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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.
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.
(S;ff): braided set.
ff ←→
=
(Reidemeister III)
1 Set-theoretic Yang–Baxter equation
✓ set S,✓ ff : Sˆ2→Sˆ2 (x;y) 7→ (xy;xy)
Yang-Baxter equation (YBE)
ff1‹ff2‹ff1=ff2‹ff1‹ff2 : Sˆ3→Sˆ3
where ff1=ffˆ IdS, ff2= IdSˆff.
Origins: Drinfel 0d 1990.
(S;ff): braided set.
ff ←→
x y
xy xy
=
(Reidemeister III)
1 Set-theoretic Yang–Baxter equation
✓ set S,✓ ff : Sˆ2→Sˆ2 (x;y) 7→ (xy;xy)
Yang-Baxter equation (YBE)
ff1‹ff2‹ff1=ff2‹ff1‹ff2 : Sˆ3→Sˆ3
where ff1=ffˆ IdS, ff2= IdSˆff.
Origins: Drinfel 0d 1990.
(S;ff): braided set.
➺ Left non-degenerate:x 7→xy bijection for all y.
ff ←→
x y
xy xy
=
(Reidemeister III)
y˜́x y
x ´y x
1 Set-theoretic Yang–Baxter equation
✓ set S,✓ ff : Sˆ2→Sˆ2 (x;y) 7→ (xy;xy)
Yang-Baxter equation (YBE)
ff1‹ff2‹ff1=ff2‹ff1‹ff2 : Sˆ3→Sˆ3
where ff1=ffˆ IdS, ff2= IdSˆff.
Origins: Drinfel 0d 1990.
(S;ff): braided set.
➺ 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 ´y 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
3 Self-distributive structures
Shelf: set S & SˆS�→S s.t.
(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
3 Self-distributive structures
Shelf: set S & SˆS�→S s.t.
(x�y)�z=(x�z)� (y�z)
Rack: & 8y, x 7→x�y bijective.
Quandle: & x�x=x.
⇔ ff�=
x y
y x�y
is a braiding on S
⇔ LND ⇔ birack
⇔ (x;x)ff�
7→ (x;x).
3 Self-distributive structures
Shelf: set S & SˆS�→S s.t.
(x�y)�z=(x�z)� (y�z)
Rack: & 8y, x 7→x�y bijective.
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;
3 Self-distributive structures
Shelf: set S & SˆS�→S s.t.
(x�y)�z=(x�z)� (y�z)
Rack: & 8y, x 7→x�y bijective.
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;
➺ group S & x�y=y−1xy:
3 Self-distributive structures
Shelf: set S & SˆS�→S s.t.
(x�y)�z=(x�z)� (y�z)
Rack: & 8y, x 7→x�y bijective.
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;
➺ group S & x�y=y−1xy:
rack = “(w)rack and ruin of a group”
3 Self-distributive structures
Shelf: set S & SˆS�→S s.t.
(x�y)�z=(x�z)� (y�z)
Rack: & 8y, x 7→x�y bijective.
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;
➺ group S & x�y=y−1xy:
rack = “(w)rack and ruin of a group”
adjunction Groups⇄Quandles
3 Self-distributive structures
Shelf: set S & SˆS�→S s.t.
(x�y)�z=(x�z)� (y�z)
Rack: & 8y, x 7→x�y bijective.
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;
➺ group S & x�y=y−1xy:
rack = “(w)rack and ruin of a group”
adjunction Groups⇄Quandles
GS;ff�= hS |x�y=y−1xy i
3 Self-distributive structures
Shelf: set S & SˆS�→S s.t.
(x�y)�z=(x�z)� (y�z)
Rack: & 8y, x 7→x�y bijective.
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
is an involutive braiding on S
⇒ LND
4 Monoids
For a monoid (S;⋆; 1),the associativity of ⋆
S is a group
⇔ ff⋆=
x y
1 x⋆y
is an involutive braiding on S
⇒ LND
SGS;ff⋆∼→S;
Sˆk3x1 ´ ´ ´xk 7→x1 ⋆ ´ ´ ´⋆xk:
4 Monoids
For a monoid (S;⋆; 1),the associativity of ⋆
S is a group
⇔ ff⋆=
x y
1 x⋆y
is an involutive braiding on S
⇒ 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
For a monoid (S;⋆; 1),the associativity of ⋆
S is a group
⇔ ff⋆=
x y
1 x⋆y
is an involutive braiding on S
⇒ 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 ´y 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
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
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
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
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
�ff is a “shadow” of ff
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
�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
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)
x2 x3 x4 x5
J2(x)=
xx3x4x52
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)
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
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)
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.
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 .
J(xy)=J(x)yJ(y)
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 .
J(xy)=J(x)yJ(y)
(x1; : : : ;xn)y=
(xy1; : : : ;x
yn)
x y
xy
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 .
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 ,
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 .
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 , where
➺ J(n) is extended to (StS−1)ˆn by
x1 x2−1 x3−1
y1
y2−1
y3−1
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 .
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 , 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.
9 Associated shelf: examples
✓ 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.
9 Associated shelf: examples
Cycle set: set S & SˆS´→S s.t.
(x ´y) ´ (x ´z)= (y ´x) ´ (y ´z)
& 8x, y 7→x ´y bijective.⇔ ff´=
y ´x y
x ´y x is a LNDbraidingon S
with ff2´ = Id
9 Associated shelf: examples
Cycle set: set S & SˆS´→S s.t.
(x ´y) ´ (x ´z)= (y ´x) ´ (y ´z)
& 8x, y 7→x ´y bijective.
Non-degenerate CS:& x 7→x ´x bijective.
⇔ ff´=
y ´x y
x ´y x is a LNDbraidingon S
with ff2´ = Id
⇔ birack
9 Associated shelf: examples
Cycle set: set S & SˆS´→S s.t.
(x ´y) ´ (x ´z)= (y ´x) ´ (y ´z)
& 8x, y 7→x ´y bijective.
Non-degenerate CS:& x 7→x ´x bijective.
⇔ ff´=
y ´x y
x ´y x is a LNDbraidingon S
with ff2´ = Id
⇔ birack
(Etingof–Schedler–Soloviev 1999, Rump 2005)
9 Associated shelf: examples
Cycle set: set S & SˆS´→S s.t.
(x ´y) ´ (x ´z)= (y ´x) ´ (y ´z)
& 8x, y 7→x ´y bijective.
Non-degenerate CS:& x 7→x ´x bijective.
⇔ ff´=
y ´x y
x ´y x is a LNDbraidingon S
with ff2´ = Id
⇔ birack
(Etingof–Schedler–Soloviev 1999, Rump 2005)
Examples: x ´y=f(y) for any f : S∼→S + glueing.
9 Associated shelf: examples
Cycle set: set S & SˆS´→S s.t.
(x ´y) ´ (x ´z)= (y ´x) ´ (y ´z)
& 8x, y 7→x ´y bijective.
Non-degenerate CS:& x 7→x ´x bijective.
⇔ ff´=
y ´x y
x ´y x is a LNDbraidingon S
with ff2´ = Id
⇔ birack
(Etingof–Schedler–Soloviev 1999, Rump 2005)
Examples: x ´y=f(y) for any f : S∼→S + glueing.
➺ x�ff´ y=x,
➺ J : ff´↔ flip
x y
y x,
9 Associated shelf: examples
Cycle set: set S & SˆS´→S s.t.
(x ´y) ´ (x ´z)= (y ´x) ´ (y ´z)
& 8x, y 7→x ´y bijective.
Non-degenerate CS:& x 7→x ´x bijective.
⇔ ff´=
y ´x y
x ´y x is a LNDbraidingon S
with ff2´ = Id
⇔ birack
(Etingof–Schedler–Soloviev 1999, Rump 2005)
Examples: x ´y=f(y) for any f : S∼→S + glueing.
➺ 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)
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;
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);
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);
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);
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);
✓ applications to the computation of group and Hochschild(co)homology for factorizable groups and for Youngtableaux (L. 2016).