set theoretic solutions of the yang-baxter equation

Set theoretic solutions of the Yang-Baxter equation

Jan OkninskiUniversity of Warsaw

Stuttgart, June 2012

Plan of the talk

This is Part 1 of a series of two talks on the relations between settheoretic solutions of the quantum Yang-Baxter equation andcertain classes of groups and certain problems on groups and grouprings.

Part 1:

1. set theoretic solutions2. two types of groups associated to solutions3. fundamental structural results4. some examples5. some open problems6. two strategies for solving the problems7. some partial solutions

Jan Okninski University of Warsaw Set theoretic solutions of the Yang-Baxter equation

Plan of the talk

This is Part 1 of a series of two talks on the relations between settheoretic solutions of the quantum Yang-Baxter equation andcertain classes of groups and certain problems on groups and grouprings.

Part 1:

1. set theoretic solutions2. two types of groups associated to solutions3. fundamental structural results4. some examples5. some open problems6. two strategies for solving the problems7. some partial solutions

Some references

[CJR] F. Cedo, E. Jespers and A. del Rıo, Involutive Yang-BaxterGroups, Trans. Amer. Math. Soc. 362 (2010), 541-2558.

[CJO] F. Cedo, E. Jespers and J. Okninski, Retractability of settheoretic solutions of the Yang-Baxter equation, Advances Math.224 (2010), 2472-2484.

[D] V.G. Drinfeld, On some unsolved problems in quantum grouptheory, Lect. Notes in Math., vol. 1510, pp. 1–8, 1992.

[ESS] P. Etingof, T. Schedler and A. Soloviev, Set-theoreticalsolutions of the quantum Yang-Baxter equation, Duke Math. J.100 (1999), 169–209.

[G] T. Gateva-Ivanova, A combinatorial approach to theset-theoretic solutions of the Yang-Baxter equation, J. Math.Phys. 45 (2004), 3828–3858.

[GV] T. Gateva-Ivanova and M. Van den Bergh, Semigroups ofI -type, J. Algebra 206 (1998), 97–112.

[JO1] E. Jespers and J. Okninski, Monoids and groups of I -type,Algebras Repres. Theory 8 (2005), 709–729.

[JO2] E. Jespers and J. Okninski, Noetherian Semigroup Algebras,Springer, 2007.

[R] W. Rump, A decomposition theorem for square-free unitarysolutions of the quantum Yang-Baxter equation, Advances Math.193 (2005), 40–55.

Set theoretic solutions

Definition

Let X be a nonempty set and r : X 2 −→ X 2 a bijective map. Onesays that r is a set theoretic solution of the Yang-Baxter equationif it satisfies on X 3 the braid relation

r12r23r12 = r23r12r23

where rij stands for the mapping X 3 −→ X 3 obtained byapplication of r in the components i , j .A solution is often denoted by (X , r).

If V is a vector space over a field K with basis X , then r induces alinear operator R on V ⊗K V , so that in the automorphism groupof V ⊗K V ⊗K V we have:

(R⊗ idV ) (idV ⊗R) (R⊗ idV ) = (idV ⊗R) (R⊗ idV ) (idV ⊗R)

Set theoretic solutions

Definition

Let X be a nonempty set and r : X 2 −→ X 2 a bijective map. Onesays that r is a set theoretic solution of the Yang-Baxter equationif it satisfies on X 3 the braid relation

r12r23r12 = r23r12r23

where rij stands for the mapping X 3 −→ X 3 obtained byapplication of r in the components i , j .A solution is often denoted by (X , r).

If V is a vector space over a field K with basis X , then r induces alinear operator R on V ⊗K V , so that in the automorphism groupof V ⊗K V ⊗K V we have:

(R⊗ idV ) (idV ⊗R) (R⊗ idV ) = (idV ⊗R) (R⊗ idV ) (idV ⊗R)

Definition

Invertible linear maps with this property are called solutions of theYang-Baxter equation.

Denote by τ : X 2 −→ X 2 the map defined by τ(x , y) = (y , x). Itcan be verified that r is a set theoretic solution of the Yang-Baxterequation if and only if R = τ ◦ r is a solution of the quantumYang-Baxter equation, that is,

R12R13R23 = R23R13R12,

where Rij denotes R acting on the i-th and j-th component.

Origin of the quantum Yang-Baxter equation - paper by Yang(1967) on statistical mechanics.

It turned out to be one of the basic equations in mathematicalphysics.

It also lies at the foundations of the theory of quantum groups.

One of the important problems is to discover all solutions of thequantum Yang-Baxter equation. Drinfeld (1992) posed thequestion of finding all set theoretic solutions.

For a solution (X , r), let

σx , γy : X −→ X

be the maps defined by

r(x , y) = (σx(y), γy (x)).

Definition

A set theoretic solution (X , r) is called:

(1) involutive if r 2 = idX×X ,

(2) right (left) non-degenerate if all maps σx (all γy , respectively)are bijections,

(3) square free if r(x , x) = (x , x) for every x ∈ X .

Theorem (JO)

If X is a finite set then an involutive solution (X , r) is leftnon-degenerate if and only if it is right non-degenerate.

σx , γy : X −→ X

r(x , y) = (σx(y), γy (x)).

Definition

Theorem (JO)

σx , γy : X −→ X

r(x , y) = (σx(y), γy (x)).

Definition

Theorem (JO)

Related groups; and monoids and algebras

(X , r) will stand for an involutive non-degenerate solution.

Definition

Let (X , r) be a solution. Denote by G(X , r) the subgroup〈σx | x ∈ X 〉 of the symmetric group SymX .

It can be shown that G(X , r) and the group 〈γx | x ∈ X 〉 areconjugate in SymX and they are equal if (X , r) is square free.

Definition

A group H is called an involutive Yang-Baxter group (IYB-group) ifit is isomorphic to G(X , r) for a solution (X , r) with a finite set X .

Related groups; and monoids and algebras

(X , r) will stand for an involutive non-degenerate solution.

Definition

Let (X , r) be a solution. Denote by G(X , r) the subgroup〈σx | x ∈ X 〉 of the symmetric group SymX .

It can be shown that G(X , r) and the group 〈γx | x ∈ X 〉 areconjugate in SymX and they are equal if (X , r) is square free.

Definition

A group H is called an involutive Yang-Baxter group (IYB-group) ifit is isomorphic to G(X , r) for a solution (X , r) with a finite set X .

Definition

Let S(X , r) and G (X , r) be the monoid and the group definedby the presentation

〈X | xy = σx(y)γy (x) if r(x , y) = (σx(y), γy (x))〉.

It is called the structure monoid, resp. the structure group, ofthe solution (X , r).

For a field K , the algebra defined by this presentation is thesemigroup algebra K [S(X , r)].

Let ZX denote the additive free abelian group with basis X . Thenatural action of SymX on ZX leads to a semidirect productZX o SymX .

Theorem (ESS)

If (X , r) is a solution, then G (X , r) is isomorphic to a subgroup ofZX o SymX of the form

{(a, φ(a)) | a ∈ ZX},

for a function φ : ZX −→ SymX .In fact, φ(x) = σx for all x ∈ X , where r(x , y) = (σx(y), γy (x)),and the map x 7→ (x , σx) from X to ZX o SymX induces anisomorphism from G (X , r) to {(a, φ(a)) | a ∈ ZX}.

Monoids and groups of I -type [GV]

Notation: FaMn = 〈u1, . . . , un〉 - free abelian (multiplicative)monoid of rank n.

Definition

A monoid S with generators x1, . . . , xn is of I -type if there is abijective map v : FaMn −→ S such that v(1) = 1 and for alla ∈ FaMn

{v(u1a), . . . , v(una)} = {x1v(a), . . . , xnv(a)}.

This comes from [Tate, Van den Bergh] - Homological propertiesof Sklyanin algebras, Invent. Math. 124 (1996), 619-647.(homological methods in the study of certain associative algebras,playing an important role in noncommutative geometry).

Monoids and groups of I -type [GV]

Notation: FaMn = 〈u1, . . . , un〉 - free abelian (multiplicative)monoid of rank n.

Definition

A monoid S with generators x1, . . . , xn is of I -type if there is abijective map v : FaMn −→ S such that v(1) = 1 and for alla ∈ FaMn

{v(u1a), . . . , v(una)} = {x1v(a), . . . , xnv(a)}.

This comes from [Tate, Van den Bergh] - Homological propertiesof Sklyanin algebras, Invent. Math. 124 (1996), 619-647.(homological methods in the study of certain associative algebras,playing an important role in noncommutative geometry).

Theorem (GV)

A monoid S = 〈x1, . . . , xn〉 is a monoid of I -type iff it is thestructure monoid of a non-degenerate, involutive solution (X , r) fora finite set X with |X | = n.

Nice properties of the algebra, similar to the polynomial algebra,have been proved (based on methods of Tate and Van den Bergh).

Theorem (GV)

Assume S of I -type. Then

1 K [S ] is a Noetherian PI -domain,

2 The growth functions of K [S ] and K [x1, . . . , xn] are equal,

3 the global dimension of K [S ] is n,

4 K [S ] has some some homological ”regularity” conditions(Cohen-Macaulay, Artin-Schelter regular, a Koszul algebra),

5 S = S(X , r) has a group of quotients. It is of the formSS−1 = G (X , r) and G (X , r) is a central localization of S.

Corollary

The structure group G (X , r) is solvable, torsion free andabelian-by-finite. So it is a Bieberbach group.

Nice properties of the algebra, similar to the polynomial algebra,have been proved (based on methods of Tate and Van den Bergh).

Theorem (GV)

Assume S of I -type. Then

1 K [S ] is a Noetherian PI -domain,

2 The growth functions of K [S ] and K [x1, . . . , xn] are equal,

3 the global dimension of K [S ] is n,

4 K [S ] has some some homological ”regularity” conditions(Cohen-Macaulay, Artin-Schelter regular, a Koszul algebra),

5 S = S(X , r) has a group of quotients. It is of the formSS−1 = G (X , r) and G (X , r) is a central localization of S.

Corollary

The structure group G (X , r) is solvable, torsion free andabelian-by-finite. So it is a Bieberbach group.

A method for verifying the Y-B equation

Let r : X 2 −→ X 2, r(x , y) = (σx(y), γy (x)), for some mapsσx , γx : X −→ X , x ∈ X .

Theorem (CJO)

(X , r) is a right non-degenerate involutive set theoretic solution iff1. r 2 = idX 2 ,2. σx ∈ SymX for x ∈ X ,3. σx ◦ σσx−1 (y) = σy ◦ σσy−1 (x) for all x , y.

Square free means σx(x) = x for every x ∈ X .

Examples of solutions

Example

S = FaMn.Let X = {x1, . . . , xn} and σi = id for every i . So G(X , r) = {id}.G (X , r) = Fan, free abelian group of rank n. This is called a trivialsolution.

Example

S = 〈x , y | x2 = y 2〉.Let X = {x , y} and σx = σy = (x , y). So G(X , r) ∼= Sym2. ThenS ↪→ FaM2 o S2. Also G (X , r) is the group defined by the same(group) presentation. It is not square free.

Examples of solutions

Example

S = FaMn.Let X = {x1, . . . , xn} and σi = id for every i . So G(X , r) = {id}.G (X , r) = Fan, free abelian group of rank n. This is called a trivialsolution.

Example

S = 〈x , y | x2 = y 2〉.Let X = {x , y} and σx = σy = (x , y). So G(X , r) ∼= Sym2. ThenS ↪→ FaM2 o S2. Also G (X , r) is the group defined by the same(group) presentation. It is not square free.

Example

S = 〈x , y , z | xy = yx , zx = yz , zy = xz〉.Let X = {x , y , z} and σx = σy = id and σz = (x , y). SoG(X , r) = 〈(x , y)〉 ∼= Sym2 and G (X , r) is defined by the same(group) presentation. It is square free.

Example

Assume that γ = σ−1 ∈ Sym(X ) and (X , r) is such that

r(x , y) = (σ(y), γ(x)).

Then (X , r) is a solution, called a permutation solution.

Example

S = 〈x , y , z | xy = yx , zx = yz , zy = xz〉.Let X = {x , y , z} and σx = σy = id and σz = (x , y). SoG(X , r) = 〈(x , y)〉 ∼= Sym2 and G (X , r) is defined by the same(group) presentation. It is square free.

Example

Assume that γ = σ−1 ∈ Sym(X ) and (X , r) is such that

r(x , y) = (σ(y), γ(x)).

Then (X , r) is a solution, called a permutation solution.

Problems

Problem

1) How to construct solutions? How to construct ”biggersolutions” from smaller ones?Is every finite solution (X , r) constructed from smallersolutions, in some sense?

2) Characterize structure groups G (X , r), in particularcharacterize groups of I -type.

3) Which solvable finite groups are IYB-groups?

4) Try to classify solutions according to their IYB-groups.

Problem 1 leads to the notions of decomposability andretractability of solutions.

First approach - decomposability of solutions [ESS]

Definition

If X1, . . . ,Xm are G(X , r)-orbits on X , then the components ofS = S(X , r) are defined by: Si = 〈xj | xj ∈ Xi 〉.They satisfy: SiSk = SkSi for every i , k, and every Si is a monoidof I -type.

Theorem (R)

If X is finite, |X | > 1 and (X , r) is square free then it isdecomposable, which means that there are ≥ 2 components. So, insome sense (X , r) is built from ”smaller” solutions. Also S(X , r)(and G (X , r)) are built from ”smaller” monoids (resp. groups) ofI -type.

We shall see that the above is not true if (X , r) is not square free.Question: how can one construct new solutions using thisapproach?

Definition

Theorem (R)

Definition

Theorem (R)

Second approach - retractability of solutions [ESS]

The relation ∼ on the set X is defined by

x ∼ y if σx = σy .

This leads to an induced solution

Ret(X , r) = (X/∼, r),

called the retraction of X , with

r([x ], [y ]) = ([σx(y)], [γy (x)]),

where [x ] denotes the ∼-class of x ∈ X .

(X , r) is called a multipermutation solution of level m if m is thesmallest integer such that Retm(X , r) has cardinality 1. Here

Retk(X , r) = Ret(Retk−1(X , r)) for k > 1.

If such m exists then we say that the solution is retractable.

Second approach - retractability of solutions [ESS]

The relation ∼ on the set X is defined by

x ∼ y if σx = σy .

This leads to an induced solution

Ret(X , r) = (X/∼, r),

called the retraction of X , with

r([x ], [y ]) = ([σx(y)], [γy (x)]),

where [x ] denotes the ∼-class of x ∈ X .

(X , r) is called a multipermutation solution of level m if m is thesmallest integer such that Retm(X , r) has cardinality 1. Here

Retk(X , r) = Ret(Retk−1(X , r)) for k > 1.

If such m exists then we say that the solution is retractable.Jan Okninski University of Warsaw Set theoretic solutions of the Yang-Baxter equation

Corollary

Assume X is finite and (X , r) is a retractable solution. Then thegroup G (X , r) is a poly-Z group.Sketch of the proof:Note that the subgroup H of G (X , r) ⊆ Fan o Symn generated byall xix

−1j with xi ∼ xj is a normal subgroup.

Then H ⊆ Fan, so it is free abelian.Thus, replace (X , r) by (X/∼, r) and continue applying Ret.

Remark

This gives a direct proof of the fact that K [G (X , r)] is a domain.Also, K [G (X , r)] has ”trivial units”: U(K [G (X , r)]) = K ∗ ·G (X , r)(so Kaplansky conjectures hold for such groups).

Corollary

Assume X is finite and (X , r) is a retractable solution. Then thegroup G (X , r) is a poly-Z group.Sketch of the proof:Note that the subgroup H of G (X , r) ⊆ Fan o Symn generated byall xix

−1j with xi ∼ xj is a normal subgroup.

Then H ⊆ Fan, so it is free abelian.Thus, replace (X , r) by (X/∼, r) and continue applying Ret.

Remark

This gives a direct proof of the fact that K [G (X , r)] is a domain.Also, K [G (X , r)] has ”trivial units”: U(K [G (X , r)]) = K ∗ ·G (X , r)(so Kaplansky conjectures hold for such groups).

Conjecture

The following conjecture was formulated by Gateva-Ivanova in”A combinatorial approach to the set-theoretic solutions of theYang-Baxter equation”, J. Math. Phys. 45 (2004), 3828–3858.

Conjecture

Every finite set theoretic involutive non-degenerate square freesolution (X , r) with X of cardinality n ≥ 2 is a multipermutationsolution of level m < n.

This had been considered in weaker forms: which of the groups ofI -type are poly-Z? [JO1]

An example

Example (JO)

S = 〈x1, x2, x3, x4 | x1x2 = x3x3, x2x1 = x4x4, x1x3 = x2x4,

x1x4 = x4x2, x2x3 = x3x1, x3x2 = x4x1〉

is a monoid of I -type. It is not decomposable. Its group ofquotients is not poly-(infinite cyclic), and thus the correspondingsolution (X , r) is not retractable. G(X , r) ∼= D8.

One uses the group P defined by

P = gr(x , y | x−1y 2x = y−2, y−1x2y = x−2).

Then A = gr(x2, y 2, (xy)2) is a normal and free abelian subgroupof P of rank 3 and P/A is the four group of Klein. Furthermore, Pis torsion free but it is not poly-(infinite cyclic).One shows that G/N ∼= Z, where G = SS−1 and P ∼= N / G .

An example

Example (JO)

S = 〈x1, x2, x3, x4 | x1x2 = x3x3, x2x1 = x4x4, x1x3 = x2x4,

x1x4 = x4x2, x2x3 = x3x1, x3x2 = x4x1〉

is a monoid of I -type. It is not decomposable. Its group ofquotients is not poly-(infinite cyclic), and thus the correspondingsolution (X , r) is not retractable. G(X , r) ∼= D8.

One uses the group P defined by

P = gr(x , y | x−1y 2x = y−2, y−1x2y = x−2).

Then A = gr(x2, y 2, (xy)2) is a normal and free abelian subgroupof P of rank 3 and P/A is the four group of Klein. Furthermore, Pis torsion free but it is not poly-(infinite cyclic).One shows that G/N ∼= Z, where G = SS−1 and P ∼= N / G .

X = {x1, x2, x3, x4}. Since there are(4

)relations and every word

xixj with 1 ≤ i , j ≤ 4 appears at most once in one of the relations,we obtain an associated bijective map r : X 2 −→ X 2. It can beverified that (X , r) is an involutive non-degenerate solution, but itis not square free. So S is a monoid of I -type.

We may consider S as a submonoid {(a, σa) | a ∈ FaM4} ofFa4 o Sym4 by identifying xi with (ui , σi ), whereFaM4 = 〈u1, u2, u3, u4〉. The permutations σi easily can bedetermined from the defining relations:

σ1 = (23), σ2 = (14), σ3 = (1243), σ4 = (1342).

In particular, the corresponding solution (X , r) is notdecomposable (only one G(X , r)-orbit).Note that G(X , r) ∼= D8.

Example shows: solutions that are not square free are notnecessarily retractable, not decomposable, their structure groupsare not necessarily poly-Z.

σ1 = (23), σ2 = (14), σ3 = (1243), σ4 = (1342).

Partial result on the conjecture

Theorem (CJO)

Assume that X is finite, |X | > 1 and (X , r) is an involutivenon-degenerate square free solution and the group G(X , r) isabelian. If r is not trivial then there exist x , y ∈ X , x 6= y suchthat σx = σy , and x , y ∈ Xk for some k ∈ {1, . . . ,m}, whereX1, . . . ,Xm are the components of X .

Motivated by this theorem, we define a relation ρ on X as follows:

(x , y) ∈ ρ if x , y ∈ Xk for some k and σx = σy .

This can be used to define a stronger version of retractability of(X , r), based on the relation ρ.

Partial result on the conjecture

Theorem (CJO)

Assume that X is finite, |X | > 1 and (X , r) is an involutivenon-degenerate square free solution and the group G(X , r) isabelian. If r is not trivial then there exist x , y ∈ X , x 6= y suchthat σx = σy , and x , y ∈ Xk for some k ∈ {1, . . . ,m}, whereX1, . . . ,Xm are the components of X .

Motivated by this theorem, we define a relation ρ on X as follows:

(x , y) ∈ ρ if x , y ∈ Xk for some k and σx = σy .

This can be used to define a stronger version of retractability of(X , r), based on the relation ρ.

Now the notion of strong retractability of (X , r) may be defined asfollows.

Definition

Let Retρ(X , r) = (X/ρ, r) denote the induced solution. We saythat (X , r) is strongly retractable if there exists m ≥ 1 such thatapplying m times the operator Retρ we get a trivial solution.

Since the IYB-group corresponding to the solution (X/ρ, r) also isabelian (as a homomorphic image) if G(X , r) is abelian, thefollowing is a direct consequence of the previous theorem. This is apartial solution of Conjecture I.

Corollary

Assume that (X , r) is a set theoretic involutive non-degeneratesquare free solution and the group G(X , r) is abelian. Then (X , r)is strongly retractable. In particular, G (X , r) is a poly-Z group.

Now the notion of strong retractability of (X , r) may be defined asfollows.

Definition

Let Retρ(X , r) = (X/ρ, r) denote the induced solution. We saythat (X , r) is strongly retractable if there exists m ≥ 1 such thatapplying m times the operator Retρ we get a trivial solution.

Since the IYB-group corresponding to the solution (X/ρ, r) also isabelian (as a homomorphic image) if G(X , r) is abelian, thefollowing is a direct consequence of the previous theorem. This is apartial solution of Conjecture I.

Corollary

Assume that (X , r) is a set theoretic involutive non-degeneratesquare free solution and the group G(X , r) is abelian. Then (X , r)is strongly retractable. In particular, G (X , r) is a poly-Z group.

Some results on Problem 3

Proposition (CJR)

1 If G is an IYB group then its Hall subgroups ( H ⊆ G suchthat |H| and [G : H] are relatively prime) are IYB.

2 The class of IYB groups is closed under direct products.

3 A wreath product of two IYB groups also is IYB.

4 Any finite solvable group is isomorphic to a subgroup of anIYB group.

5 Every finite nilpotent group of class 2 is IYB.

Not known: which finite nilpotent groups are IYB? (maybe all ofthem?)

set theoretic solutions of the yang-baxter equation

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