On set-theoretic solutionsto the Yang-Baxter equation

Victoria LEBED (Nantes)with Leandro VENDRAMIN (Buenos Aires)

Mulhouse, October 21, 2015

1 Yang-Baxter equation

✓ V: vector space,✓ σ : V⊗2→V⊗2.

Yang-Baxter equation (YBE)

σ1 ◦σ2 ◦σ1 = σ2 ◦σ1 ◦σ2 : V⊗3→V⊗3

where σ1 = σ⊗ IdV , σ2 = IdV⊗σ.

Origins:➺ factorization condition for thedispersion matrix in the 1-dim. n-bodyproblem (McGuire, Yang, 60’);➺ partition function for exactly solvablelattice models (Baxter, 70’).

1 Set-theoretic Yang-Baxter equation

✓ V: set,✓ σ : V×2→V×2

Yang-Baxter equation (YBE)

σ1 ◦σ2 ◦σ1 = σ2 ◦σ1 ◦σ2 : V×3→ V×3

where σ1 = σ× IdV , σ2 = IdV×σ.

Origins: Drinfel ′d, 1990.

1 Set-theoretic Yang-Baxter equation

✓ V: set,✓ σ : V×2→V×2, (x,y) 7→ (xy,xy)

Yang-Baxter equation (YBE)

σ1 ◦σ2 ◦σ1 = σ2 ◦σ1 ◦σ2 : V×3→ V×3

where σ1 = σ× IdV , σ2 = IdV×σ.

Origins: Drinfel ′d, 1990.

(V,σ) is called a braided set,with a braiding σ.

σ ←→

x y

xy xy

=

(Reidemeister III)

1 Set-theoretic Yang-Baxter equation

✓ V: set,✓ σ : V×2→V×2, (x,y) 7→ (xy,xy)

Yang-Baxter equation (YBE)

σ1 ◦σ2 ◦σ1 = σ2 ◦σ1 ◦σ2 : V×3→ V×3

where σ1 = σ× IdV , σ2 = IdV×σ.

Origins: Drinfel ′d, 1990.

(V,σ) is called a braided set,with a braiding σ.

Left non-degenerate braided set:for all y, x 7→ xy is a bijection X→X.

σ ←→

x y

xy xy

=

(Reidemeister III)

y ·̃ x y

x ·y x

1 Set-theoretic Yang-Baxter equation

✓ V: set,✓ σ : V×2→V×2, (x,y) 7→ (xy,xy)

Yang-Baxter equation (YBE)

σ1 ◦σ2 ◦σ1 = σ2 ◦σ1 ◦σ2 : V×3→ V×3

where σ1 = σ× IdV , σ2 = IdV×σ.

Origins: Drinfel ′d, 1990.

(V,σ) is called a braided set,with a braiding σ.

Left non-degenerate braided set:for all y, x 7→ xy is a bijection X→X.

Birack: left and right non-degeneratebraided set with invertible σ.

σ ←→

x y

xy xy

=

(Reidemeister III)

y ·̃ x y

x ·y x

2 Structure (semi)group

✓ X: set,✓ σ : X×2→X×2, (x,y) 7→ (xy,xy).

Structure (semi)group of (V,σ): (S)GX,σ = 〈X | xy=xyxy 〉

2 Structure (semi)group

✓ X: set,✓ σ : X×2→X×2, (x,y) 7→ (xy,xy).

Structure (semi)group of (V,σ): (S)GX,σ = 〈X | xy=xyxy 〉

➺ Captures properties of σ.

2 Structure (semi)group

✓ X: set,✓ σ : X×2→X×2, (x,y) 7→ (xy,xy).

Structure (semi)group of (V,σ): (S)GX,σ = 〈X | xy=xyxy 〉

➺ Captures properties of σ.

➺ A source of interesting groups and algebras:Theorem: If (X,σ) is a finite RI-compatible birack withσ2 = Id, then

✓ SGX,σ is of I-type, cancellative, Öre;✓ GX,σ is solvable, Garside;✓ kSGX,σ is Koszul, noetherian, Cohen-Macaulay,

Artin-Schelter regular(Manin, Gateva-Ivanova & Van den Bergh, Etingof-Schedler-Soloviev, Jespers-Okniński, Chouraqui, 80’-. . . ).

3 Self-distributive structures

Shelf: set X & � : X×X→X s.t.

(x� y)� z= (x� z)� (y� z) ⇔ σ� =

x y

y x� y is abraidingon X

3 Self-distributive structures

Shelf: set X & � : X×X→X s.t.

(x� y)� z= (x� z)� (y� z)

Rack: & for all y, x 7→ x� y bijective.

⇔ σ� =

x y

y x� y is abraidingon X

⇔ σ� is LND⇔ (X,σ�) is a birack

3 Self-distributive structures

Shelf: set X & � : X×X→X s.t.

(x� y)� z= (x� z)� (y� z)

Rack: & for all y, x 7→ x� y bijective.

Quandle: & x� x= x.

⇔ σ� =

x y

y x� y is abraidingon X

⇔ σ� is LND⇔ (X,σ�) is a birack

⇔ (x,x)σ�7→ (x,x).

3 Self-distributive structures

Shelf: set X & � : X×X→X s.t.

(x� y)� z= (x� z)� (y� z)

Rack: & for all y, x 7→ x� y bijective.

Quandle: & x� x= x.

⇔ σ� =

x y

y x� y is abraidingon X

⇔ σ� is LND⇔ (X,σ�) is a birack

⇔ (x,x)σ�7→ (x,x).

(Joyce, Matveev, 1982)Applications:

➺ invariants of knots and knotted surfaces;➺ Hopf algebra classification;➺ study of large cardinals.

3 Self-distributive structures

Shelf: set X & � : X×X→X s.t.

(x� y)� z= (x� z)� (y� z)

Rack: & for all y, x 7→ x� y bijective.

Quandle: & x� x= x.

⇔ σ� =

x y

y x� y is abraidingon X

⇔ σ� is LND⇔ (X,σ�) is a birack

⇔ (x,x)σ�7→ (x,x).

(Joyce, Matveev, 1982)Applications:

➺ invariants of knots and knotted surfaces;➺ Hopf algebra classification;➺ study of large cardinals.

Examples:➺ x� y= f(x) for any f : X→X;

3 Self-distributive structures

Shelf: set X & � : X×X→X s.t.

(x� y)� z= (x� z)� (y� z)

Rack: & for all y, x 7→ x� y bijective.

Quandle: & x� x= x.

⇔ σ� =

x y

y x� y is abraidingon X

⇔ σ� is LND⇔ (X,σ�) is a birack

⇔ (x,x)σ�7→ (x,x).

(Joyce, Matveev, 1982)Applications:

➺ invariants of knots and knotted surfaces;➺ Hopf algebra classification;➺ study of large cardinals.

Examples:➺ x� y= f(x) for any f : X→X;➺ group X, x� y= y−1xy.

3 Self-distributive structures

Shelf: set X & � : X×X→X s.t.

(x� y)� z= (x� z)� (y� z)

Rack: & for all y, x 7→ x� y bijective.

Quandle: & x� x= x.

⇔ σ� =

x y

y x� y is abraidingon X

⇔ σ� is LND⇔ (X,σ�) is a birack

⇔ (x,x)σ�7→ (x,x).

(Joyce, Matveev, 1982)Applications:

➺ invariants of knots and knotted surfaces;➺ Hopf algebra classification;➺ study of large cardinals.

Examples:➺ x� y= f(x) for any f : X→X;➺ group X, x� y= y−1xy.

GX,σ� = 〈X | x� y= y−1xy 〉.

3 Self-distributive structures

Shelf: set X & � : X×X→X s.t.

(x� y)� z= (x� z)� (y� z)

Rack: & for all y, x 7→ x� y bijective.

Quandle: & x� x= x.

⇔ σ�′ =

x y

y� x x is abraidingon X

⇔ σ� is RND⇔ (X,σ�) is a birack

⇔ (x,x)σ�7→ (x,x).

(Joyce, Matveev, 1982)Applications:

➺ invariants of knots and knotted surfaces;➺ Hopf algebra classification;➺ study of large cardinals.

Examples:➺ x� y= f(x) for any f : X→X;➺ group X, x� y= y−1xy.

GX,σ� = 〈X | x� y= y−1xy 〉.

4 Cycle sets

Cycle set: set X & · : X×X→X s.t.

(x ·y) · (x · z) = (y ·x) · (y · z)

& for all x, y 7→ x ·y bijective.

⇔ σ· =

y ·x y

x ·y x is a LNDbraidingon X

with σ2· = Id

4 Cycle sets

Cycle set: set X & · : X×X→X s.t.

(x ·y) · (x · z) = (y ·x) · (y · z)

& for all x, y 7→ x ·y bijective.

Non-degenerate CS:& x 7→ x ·x bijective.

⇔ σ· =

y ·x y

x ·y x is a LNDbraidingon X

with σ2· = Id

⇔ σ· is RND⇔ (X,σ·) is a birack

4 Cycle sets

Cycle set: set X & · : X×X→X s.t.

(x ·y) · (x · z) = (y ·x) · (y · z)

& for all x, y 7→ x ·y bijective.

Non-degenerate CS:& x 7→ x ·x bijective.

⇔ σ· =

y ·x y

x ·y x is a LNDbraidingon X

with σ2· = Id

⇔ σ· is RND⇔ (X,σ·) is a birack

(Etingof-Schedler- Soloviev 1999, Rump 2005)

Applications: (semi)group theory.

4 Cycle sets

Cycle set: set X & · : X×X→X s.t.

(x ·y) · (x · z) = (y ·x) · (y · z)

& for all x, y 7→ x ·y bijective.

Non-degenerate CS:& x 7→ x ·x bijective.

⇔ σ· =

y ·x y

x ·y x is a LNDbraidingon X

with σ2· = Id

⇔ σ· is RND⇔ (X,σ·) is a birack

(Etingof-Schedler- Soloviev 1999, Rump 2005)

Applications: (semi)group theory.

Examples: x ·y= f(y) for any f : X∼

→X.

4 Cycle sets

Cycle set: set X & · : X×X→X s.t.

(x ·y) · (x · z) = (y ·x) · (y · z)

& for all x, y 7→ x ·y bijective.

Non-degenerate CS:& x 7→ x ·x bijective.

⇔ σ· =

y ·x y

x ·y x is a LNDbraidingon X

with σ2· = Id

⇔ σ· is RND⇔ (X,σ·) is a birack

(Etingof-Schedler- Soloviev 1999, Rump 2005)

Applications: (semi)group theory.

Examples: x ·y= f(y) for any f : X∼

→X.

GX,σ· = 〈X | (x ·y)x= (y ·x)y 〉.

5 Monoids

For a monoid (X,⋆,1),the associativity of ⋆ ⇔ σ⋆ =

x y

1 x⋆y is a

braiding

on X

(with σ2⋆= σ⋆)

5 Monoids

For a monoid (X,⋆,1),the associativity of ⋆

X is a group

⇔ σ⋆ =

x y

1 x⋆y is a

braiding

on X

(with σ2⋆= σ⋆)

⇒ σ⋆ is LND

5 Monoids

For a monoid (X,⋆,1),the associativity of ⋆

X is a group

⇔ σ⋆ =

x y

1 x⋆y is a

braiding

on X

(with σ2⋆= σ⋆)

⇒ σ⋆ is LND

(L. 2013)

5 Monoids

For a monoid (X,⋆,1),the associativity of ⋆

X is a group

⇔ σ⋆ =

x y

1 x⋆y is a

braiding

on X

(with σ2⋆= σ⋆)

⇒ σ⋆ is LND

(L. 2013)

One has a semigroup isomorphism

SGX,σ⋆∼

→X,

x1 · · ·xk 7→ x1 ⋆ · · · ⋆xk.

6 Associated shelf

Fix a LND braided set (X,σ).Proposition (L.-V. 2015): one has a shelf (X,�σ), where

x

y

(y ·x)y =: x�σ y

y ·x

6 Associated shelf

Fix a LND braided set (X,σ).Proposition (L.-V. 2015): one has a shelf (X,�σ), where

x

y

(y ·x)y =: x�σ y

y ·x

Proof:

6 Associated shelf

Fix a LND braided set (X,σ).Proposition (L.-V. 2015): one has a shelf (X,�σ), where

x

y

(y ·x)y =: x�σ y

y ·x

Proof:

6 Associated shelf

Fix a LND braided set (X,σ).Proposition (L.-V. 2015): one has a shelf (X,�σ), where

x

y

(y ·x)y =: x�σ y

y ·x

Proof:

6 Associated shelf

Fix a LND braided set (X,σ).Proposition (L.-V. 2015): one has a shelf (X,�σ), where

x

y

(y ·x)y =: x�σ y

y ·x

Proof:

6 Associated shelf

Fix a LND braided set (X,σ).Proposition (L.-V. 2015): one has a shelf (X,�σ), where

x

y

(y ·x)y =: x�σ y

y ·x

Motto: �σ is often simpler than σ,but encodes its important properties.

6 Associated shelf

Fix a LND braided set (X,σ).Proposition (L.-V. 2015): one has a shelf (X,�σ), where

x

y

(y ·x)y =: x�σ y

y ·x

Motto: �σ is often simpler than σ,but encodes its important properties.

Proposition (L.-V. 2015):

➺ (X,�σ) is a rack ⇔ σ is invertible;➺ (X,�σ) is a trivial (x�σ y= x) ⇔ σ2 = Id;➺ x�σ x= x ⇔ σ(x ·x,x) = (x ·x,x).

7 Guitar map

J(n) : X×n∼

→X×n,

(x1, . . . ,xn) 7→ (xx2···xn1 , . . . ,x

xnn−1,xn),

where xxi+1···xni = (. . . (xxi+1i ) . . .)

xn.

x1 ··· xn

J1(x)

J2(x)

...

Jn(x)

7 Guitar map

J(n) : X×n∼

→X×n,

(x1, . . . ,xn) 7→ (xx2···xn1 , . . . ,x

xnn−1,xn),

where xxi+1···xni = (. . . (xxi+1i ) . . .)

xn.

x1 ··· xn

J1(x)

J2(x)

...

Jn(x)

x2 x3 x4 x5

J2(x)=

xx3x4x52

7 Guitar map

J(n) : X×n∼

→X×n,

(x1, . . . ,xn) 7→ (xx2···xn1 , . . . ,x

xnn−1,xn),

where xxi+1···xni = (. . . (xxi+1i ) . . .)

xn.

x1 ··· xn

J1(x)

J2(x)

...

Jn(x)

x2 x3 x4 x5

J2(x)=

xx3x4x52

Proposition (L.-V. 2015): Jσi = σ′iJ , where σ

′ = σ ′�σ.

σ ′ =

x y

y�σ x x

7 Guitar map

J(n) : X×n∼

→X×n,

(x1, . . . ,xn) 7→ (xx2···xn1 , . . . ,x

xnn−1,xn),

where xxi+1···xni = (. . . (xxi+1i ) . . .)

xn.

x1 ··· xn

J1(x)

J2(x)

...

Jn(x)

x2 x3 x4 x5

J2(x)=

xx3x4x52

Proposition (L.-V. 2015): Jσi = σ′iJ , where σ

′ = σ ′�σ.

Corollary: σ and σ ′ yield isomorphic Bn-actions on X×n.Warning: In general, (X,σ)≇ (X,σ ′) as braided sets!

8 RI-compatibility

RI-compatible braiding: ∃t : X∼

→X s.t. σ(t(x),x) = (t(x),x).

x

xt(x) =

x

x=

x

xt−1(x)

(Reidemeister I)

8 RI-compatibility

RI-compatible braiding: ∃t : X∼

→X s.t. σ(t(x),x) = (t(x),x).

x

xt(x) =

x

x=

x

xt−1(x)

(Reidemeister I)

Examples:➺ for a rack, it means x� x= x (here t(x) = x);

8 RI-compatibility

RI-compatible braiding: ∃t : X∼

→X s.t. σ(t(x),x) = (t(x),x).

x

xt(x) =

x

x=

x

xt−1(x)

(Reidemeister I)

Examples:➺ for a rack, it means x� x= x (here t(x) = x);➺ for a cycle set, it means the non-degeneracy (here

t(x) = x ·x).

9 Structure group via associated shelf

Theorem (L.-V. 2015): (1) The guitar maps induce a

bijective 1-cocycle J : SGX,σ∼

→ SGX,σ ′ , where σ′ = σ ′

�σ.

9 Structure group via associated shelf

Theorem (L.-V. 2015): (1) The guitar maps induce a

bijective 1-cocycle︸ ︷︷ ︸

J : SGX,σ∼

→ SGX,σ ′ , where σ′ = σ ′

�σ.

J(xy) = J(x)yJ(y)

9 Structure group via associated shelf

Theorem (L.-V. 2015): (1) The guitar maps induce a

bijective 1-cocycle︸ ︷︷ ︸

J : SGX,σ∼

→ SGX,σ ′ , where σ′ = σ ′

�σ.

J(xy) = J(x)yJ(y)

(x1, . . . ,xn)y =

(xy1 , . . . ,x

yn)

x y

xy

9 Structure group via associated shelf

Theorem (L.-V. 2015): (1) The guitar maps induce a

bijective 1-cocycle︸ ︷︷ ︸

J : SGX,σ∼

→ SGX,σ ′ , where σ′ = σ ′

�σ.

J(xy) = J(x)yJ(y)

(x1, . . . ,xn)y =

(xy1 , . . . ,x

yn)

x y

xy

(2) If (X,σ) is an RI-compatible birack, then the maps

K×nJ(n) induce a bijective 1-cocycle GX,σ∼

→GX,σ ′ ,

9 Structure group via associated shelf

Theorem (L.-V. 2015): (1) The guitar maps induce a

bijective 1-cocycle︸ ︷︷ ︸

J : SGX,σ∼

→ SGX,σ ′ , where σ′ = σ ′

�σ.

J(xy) = J(x)yJ(y)

(x1, . . . ,xn)y =

(xy1 , . . . ,x

yn)

x y

xy

(2) If (X,σ) is an RI-compatible birack, then the maps

K×nJ(n) induce a bijective 1-cocycle GX,σ∼

→GX,σ ′ , where

➺ J(n) is extended to (X⊔X−1)×n by

x1 x2−1 x3

−1

y1

y2−1

y3−1

9 Structure group via associated shelf

Theorem (L.-V. 2015): (1) The guitar maps induce a

bijective 1-cocycle︸ ︷︷ ︸

J : SGX,σ∼

→ SGX,σ ′ , where σ′ = σ ′

�σ.

J(xy) = J(x)yJ(y)

(x1, . . . ,xn)y =

(xy1 , . . . ,x

yn)

x y

xy

(2) If (X,σ) is an RI-compatible birack, then the maps

K×nJ(n) induce a bijective 1-cocycle GX,σ∼

→GX,σ ′ , where

➺ J(n) is extended to (X⊔X−1)×n by

➺ K(x) = x, K(x−1) = t(x)−1.x1 x2

−1 x3−1

y1

y2−1

y3−1

10 Associated shelf: examples

✓ For a rack (X,�)➺ �σ�=�,➺ J : σ�↔ σ

′�.

10 Associated shelf: examples

✓ For a rack (X,�)➺ �σ�=�,➺ J : σ�↔ σ

′�.

✓ For a cycle set (X, ·)➺ x�σ· y= x,

10 Associated shelf: examples

✓ For a rack (X,�)➺ �σ�=�,➺ J : σ�↔ σ

′�.

✓ For a cycle set (X, ·)➺ x�σ· y= x,

➺ J : σ·↔ flipx y

y x,

➺ (S)GX,σ ′· is the free abelian (semi)group on X.

10 Associated shelf: examples

✓ For a rack (X,�)➺ �σ�=�,➺ J : σ�↔ σ

′�.

✓ For a cycle set (X, ·)➺ x�σ· y= x,

➺ J : σ·↔ flipx y

y x,

➺ (S)GX,σ ′· is the free abelian (semi)group on X.

✓ For a group (X,⋆,1)➺ x�σ⋆ y= y,

10 Associated shelf: examples

✓ For a rack (X,�)➺ �σ�=�,➺ J : σ�↔ σ

′�.

✓ For a cycle set (X, ·)➺ x�σ· y= x,

➺ J : σ·↔ flipx y

y x,

➺ (S)GX,σ ′· is the free abelian (semi)group on X.

✓ For a group (X,⋆,1)➺ x�σ⋆ y= y,➺ J(x1, . . . ,xn−1,xn) = (x1 ⋆ · · · ⋆xn, . . . , xn−1 ⋆xn, xn),

10 Associated shelf: examples

✓ For a rack (X,�)➺ �σ�=�,➺ J : σ�↔ σ

′�.

✓ For a cycle set (X, ·)➺ x�σ· y= x,

➺ J : σ·↔ flipx y

y x,

➺ (S)GX,σ ′· is the free abelian (semi)group on X.

✓ For a group (X,⋆,1)➺ x�σ⋆ y= y,➺ J(x1, . . . ,xn−1,xn) = (x1 ⋆ · · · ⋆xn, . . . , xn−1 ⋆xn, xn),

➺ J : σ⋆↔

x y

x x,

➺ SGX,σ ′⋆

∼

→X, x1 · · ·xk 7→ x1.

YBE: preliminariesFamilies of solutionsAssociated shelves