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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···xn
1 , . . . ,xxn
n−1,xn),
where xxi+1···xn
i = (. . . (xxi+1
i ) . . .)xn.
x1 ··· xn
J1(x)
J2(x)
...
Jn(x)
7 Guitar map
J(n) : X×n ∼
→X×n,
(x1, . . . ,xn) 7→ (xx2···xn
1 , . . . ,xxn
n−1,xn),
where xxi+1···xn
i = (. . . (xxi+1
i ) . . .)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···xn
1 , . . . ,xxn
n−1,xn),
where xxi+1···xn
i = (. . . (xxi+1
i ) . . .)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···xn
1 , . . . ,xxn
n−1,xn),
where xxi+1···xn
i = (. . . (xxi+1
i ) . . .)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.
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