On set-theoretic solutions to the Yang–Baxter equationlebed/Lebed BA.pdf · 2016-09-27 · On set-theoretic solutions to the Yang–Baxter equation Victoria LEBED (Nantes, France

On set-theoretic solutionsto the Yang–Baxter equation

Victoria LEBED (Nantes, France & Dublin, Ireland)with Leandro VENDRAMIN (Buenos Aires)

XXICLA, Buenos Aires, July 2016

1 Set-theoretic Yang–Baxter equation

✓ set S,✓ ff : Sˆ2→Sˆ2

Yang-Baxter equation (YBE)

ff1‹ff2‹ff1=ff2‹ff1‹ff2 : Sˆ3→Sˆ3

where ff1=ffˆ IdS, ff2= IdSˆff.

Origins: Drinfel 0d 1990.


✓ set S,✓ ff : Sˆ2→Sˆ2





(S;ff): braided set.

ff ←→

=

(Reidemeister III)


✓ set S,✓ ff : Sˆ2→Sˆ2 (x;y) 7→ (xy;xy)






ff ←→

x y

xy xy

=

(Reidemeister III)








➺ Left non-degenerate:x 7→xy bijection for all y.

ff ←→

x y

xy xy

=

(Reidemeister III)

y˜́x y

x ý x








➺ Left non-degenerate:x 7→xy bijection for all y.

➺ Birack: ff invertible andleft & right non-degenerate.

ff ←→

x y

xy xy

=

(Reidemeister III)

y˜́x y

x ý x

2 Structure (semi)group

Structure (semi)group of (S;ff): (S)GS;ff= hS |xy=xyxy i

braided sets groups & algebras

methodsww

2 Structure (semi)group

Structure (semi)group of (S;ff): (S)GS;ff= hS |xy=xyxy i

braided sets

examples

66groups & algebras

methodsww

Theorem: (S;ff) a finite RI-compatible birack, ff2= Id =⇒

✓ SGS;ff is of I-type, cancellative, Öre;✓ GS;ff is solvable, Garside;✓ kSGS;ff 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 S & SˆS�→S s.t.

(x�y)�z=(x�z)� (y�z)

⇔ ff�=

x y

y x�y

is a braiding on S



(x�y)�z=(x�z)� (y�z)

Rack: & 8y, x 7→x�y bijective.

⇔ ff�=

x y

y x�y

is a braiding on S

⇔ LND ⇔ birack



(x�y)�z=(x�z)� (y�z)


Quandle: & x�x=x.

⇔ ff�=

x y

y x�y

is a braiding on S

⇔ LND ⇔ birack

⇔ (x;x)ff�

7→ (x;x).



(x�y)�z=(x�z)� (y�z)


Quandle: & x�x=x.

⇔ ff�=

x y

y x�y

is a braiding on S

⇔ LND ⇔ birack

⇔ (x;x)ff�

7→ (x;x).

Examples:

➺ Z[t]-module S & a�b= ta+(1− t)b;



(x�y)�z=(x�z)� (y�z)


Quandle: & x�x=x.

⇔ ff�=

x y

y x�y

is a braiding on S

⇔ LND ⇔ birack

⇔ (x;x)ff�

7→ (x;x).

Examples:


➺ group S & x�y=y−1xy:



(x�y)�z=(x�z)� (y�z)


Quandle: & x�x=x.

⇔ ff�=

x y

y x�y

is a braiding on S

⇔ LND ⇔ birack

⇔ (x;x)ff�

7→ (x;x).

Examples:



rack = “(w)rack and ruin of a group”



(x�y)�z=(x�z)� (y�z)


Quandle: & x�x=x.

⇔ ff�=

x y

y x�y

is a braiding on S

⇔ LND ⇔ birack

⇔ (x;x)ff�

7→ (x;x).

Examples:




adjunction Groups⇄Quandles



(x�y)�z=(x�z)� (y�z)


Quandle: & x�x=x.

⇔ ff�=

x y

y x�y

is a braiding on S

⇔ LND ⇔ birack

⇔ (x;x)ff�

7→ (x;x).

Examples:




adjunction Groups⇄Quandles

GS;ff�= hS |x�y=y−1xy i



(x�y)�z=(x�z)� (y�z)


Quandle: & x�x=x.

⇔ ff�=

x y

y x�y

is a braiding on S

⇔ LND ⇔ birack

⇔ (x;x)ff�

7→ (x;x).

Applications:

➺ invariants of knots and knotted surfaces(Joyce & Matveev 1982);

➺ study of large cardinals(Laver 1980s);

➺ Hopf algebra classification(Andruskiewitsch–Graña 2003).

4 Monoids

For a monoid (S;⋆; 1),the associativity of ⋆ ⇔ ff⋆=

x y

1 x⋆y

is an involutive braiding on S

4 Monoids

For a monoid (S;⋆; 1),the associativity of ⋆

S is a group

⇔ ff⋆=

x y

1 x⋆y


⇒ LND

4 Monoids


S is a group

⇔ ff⋆=

x y

1 x⋆y


⇒ LND

SGS;ff⋆∼→S;

Sˆk3x1 ´ ´ ´xk 7→x1 ⋆ ´ ´ ´⋆xk:

4 Monoids


S is a group

⇔ ff⋆=

x y

1 x⋆y


⇒ LND

SGS;ff⋆∼→S;

Sˆk3x1 ´ ´ ´xk 7→x1 ⋆ ´ ´ ´⋆xk:

Generalization: monoid factorization G=HK,S=H[K, ff(x;y)= (h;k), h2H, k2K, hk=xy;

S’SGS;ff=1S=1SG.

4 Monoids


S is a group

⇔ ff⋆=

x y

1 x⋆y


⇒ LND

SGS;ff⋆∼→S;

Sˆk3x1 ´ ´ ´xk 7→x1 ⋆ ´ ´ ´⋆xk:

Generalization: monoid factorization G=HK,S=H[K, ff(x;y)= (h;k), h2H, k2K, hk=xy;

S’SGS;ff=1S=1SG.

Other examples:✓ cycle sets, braces;✓ Young tableaux;✓ distributive lattices.

5 Associated shelf

Fix an LND braided set (S;ff).

ff ←→

x y

xy xy

y˜́x y

x ý x

Proposition (L.-V. 2015): one has a shelf (S;�ff), where

x

y

(y ´x)y=:x�ff y

y ´x

5 Associated shelf

Fix an LND braided set (S;ff).Proposition (L.-V. 2015): one has a shelf (S;�ff), where

x

y

(y ´x)y=:x�ff y

y ´x

Proof:

5 Associated shelf


x

y

(y ´x)y=:x�ff y

y ´x

Proof:

5 Associated shelf


x

y

(y ´x)y=:x�ff y

y ´x

Proof:

5 Associated shelf


x

y

(y ´x)y=:x�ff y

y ´x

Proof:

5 Associated shelf


x

y

(y ´x)y=:x�ff y

y ´x

�ff is a “shadow” of ff

5 Associated shelf


x

y

(y ´x)y=:x�ff y

y ´x

�ff is a “shadow” of ff

Proposition (L.-V. 2015):

➺ (S;�ff) is a rack ⇔ ff is invertible;➺ (S;�ff) is a trivial (x�ff y=x) ⇔ ff2= Id;➺ x�ffx=x ⇔ ff(x ´x;x)= (x ´x;x).

6 Guitar map

J(n) : Sˆn∼→Sˆn;

(x1; : : : ;xn) 7→ (xx2´´´xn1 ; : : : ;x

xnn−1;xn);

where xxi+1´´´xni =(: : :(x

xi+1i ) : : :)xn.

x1 ´´´ xn

J1(x)

J2(x)

...

Jn(x)

6 Guitar map


(x1; : : : ;xn) 7→ (xx2´´´xn1 ; : : : ;x

xnn−1;xn);


xi+1i ) : : :)xn.

x1 ´´´ xn

J1(x)

J2(x)

...

Jn(x)

x2 x3 x4 x5

J2(x)=

xx3x4x52

6 Guitar map


(x1; : : : ;xn) 7→ (xx2´´´xn1 ; : : : ;x

xnn−1;xn);


xi+1i ) : : :)xn.

x1 ´´´ xn

J1(x)

J2(x)

...

Jn(x)

x2 x3 x4 x5

J2(x)=

xx3x4x52

Proposition (L.-V. 2015): Jffi=ff0iJ .

ff=

x y

xy xy

ff 0=

x y

y�ff x x

6 Guitar map


(x1; : : : ;xn) 7→ (xx2´´´xn1 ; : : : ;x

xnn−1;xn);


xi+1i ) : : :)xn.

x1 ´´´ xn

J1(x)

J2(x)

...

Jn(x)

x2 x3 x4 x5

J2(x)=

xx3x4x52

Proposition (L.-V. 2015): Jffi=ff0iJ .

Corollary: ff and ff 0 yield isomorphic Bn-actions on Sˆn.

Warning: In general, (S;ff)≇ (S;ff 0) as braided sets!

7 RI-compatibility

RI-compatible braiding: 9t : S∼→S s.t. ff(t(x);x)= (t(x);x).

x

xt(x) =

x

x=

x

xt−1(x)

(Reidemeister I)

7 RI-compatibility

RI-compatible braiding: 9t : S∼→S s.t. ff(t(x);x)= (t(x);x).

x

xt(x) =

x

x=

x

xt−1(x)

(Reidemeister I)

Example:for a rack, it means x�x=x (here t(x)=x).

8 Structure group via associated shelf

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

bijective 1-cocycle J : SGS;ff∼→SGS;ff 0 , where ff 0=ff 0�ff .

Reminder: SGS;ff= hS |xy=xyxy i.



bijective 1-cocycle︸︷︷︸

J : SGS;ff∼→SGS;ff 0 , where ff 0=ff 0�ff .

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





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

(x1; : : : ;xn)y=

(xy1; : : : ;x

yn)

x y

xy





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

(x1; : : : ;xn)y=

(xy1; : : : ;x

yn)

x y

xy

(2) If (S;ff) is an RI-compatible birack, then the maps

KˆnJ(n) induce a bijective 1-cocycle GS;ff∼→GS;ff 0 ,





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

(x1; : : : ;xn)y=

(xy1; : : : ;x

yn)

x y

xy


KˆnJ(n) induce a bijective 1-cocycle GS;ff∼→GS;ff 0 , where

➺ J(n) is extended to (StS−1)ˆn by

x1 x2−1 x3−1

y1

y2−1

y3−1





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

(x1; : : : ;xn)y=

(xy1; : : : ;x

yn)

x y

xy


KˆnJ(n) induce a bijective 1-cocycle GS;ff∼→GS;ff 0 , where

➺ J(n) is extended to (StS−1)ˆn by

➺ K(x)=x, K(x−1)= t(x)−1.x1 x2−1 x3

−1

y1

y2−1

y3−1

9 Associated shelf: examples

✓ For a rack (S;�)

➺ �ff�=�,

➺ J :

x y

y x�y↔

x y

y�x x.


✓ For a rack (S;�)

➺ �ff�=�,

➺ J :

x y

y x�y↔

x y

y�x x.

✓ For a group (S;⋆; 1)

➺ x�ff⋆ y=y,

➺ J : ff⋆↔

x y

x x,

➺ SGS;ff 0⋆

∼→S,

x1 ´ ´ ´xk 7→x1.


Cycle set: set S & SˆS´→S s.t.

(x ý) ´ (x ´z)= (y ´x) ´ (y ´z)

& 8x, y 7→x ý bijective.⇔ ff´=

y ´x y

x ý x is a LNDbraidingon S

with ff2´ = Id



(x ý) ´ (x ´z)= (y ´x) ´ (y ´z)

& 8x, y 7→x ý bijective.

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

⇔ ff´=

y ´x y


with ff2´ = Id

⇔ birack



(x ý) ´ (x ´z)= (y ´x) ´ (y ´z)



⇔ ff´=

y ´x y


with ff2´ = Id

⇔ birack

(Etingof–Schedler–Soloviev 1999, Rump 2005)



(x ý) ´ (x ´z)= (y ´x) ´ (y ´z)



⇔ ff´=

y ´x y


with ff2´ = Id

⇔ birack


Examples: x ý=f(y) for any f : S∼→S + glueing.



(x ý) ´ (x ´z)= (y ´x) ´ (y ´z)



⇔ ff´=

y ´x y


with ff2´ = Id

⇔ birack



➺ x�ff´ y=x,

➺ J : ff´↔ flip

x y

y x,



(x ý) ´ (x ´z)= (y ´x) ´ (y ´z)



⇔ ff´=

y ´x y


with ff2´ = Id

⇔ birack



➺ x�ff´ y=x,

➺ J : ff´↔ flip

x y

y x,

➺ (S)GS;ff 0´ is the free abelian (semi)group on S.

10 Braided (co)homology

(co)homology of (S;ff) // (co)homology of SGS;ffoo

small complexes more tools

(Carter–Elhamdadi–Saito 2004)





✓ unifies group & rack (co)homologies, and many more;






✓ a new theory for cycle sets and braces (L.-V. 2015, 2016);







✓ graphical calculus;

✓ a lot of structure (cup product etc.; Farinati– García-Galofre 2016);









✓ for LND braidings, a simpler form via the guitar map(L.-V. 2015);









✓ for LND braidings, a simpler form via the guitar map(L.-V. 2015);

✓ applications to the computation of group and Hochschild(co)homology for factorizable groups and for Youngtableaux (L. 2016).

On set-theoretic solutions to the Yang–Baxter equationlebed/Lebed BA.pdf · 2016-09-27 · On set-theoretic solutions to the Yang–Baxter equation Victoria LEBED (Nantes, France

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