New aspects of the Yang-Baxter equation

New aspects of the Yang-Baxter equation

Victoria LEBED

Jean Leray Mathematics Institute, University of Nantes

Symposium on Mathematical Physics

November 10, 2014

←→

1 Yang-Baxter equation

✓ A vector space V (or an object in any

monoidal category)

✓ σ : V⊗2→ V⊗2

Yang-Baxter equation (YBE):

σ1 ◦σ2 ◦σ1 = σ2 ◦σ1 ◦σ2 : V⊗3→V⊗3

where σi = Id⊗i−1V ⊗σ⊗ Id⊗···

V .

A map σ satisfying YBE is a braiding.

σ ←→

V⊗V

V⊗V

=

(Reidemeister III)



monoidal category)

✓ σ : V⊗2→ V⊗2


σ1 ◦σ2 ◦σ1 = σ2 ◦σ1 ◦σ2 : V⊗3→V⊗3


V .


(V,σ) ;

σ ←→

V⊗V

V⊗V

=

(Reidemeister III)

rep. of B+n (pos.

braid monoid):

i i+1 n

7→ σi



monoidal category)

✓ σ : V⊗2→ V⊗2


σ1 ◦σ2 ◦σ1 = σ2 ◦σ1 ◦σ2 : V⊗3→V⊗3


V .


(V,σ) ;

σ is invertible ;

σ ←→

V⊗V

V⊗V

=

(Reidemeister III)

rep. of B+n (pos.

braid monoid):

i i+1 n

7→ σi

rep. of Bn (braid

group)

7→ σ−1i

2 YBE in physics

✓ Particle physics: factorization condition for the dispersion

matrix in the 1-dim. n-body problem (McGuire, Yang, 60’).

collisions ←→

2 YBE in physics



collisions ←→

✓ Statistical mechanics: partition function for exactly

solvable lattice models (Onsager, 1944, Ising model;

Baxter, 70’, 8-vertex, hard hexagon & chiral Potts models).

Boltzmann

weights

2 YBE in physics



collisions ←→

✓ Statistical mechanics: partition function for exactly

solvable lattice models (Onsager, 1944, Ising model;

Baxter, 70’, 8-vertex, hard hexagon & chiral Potts models).

✓ Quantum inverse scattering method for completely

integrable systems (Faddeev et al., 1979).

✓ Factorizable S-matrices in 2-dim. quantum field theory

(Zamolodchikov, 1979).

✓ Quantum group (Drinfel ′d, 80’).

✓ C∗ algebras (Woronowicz, 80’).

✓ Conformal field theory.

3 A homology theory for the YBE

Aim: Unify homology theories for basic algebraic structures.



Tools: Graphical calculus.




Ingredients:

✓ A braided vector space (V,σ);

✓ a left braided V-module: (M, ρ : M⊗V→M) s.t.

ρ◦ρ1 = ρ◦ρ1 ◦σ2 :

M⊗V⊗V→M

ρ

ρ

M⊗V⊗V

=

ρ

ρ

M⊗V⊗V

σ

✓ a right braided V-module (N, λ : V⊗N→N).




Ingredients:

✓ A braided vector space (V,σ);

✓ a left braided V-module: (M, ρ : M⊗V→M) s.t.

ρ◦ρ1 = ρ◦ρ1 ◦σ2 :

M⊗V⊗V→M

ρ

ρ

M⊗V⊗V

=

ρ

ρ

M⊗V⊗V

σ

✓ a right braided V-module (N, λ : V⊗N→N).

Theorem (L. 2013): M⊗T(V)⊗N carries a family of

differentials δ(α,β) = α•δ + βδ•, α,β∈ k.

(I.e., δ(α,β) ◦δ(α,β) = 0.)




•δ=∑

(−1)i−1

ρσ

σ

1 ... i ...n

V ... VM N

δ• =∑

(−1)i−1

λ

σ

σ

1 ... i ... n

V ... VM N




•δ=∑

(−1)i−1

ρσ

σ

1 ... i ...n

V ... VM N

δ• =∑

(−1)i−1

λ

σ

σ

1 ... i ... n

V ... VM N

Proof:

YBE=

br. mod.=

& sign =

(−1)#cross.




•δ=∑

(−1)i

ρσ

σ

1 ... i ...n

V ... VM N

δ• =∑

(−1)i

λ

σ

σ

1 ... i ... n

V... VM N

Remarks:

✓ Functoriality.

✓ Interpretation in terms of quantum shuffles (Rosso, 1995).

✓ Duality ; a cohomology theory.

✓ Pre-cubical structure.




Remarks:

✓ Functoriality.



✓ Pre-cubical structure.

✓ Degeneracies.

Braided coalgebra: br. v. sp. (V,σ) & ∆ : V→ V⊗V s.t.

= = =

B Cf. Reidemeister moves for knotted 3-valent graphs!




Remarks:

✓ Functoriality.



✓ Cubical structure.

✓ Degeneracies.

Braided coalgebra: br. v. sp. (V,σ) & ∆ : V→ V⊗V s.t.

= = =

B Cf. Reidemeister moves for knotted 3-valent graphs!

Theorem (L. 2013): All δ(α,β) restrict to∑

i Im(∆i).

; normalization

4 Alg. structures via braidings

A Associative algebras



(V, µ : V⊗V→ V, ξ : k→V), ξ(α) =α1V , s.t.

Associativity:µ◦µ1 = µ◦µ2

=

Unit axiom:µ◦ξ1 = µ◦ξ2 = IdV

= =



(V, µ : V⊗V→ V, ξ : k→V), ξ(α) =α1V , s.t.


=


= =

; “Associativity braiding”

σAss = ξ⊗µ : v⊗w 7→ 1⊗v ·w

✓ YBE for σAss ⇐⇒(un. ax.)

associativity for µ



(V, µ : V⊗V→ V, ξ : k→V), ξ(α) =α1V , s.t.


=


= =

; “Associativity braiding”

σAss = ξ⊗µ : v⊗w 7→ 1⊗v ·w

✓ YBE for σAss ⇐⇒(un. ax.)


Proof: =



(V, µ, ξ) ; σAss = ξ⊗µ : v⊗w 7→ 1⊗v ·w

✓ YBE for σAss ⇐⇒(un.ax.)


✓ A fully faithful functorAlg −֒→ Br•

.



(V, µ, ξ) ; σAss = ξ⊗µ : v⊗w 7→ 1⊗v ·w




.

✓ Duality: Alg −֒→ Br

•

•←−֓ oAlg.



(V, µ, ξ) ; σAss = ξ⊗µ : v⊗w 7→ 1⊗v ·w




.


•

•←−֓ oAlg.

✓ σAss ◦σAss = σAss =⇒ highly non-invertible.

✓ Braided modules for (V, σAss) ←→ modules for (V, µ, ξ).



(V, µ, ξ) ; σAss = ξ⊗µ : v⊗w 7→ 1⊗v ·w




.


•

•←−֓ oAlg.



✓ ∆Ass = ξ1 : v 7→ 1⊗v ; braided coalgebra.



(V, µ, ξ) ; σAss = ξ⊗µ : v⊗w 7→ 1⊗v ·w




.


•

•←−֓ oAlg.



✓ ∆Ass = ξ1 : v 7→ 1⊗v ; braided coalgebra.

✓ Braided homologies for (V, σAss) include

➺ bar differential; ➺ Hochschild; ➺ group hom.


B Leibniz algebras

(V, µ : V⊗V→ V, ξ : k→V), ξ(α) =α1V , s.t.

Leibniz identity: µ◦µ2 = µ◦µ1−µ◦µ1 ◦τ, where τ : w⊗u→u⊗w

[v, [w,u]] = [[v,w],u]− [[v,u],w]

Lie unit axiom: µ◦ξ2 = µ◦ξ1 = 0

[1,v] = [v,1] = 0

(Bloh 1965, Loday & Cuvier 1991: a non-commutative

generalization of Lie algebras.)


B Leibniz algebras

(V, µ : V⊗V→ V, ξ : k→V), ξ(α) =α1V , s.t.


[v, [w,u]] = [[v,w],u]− [[v,u],w]


[1,v] = [v,1] = 0



; “Leibniz braiding” σLei = τ+ξ⊗µ : v⊗w 7→w⊗v+1⊗ [v,w]

✓ YBE for σLei ⇐⇒(Lie un. ax.)

Leibniz identity for µ

✓ A fully faithful functorLei −֒→ Br•

.


B Leibniz algebras

(V, µ : V⊗V→ V, ξ : k→V), ξ(α) =α1V , s.t.


[v, [w,u]] = [[v,w],u]− [[v,u],w]


[1,v] = [v,1] = 0



; “Leibniz braiding” σLei = τ+ξ⊗µ : v⊗w 7→w⊗v+1⊗ [v,w]

✓ YBE for σLei ⇐⇒(Lie un. ax.)

Leibniz identity for µ

✓ A fully faithful functorLei −֒→ Br•

.

✓ σLei is invertible.

✓ Braided mod. for (V, σLei) ←→ Leibniz mod. for (V, µ, ξ).


B Leibniz algebras

(V, µ : V⊗V→ V, ξ : k→V), ξ(α) =α1V , s.t.

[v, [w,u]] = [[v,w],u]− [[v,u],w], [1,v] = [v,1] = 0.

; “Leibniz braiding” σLei : v⊗w 7→w⊗v+1⊗ [v,w]

✓ Suppose that V =V ′⊕k1, [V ′,V ′]⊆ V ′. Then

∆Lei = ξ1+ξ2 : v 7→ 1⊗v+v⊗1 for v∈ V ′ and ∆Lei(1) = 1⊗1

turns (V, σLei) into a braided coalgebra.


B Leibniz algebras

(V, µ : V⊗V→ V, ξ : k→V), ξ(α) =α1V , s.t.

[v, [w,u]] = [[v,w],u]− [[v,u],w], [1,v] = [v,1] = 0.





✓ Braided homologies for (V, σLei) include Leibniz homology.

(M⊗T(V),dLei)

anti-

symm.��

Cuvier-Loday

Lie

V� // (M⊗Λ(V),dCE) Chevalley-Eilenberg


B Leibniz algebras

(V, µ : V⊗V→ V, ξ : k→V), ξ(α) =α1V , s.t.

[v, [w,u]] = [[v,w],u]− [[v,u],w], [1,v] = [v,1] = 0.






Lei

anti-

symm.��

V� // (M⊗T(V),dLei)

anti-

symm.��

Cuvier-Loday

Lie

?�

OO

V_

OO

� // (M⊗Λ(V),dCE) Chevalley-Eilenberg


B Leibniz algebras

(V, µ : V⊗V→ V, ξ : k→V), ξ(α) =α1V , s.t.

[v, [w,u]] = [[v,w],u]− [[v,u],w], [1,v] = [v,1] = 0.






Lei

anti-

symm.��

V� // (M⊗T(V),dLei)

anti-

symm.��

Cuvier-Loday

Lie

?�

OO

V_

OO

� // (M⊗Λ(V),dCE) Chevalley-Eilenberg

✓ Explains the choice of the lift of the Jacobi identity.


C Self-distributive structures



b a�b

a b

colorings

by (S,�)

“SD braiding”

σSD : (a,b) 7→ (b,a�b) ab

c

cb� c (a�b)�c

a�b

RIII←→

ab

c

cb� c (a�c)�(b�c)

a�c

YBE ←→ RIII ←→ (a�b)� c= (a� c)� (b� c) (SD)



b a�b

a b

colorings

by (S,�)

“SD braiding”


c

cb� c (a�b)�c

a�b

RIII←→

ab

c


a�c


∃σ−1SD ←→ RII ←→ a 7→ a�b is bijective (Inv)

←→ RI ←→ a�a= a (Idem)

RII←→

RI←→



b a�b

a b

colorings

by (S,�)

“SD braiding”


c

cb� c (a�b)�c

a�b

RIII←→

ab

c


a�c


∃σ−1SD ←→ RII ←→ a 7→ a�b is bijective (Inv)

= ←→ RI ←→ a�a= a (Idem)

RII←→

RI←→

∆SD : a 7→ (a,a)



b a�b

a b

colorings

by (S,�)

Joyce, Matveev 1982:

knot invariantscolorings

; quandle

pos. braids RIII ←→ (a�b)� c= (a� c)� (b� c) shelf

braids RII ←→ a 7→ a�b is bijective rack

knots RI ←→ a�a= a quandle

RII←→

RI←→



b a�b

a b

colorings

by (S,�)



; quandle




Ex.: ➺ Conjugation quandles: (group G, g�h= h−1gh)

coloring rule ←→Wirtinger presentation rule,

colorings ←→ Rep(π1(R3\K),G).



b a�b

a b

colorings

by (S,�)



; quandle







Ex.: ➺ Dihedral quandles: (Zn, a�b= 2b−a)

colorings ←→ n-Fox colorings.



b a�b

a b

colorings

by (S,�)



; quandle







Ex.: ➺ Dihedral quandles: (Zn, a�b= 2b−a)

colorings ←→ n-Fox colorings.

n= 36=



diagrams: DR-move D ′

colorings: C C ′

coloring sets: Col S(D)bij.←→ Col S(D

′)

counting invariants: #Col S(D) = #Col S(D′)




colorings: C C ′


′)


Question: Extract more information?

Idea: Some “weight” ω s.t. ω(C) =ω(C ′)

=⇒ {ω(C) |C ∈ Col S(D) }= {ω(C ′) |C ′ ∈ Col S(D′) }.




colorings: C C ′


′)


Question: Extract more information?

Idea: Some “weight” ω s.t. ω(C) =ω(C ′)

=⇒ {ω(C) |C ∈ Col S(D) }= {ω(C ′) |C ′ ∈ Col S(D′) }.

Answer: quandle cocycle invariants (Carter-Jelsovsky-

Kamada-Langford-Saito 1999).

φ : S×S→A ;

Boltzmann weight:ωφ(C) =

∑

a

b

±φ(a,b)



Rack & quandle cohomology theories

(Fenn-Rourke-Sanderson 1995, Carter et al. 1999)

Motivation:

➺ {ωφ(C) | C∈ Col S(D) } yields a braid / knot invariant when φ

is a rack / quandle 2-cocycle;

➺ this invariant is trivial when φ is a coboundary;





Motivation:




➺ adding coefficients allows to color diagram regions:a

rackmod.

M∋ m m ·a ωφ(C) =∑

a

b

m

±φ(m,a,b)

➺ everything generalizes to Kn−1 →֒ Rn+1.





Motivation:





rackmod.

M∋ m m ·a ωφ(C) =∑

a

b

m

±φ(m,a,b)


Inspiration: (Co-)homology theories for associativity.

Question (Przytycki): Explain the parallels between the

associative and the SD worlds?





Motivation:





rackmod.

M∋ m m ·a ωφ(C) =∑

a

b

m

±φ(m,a,b)


Inspiration: (Co-)homology theories for associativity.

Question (Przytycki): Explain the parallels between the

associative and the SD worlds?

Answer: Common braided interpretation.



Shelf (S, �) ; σSD : (a,b) 7→ (b,a�b)

✓ YBE for σSD ⇐⇒ SD for �

✓ A fully faithful functorShelf −֒→Br

.

✓ σSD is invertible ⇐⇒ (S, �) is a rack.

✓ Braided modules for (V, σSD) ←→ rack modules for (S, �).

✓ ∆SD : a 7→ (a,a) ; weak braided coalgebra if (S, �) is a

quandle.

✓ Braided homologies for (V, σSD) include rack, quandle,

and other SD homologies.

5 Multi-component braidings

Question: How to treat more complicated structures?



Braided system: V1,V2, . . . ,Vr and σi,j : Vi⊗Vj→ Vj⊗Vi, i6 j,

satisfying the colored Yang-Baxter equation (cYBE):

σj,k1 ◦σ

i,k2 ◦σi,j

1 = σi,j2 ◦σ

i,k1 ◦σ

j,k2

Vi⊗Vj⊗Vk→ Vk⊗Vj⊗Vi, i6 j6 k Vi Vj Vk

←→

Vi Vj Vk

The collection (σi) satisfying cYBE is a multi-braiding.





σj,k1 ◦σ

i,k2 ◦σi,j

1 = σi,j2 ◦σ

i,k1 ◦σ

j,k2


←→

Vi Vj Vk


Left braided V-module:

(M, (ρi : M⊗Vi→M)) s.t.

ρjρi

M Vi Vj

= ρj

ρiσi,j

M Vi Vj





σj,k1 ◦σ

i,k2 ◦σi,j

1 = σi,j2 ◦σ

i,k1 ◦σ

j,k2


←→

Vi Vj Vk


Left braided V-module:

(M, (ρi : M⊗Vi→M)) s.t.

ρjρi

M Vi Vj

= ρj

ρiσi,j

M Vi Vj

Theorem (L., 2013): M⊗T(V1)⊗·· ·⊗T(Vr)⊗N carries a

family of differentials δ(α,β) =α•δ + βδ•, α,β∈ k.


Finite-dim. bialgebra H ;

(H,H∗; σH,H = σrAss(H),σH∗,H∗ = σAss(H

∗),σH,H∗ = σYD)

σH,H = σH∗,H∗ = σH,H∗ =ev

h⊗ l 7→ 〈l(1),h(2)〉l(2)⊗h(1)

✓ YBE on H⊗H∗⊗H∗ ⇐⇒(un. ax.)

bialgebra compatibility






h⊗ l 7→ 〈l(1),h(2)〉l(2)⊗h(1)



✓ A fully faithful functor ∗Bialg −֒→ ∗

2BrSyst•

•.






h⊗ l 7→ 〈l(1),h(2)〉l(2)⊗h(1)




2BrSyst•

•.

✓ σH,H∗ is invertible ⇐⇒ H is a Hopf algebra .






h⊗ l 7→ 〈l(1),h(2)〉l(2)⊗h(1)




2BrSyst•

•.


✓ Braided modules ←→ Hopf modules over H.






h⊗ l 7→ 〈l(1),h(2)〉l(2)⊗h(1)




2BrSyst•

•.


✓ Braided modules ←→ Hopf modules over H.

✓ Braided homologies include

➺ Gerstenhaber-Schack; ➺ Panaite-Ştefan.


Finite-dim. bialgebra H ; (H,Hop,H∗,(H∗)op; . . .).


4BrSyst•

•.




4BrSyst•

•.

✓ Braided modules ←→ Hopf bimodules over H.




4BrSyst•

•.


Application:

➺ Hopf bimodules are modules over the Heisenberg double

H (H) =H⊗H∗

➺ Cibils-Rosso 1998: “Hopf bimodules are modules” over

X (H) = (H⊗Hop)⊗(H∗⊗ (H∗)op)




4BrSyst•

•.


Application:


H (H) =H⊗H∗


X (H) = (H⊗Hop)⊗(H∗⊗ (H∗)op)

➺ Panaite 2002: “Hopf bimodules are modules over ...”

Y (H) =H∗#(Hop⊗H)#(H∗)op &

Z (H) = (H∗⊗ (H∗)op) ⊲⊳ (Hop⊗H)




4BrSyst•

•.


Application:


H (H) =H⊗H∗


X (H) = (H⊗Hop)⊗(H∗⊗ (H∗)op)


Y (H) =H∗#(Hop⊗H)#(H∗)op &

Z (H) = (H∗⊗ (H∗)op) ⊲⊳ (Hop⊗H)

➺ Theorem (L. 2013): Hopf bimodules are modules over

4!= 24 pairwise isomorphic algebras.




4BrSyst•

•.


Application:


H (H) =H⊗H∗


X (H) = (H⊗Hop)⊗(H∗⊗ (H∗)op)


Y (H) =H∗#(Hop⊗H)#(H∗)op &

Z (H) = (H∗⊗ (H∗)op) ⊲⊳ (Hop⊗H)

➺ Theorem (L. 2013): Hopf bimodules are modules over

4!= 24 pairwise isomorphic algebras.

✓ Braided homologies include the Ospel-Taillefer theory.

6 The unifying role of the YBE

(multi-)braiding 7→algebraic structure

rBrSyst ←֓ Stru

(c)YBE ⇔ the defining relations

invertibility ⇔ algebraic properties

braided modules ↔ usual modules

braided hom. theory ⊇ usual hom. theories



rBrSyst ←֓ Stru





Other “braidable” structures:

➺ module algebras (Yau);

➺ Yetter-Drinfel ′d modules (Panaite-Ştefan);



rBrSyst ←֓ Stru








➺ (non-commutative) Poisson algebras (Fresse);



rBrSyst ←֓ Stru








➺ (non-commutative) Poisson algebras (Fresse);

➺ multiple conjugation quandles (Ishii)

knotted

handle-bodies←→

New aspects of the Yang-Baxter equation

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