊗ v)j = u ⊗ e, whence w = u and uv = e.

Dually, ∃ v′ ∈ A1 . v′u = e:

Quantum loops as Hopf algebras

Theorem: If (A, ∇, ∆, η, ε) is an associative, coassociative finite-dimensional quantum loop in K, and (A, ∆, ε) is pointed, then (A, ∇, ∆, η, ε) is a Hopf algebra. Proof: Finite set A1 = {x ∈ A | a∆ = a ⊗ a , aε = 1} of setlike elements of A forms a monoid under multiplication, identity 1η = e. For x, y ∈ A1, injective left composite

x ⊗ y ✤∆⊗1A x ⊗ x ⊗ y ✤1A⊗∇ x ⊗ xy gives bijective j : A1 ⊗ A1 → A1 ⊗ A1.

So

∀ u ∈ A1 , ∃ v, w ∈ A1 . w ⊗ wv = (w ⊗ v)j = u ⊗ e, whence w = u and uv = e.

Dually, ∃ v′ ∈ A1 . v′u = e: Each setlike element is invertible.

Quantum loops as Hopf algebras

Theorem: If (A, ∇, ∆, η, ε) is an associative, coassociative finite-dimensional quantum loop in K, and (A, ∆, ε) is pointed, then (A, ∇, ∆, η, ε) is a Hopf algebra. Proof: Finite set A1 = {x ∈ A | a∆ = a ⊗ a , aε = 1} of setlike elements of A forms a monoid under multiplication, identity 1η = e. For x, y ∈ A1, injective left composite

x ⊗ y ✤∆⊗1A x ⊗ x ⊗ y ✤1A⊗∇ x ⊗ xy gives bijective j : A1 ⊗ A1 → A1 ⊗ A1.

So

∀ u ∈ A1 , ∃ v, w ∈ A1 . w ⊗ wv = (w ⊗ v)j = u ⊗ e, whence w = u and uv = e.

Dually, ∃ v′ ∈ A1 . v′u = e: Each setlike element is invertible. Since the bimonoid (A, ∇, ∆, η, ε) is pointed, Radford [Proposition 7.6.3] gives a Hopf algebra (A, ∇, ∆, η, ε, S).

Thank you for your attention!