Folding of set-theoretical solutions of the Yang-Baxter equation
aa r X i v : . [ m a t h . G R ] S e p Folding of set-theoretical solutions of theYang-Baxter equation
Fabienne Chouraqui ∗ and Eddy Godelle ∗ Abstract
We establish a correspondence between the invariant subsets ofa non-degenerate symmetric set-theoretical solution of the quantumYang-Baxter equation and the parabolic subgroups of its structuregroup, equipped with its canonical Garside structure. Moreover, weintroduce the notion of a foldable solution, which extends the one of adecomposable solution.
AMS Subject Classification: 16T25, 20F36.Keywords: set-theoretical solution of the quantum Yang-Baxter equation;parabolic subgroups of Garside groups; folding.
The
Quantum Yang-Baxter Equation ( QYBE for short) is an importantequation in the field of mathematical physics, and it lies in the foundationof the theory of quantum groups. Finding all the solutions of this equationis an important issue.Let V be a vector space with base X and S : X × X → X × X be abijection. The pair ( X, S ) is said to be a set-theoretical solution of the QYBEif the linear operator R : V ⊗ V → V ⊗ V induced by S is a solution to theQYBE. The question of the classification of the set-theoretical solutions wasraised by Drinfeld in [11], and since has been the object of numerous recentarticles [2, 4, 12, 15, 16, 17]. In [12], Etingof, Soloviev and Schedler, focus onset-theoretical solutions that are non-degenerate and symmetric. In orderto approach the classification problem, they introduce and use the notion ∗ Both authors are partially supported by the
Agence Nationale de la Recherche ( projetTh´eorie de Garside , ANR-08-BLAN-0269-03). The first author is also supported by theAffdu-Elsevier fellowship.
1f an invariant subset . They associate a group called the structure group to each non-degenerate and symmetric set-theoretical solution. In [4], thefirst author establishes a one-to-one correspondence between non-degenerateand symmetric set-theoretical solutions of the QYBE and
Garside grouppresentations which satisfy some additional conditions. The notion of aGarside group is also the object of numerous articles [1, 7, 18, 20] and is anatural generalisation of the notion of an Artin-Tits group. In particular thesecond author proves in [18] that the classical notion of a standard parabolicsubgroup of an Artin-Tits group can be extended to the general framework ofGarside groups. The first objective of the present paper is to show that thetwo notions of an invariant subset and of a standard parabolic subgroup aredeeply related in the context of non-degenerate symmetric set-theoreticalsolutions of the QYBE. More precisely (see next section for definitions andnotations) we prove:
Theorem 1.
Let X be a finite set, and ( X, S ) be a non-degenerate sym-metric set-theoretical solution of the quantum Yang-Baxter equation. Let G be the structure group of ( X, S ) . For Y ⊆ X , denote by G Y the subgroup of G generated by Y . The map Y G Y induces a one-to-one correspondancebetween the set of invariant non-degenerate subsets of ( X, S ) and the set ofstandard parabolic subgroups of G . Indeed, in order to classify non-degenerate symmetric set-theoretical so-lution of the quantum Yang-Baxter equation, Etingof, Soloviev and Schedlerintroduced in [12] the notion of a decomposable solution . This is a solutionwhich is the union of two disjoint non-degenerate invariant subsets. In thelast part of the article, we extend the notion of a folding of Coxeter graph introduced by Crisp to study morphisms between Artin-Tits groups to thecontext of set-theoretical solutions of the QYBE. The notion of a foldablesolution can be seen as a generalisation of the notion of decomposable so-lution. We prove that every Garside subgroup, that verifies some obviousnecessary condition, is associated to a set-theoretical solution, and
Theorem 2.
Let X be a finite set, and ( X, S ) be a non-degenerate, symmet-ric set-theoretical solution of the QYBE. The pair ( X, S ) is decomposable ifand only if it has a strong folding ( X ′ , S ′ ) which a trivial solution and suchthat X ′ = 2 . The paper is organized as follows. In Section 2, we introduce the QYBE,the notion of a non-degenerate symmetric set-theoretical solution, and itsstructure group. In Section 3, we provide the necessary background on Gar-2ide groups and their parabolic subgroups. In Section 4, we prove Theorem 1.In Section 5, we introduce the notion of a folding and prove Theorem 2.
Acknowledgment.
The first author is very grateful to Arye Juhasz forfruitful conversations. The authors are also grateful to Pavel Etingof andTravis Schedler for personnal communications [13].
In this section, we introduce basic definitions and useful notions related tothe Quantum Yang-Baxter Equation. We follow [12] and refer to it for moredetails.For all the section, we fix a vector space V . The Quantum Yang-BaxterEquation on V is the equality R R R = R R R of linear transformations on V ⊗ V ⊗ V where the indeterminate is a lineartransformation R : V ⊗ V → V ⊗ V , and R ij means R acting on the i th and j th components. A set-theoretical solution of this equation is a pair ( X, S )such that X is a basis for V , and S : X × X → X × X is a bijective mapthat induces a solution R of the QYBE. Consider a set-theoretical solution (
X, S ) of the QYBE. Following [12], for x, y in X , we set S ( x, y ) = ( g x ( y ) , f y ( x )). Definition 2.1. (i) The pair (
X, S ) is nondegenerate if the maps f x and g x are bijections for any x ∈ X .(ii) The pair ( X, S ) is involutive if S ◦ S = Id X .(iii) The pair ( X, S ) is braided if S satisfies the braid relation S S S = S S S , where the map S ii +1 : X n → X n is defined by S ii +1 = id X i − × S × id X n − i − , i < n .(iv) The pair ( X, S ) is symmetric if (
X, S ) is involutive and braided.Let α : X × X → X × X be the permutation map, i.e α ( x, y ) = ( y, x ),and let R = α ◦ S . The map R is called the R − matrix corresponding to S .Etingof, Soloviev and Schedler show in [12], that ( X, S ) is a braided pair if3nd only if R satisfies the QYBE, and that ( X, S ) is a symmetric pair if andonly if in addition R satisfies the unitary condition R R = 1. A solution( X, S ) is a trivial solution if the maps f x and g x are the identity on X forall x ∈ X , that is S is the permutation map on X × X . In the sequel we areinterested in non-degenerate symmetric pairs. Definition 2.2.
Assume (
X, S ) is non-degenerate and symmetric. The structure group of (
X, S ) is defined to be the group G ( X, S ) with the fol-lowing group presentation: h X | ∀ x, y ∈ X, xy = g x ( y ) f y ( x ) i Since the maps g x are bijective and S is involutive, one can deduce thatfor each x in X there is a unique y such that S ( x, y ) = ( x, y ). Therefore,the presentation of G ( X, S ) contains n ( n − non-trivial relations. Example . Let X be the set { x , · · · , x } , and S be the map defined by S ( x i , x j ) = ( x σ i ( j ) , x τ j ( i ) ) where σ i and τ j are the following permutationson { , · · · , } : τ = σ = τ = σ = (1 , , , τ = σ = τ = σ =(1 , , ,
2) and τ = σ = id { , ··· , } . A case-by-case analysis shows that ( X, S )is a non-degenerate symmetric solution. Its structure group is generated bythe set X and defined by the 10 following relations: x = x ; x x = x x ; x x = x x ; x x = x x ; x x = x x ; x = x ; x x = x x ; x x = x x ; x x = x x ; x x = x x . Here we introduce the crutial notion of a decomposable solution . Definition 2.4. [12] ( i ) A subset Y of a non-degenerate symmetric set-theoretical solution X is said to be an invariant subset if S ( Y × Y ) ⊆ Y × Y .( ii ) An invariant subset Y is said to be non-degenerate if ( Y, S | Y × Y ) is anon-degenerate and symmetric set.( iii ) A non-degenerate and symmetric solution ( X, S ) is said to be decom-posable if X is a union of two nonempty disjoint non-degenerate invariantsubsets. Otherwise, ( X, S ) is said to be indecomposable .In [12], Etingof et al show that if X is finite, then any invariant sub-set Y of X is non-degenerate. Moreover, they show that if ( X, S ) is non-degenerate and braided then the assignment x → f x is a right action of G ( X, S ) on X and that ( X, S ) is indecomposable if and only if G ( X, S ) actstransitively on X . They give a classification of non-degenerate symmetric4olutions with X up to 8 elements, considering their decomposability andother properties. Gateva-Ivanova conjectured that every square-free, non-degenerate symmetric solution ( X, S ) is decomposable whenever X is finite.This has been proved by Rump in [22]. He also prove that the extension toan infinite set X is false. Finally, in [4], the first author finds a criterion fordecomposability of the solution involving the Garside structure of the struc-ture group. She proves that a non-degenerate symmetric solution ( X, S ) isindecomposable if and only if its structure group is ∆-pure Garside.
We turn now to the notion of a Garside group. We only recall the basicmaterial that we need, and refer to [7], [8] and [18] for more details.
We start with some preliminaries. If M is a monoid generated by a set X , and if x ∈ M is the image of the word w by the canonical morphismfrom the free monoid on X onto M , then we say that w represents x . Amonoid M is cancellative if for every x, y, z, t in M , the equality xyz = xtz implies y = t . The element x is a left divisor ( resp. a right divisor ) of z if there is an element t in M such that z = xt ( resp. z = tx ). It is leftnoetherian ( resp. right noetherian ) if every sequence ( x n ) n ∈ N of elements of M such that x n +1 is a left divisor ( resp. a right divisor) of x n stabilizes. Itis noetherian if it is both left and right noetherian. An element ∆ is said tobe balanced if it has the same set of right and left divisors. In this case, wedenote by Div(∆) its set of divisors. If M is a cancellative and noetherianmonoid, then left and right divisibilities are partial orders on M . Definition 3.1. (i) A locally Garside monoid is a cancellative noethe-rian monoid such that any two elements have a common multiple for left-divisibility and right-divisibility if and only if they have a least commonmultiple for left-divisibility and right-divisibility, respectively.(ii) A
Garside element of a locally Garside monoid is a balanced element ∆whose set of divisors Div(∆) generates the whole monoid. When such anelement exists, then we say that the monoid is a
Garside monoid .(iii) A ( locally ) Garside group G ( M ) is the enveloping group of a (locally)Garside monoid M .Garside groups have been first introduced in [9]. The seminal exampleare the so-called Artin-Tits groups . Among these groups,
Spherical type rtin-Tits groups are special examples of Garside groups. We refer to [10]for general results on locally Garside groups. Recall that an element x = 1 ina monoid is called an atom if the equality x = yz implies y = 1 or z = 1. Itfollows from the definining properties of a Garside monoid that the followingproperties holds for a Garside monoid M : The monoid M is generated by itsset of atoms, and every atom divides the Garside elements. Any two elementsin M have a left and right gcd and lcm; in particular, M verifies the Ore’sconditions, so it embeds in its group of fractions [5]. The left and right lcmof two Garside elements are Garside elements and coincide; therefore, bythe noetherianity property there exists a unique minimal Garside elementfor both left and right divisibilities. This element will be called the Garsideelement of the monoid in the sequel.
Example . [4] Consider the notation of Example 2.3. The group G ( X, S )is a Garside group. The Garside element ∆ is the right and left lcm of X ,and is represented by x x , x x , x x , x x , and others. The notion of a parabolic subgroup of an Artin-Tits group is well-known.In [18], the second author extends this notion to the wider context of Garsidegroups.
Definition 3.3.
Let M be a Garside monoid with Garside element ∆.(i) A submonoid N of M is standard parabolic if there exists δ in Div(∆) thatis balanced and such that Div( δ ) generates N with N ∩ Div(∆) = Div( δ ).(ii) A standard parabolic subgroup of the Garside group G ( M ) is a subgroupgenerated by a parabolic submonoid.In the sequel, we denote by M δ the monoid N in the above definition. Lemma 3.4. [18] Let M be a Garside monoid with Garside element ∆ ,and consider a standard parabolic submonoid M δ . Then M δ is a Garsidemonoid with δ as a Garside element. Moreover, the Garside group G ( M δ ) is isomorphic to the parabolic subgroup of G ( M ) generated by M δ .Example . If M is an Artin-Tits monoid, then its classical parabolicsubmonoids are the parabolic submonoids defined by the associated Garsidestructure. More precisely, these are the submonoids generated by any set ofatoms of M . Example . Consider the notation of Example 2.3. There are two non-trivial standard parabolic subgroups. One is generated by { x , x , x , x } ,and the other is generated by { x } . 6 Parabolic subgroups of the structure group
The following result explains the deep connection between the theory ofset-theoretical solutions of QYBE and that of Garside groups.
Theorem 4.1. [4, Thm.1] (i) Assume that
Mon h X | R i is a Garside monoidsuch that:(a) the cardinality of R is n ( n − / , where n is the cardinality of X andeach side of a relation in R has length 2 and(b) if the word x i x j appears in R , then it appears only once.Then, there exists a function S : X × X → X × X such that ( X, S ) is a non-degenerate symmetric set-theoretical solution and Gp h X | R i is its structuregroup.(ii) For every non-degenerate symmetric set-theoretical solution ( X, S ) , thestructure group G ( X, S ) is a Garside group, whose Garside monoid is asabove. Our objective in this section is to show that the connection is evendeeper. Indeed, we prove Theorem 1. As a first step, we show that an invari-ant subset generates a standard parabolic subgroup (see Proposition 4.6).In a second step, we prove that every such subgroup arises in this way (seeProposition 4.8).
For all this section, we assume X is a finite set, and ( X, S ) is a non-degenerate symmetric set-theoretical solution of the QYBE. For short, wewrite G for G ( X, S ). We denote by M the submonoid of G generated by X ,and by ∆ the Garside element of M . We recall that we denote by Div(∆)the set of divisors of ∆. We fix an invariant subset Y of X , we denote by δ the right lcm of Y in M , and by M Y and G Y the submonoid and thesubgroup, respectively, of G generated Y . Proposition 4.2 ([4]) . (i) The Garside element ∆ is the lcm of X for bothleft and right divisibilities.(ii) Let s belong to M . Then, s belongs to Div(∆) ⇐⇒ ∃ X ℓ ⊆ X such that s is the right lcm of X ℓ ⇐⇒ ∃ X r ⊆ X such that s is the left lcm of X r . iii) If s belongs to Div(∆) then the subsets X ℓ and X r defined in Point (ii)are unique and have the same cardinality.Proof. Point (i) is proved in [4]. Only the first equivalency of (ii) is provedin [4, Prop. 4.4], but the second one is implicit there. Finally, Point (iii) isalso a direct consequence of [4]: Indeed, from the proof of [4, Theorem 4.7],the length of the right lcm of k distinct elements of X is equal to k and ina dual way, the same result holds for the left lcm.Note that X ℓ and X r may be different. In the sequel, for s in Div(∆),we denote by X ℓ ( s ) and X r ( s ) the subsets X ℓ and X r defined in Lemma 4.2.For instance, X ℓ (∆) = X r (∆) = X , and X ℓ ( δ ) = Y . It is interesting tonote that the equality X ℓ ( s ) = X r ( s ) does not imply that s is balanced. InExample 3.2, it holds that x is the left and right lcm of the generators x , x , and x , but x is not balanced since x x is a left divisor but not a rightdivisor. Lemma 4.3. (i) If s belongs to M Y , then all the letters in a word thatrepresents s belong to Y . In particular, every left or right divisor of s liesin M Y .(ii) Let s belong to Div(∆) . Then s ∈ M Y ⇐⇒ X ℓ ( s ) ⊆ Y ⇐⇒ X r ( s ) ⊆ Y. In particular, δ belongs to M Y .(iii) The monoid M Y is equal to M ∩ G Y .Proof. The following is the key argument in the proof: if Y is invariant,then S ( Y, Y ) is included in Y × Y , which means that the defining relationsinvolving two generators from Y in one-hand side of the relation have nec-essarily the form y y = y y , where y i ∈ Y , 1 ≤ i ≤
4. Let s belong to M Y ,then Point (i) follows directly from the above. Let us prove (ii). If addi-tionally s belongs to Div(∆), then X ℓ ( s ) ⊆ Y and X r ( s ) ⊆ Y . Conversely,if s ∈ Div(∆), X ℓ ( s ) ⊆ Y and X r ( s ) ⊆ Y then using the same argument andthe reversing process (see [7]) to compute the right (or left) lcm of a subsetof Y , we have s ∈ M Y . It remains to show that (iii) holds. Clearly, M Y isincluded in M ∩ G Y and by applying the double reversing process on a wordon Y ± that represents an element in M ∩ G Y , we obtain an element in M ,whose letters belong to Y (still by the same argument).We recall that if s is a balanced element in Div(∆), then its support Supp( s )is defined to be the set X ∩ Div( s ). It is shown in [18] that the atoms set ofa standard parabolic subgroup G δ is Supp( δ ).8 emma 4.4. The element δ lies in Div(∆) , is balanced, and
Supp( δ ) = Y .Proof. The right lcm of Y is δ and Y ⊆ X , so by Proposition 4.2(i), theelement δ is a left divisor of ∆ and δ lies in Div(∆). From lemma 4.3 (ii),the equality X ℓ ( δ ) = Y implies that δ lies in M Y and X r ( δ ) ⊆ Y . But thesets X ℓ ( δ ) and X r ( δ ) have the same cardinality by Proposition 4.2(iii), sowe have X ℓ ( δ ) = X r ( δ ) = Y . Now, let u be a left divisor of δ . We showthat u is also a right divisor of δ . Since δ belongs to Div(∆) ∩ M Y , it followsfrom lemma 4.3 (i) that u lies in Div(∆) ∩ M Y . So, by Lemma 4.2(ii), weget X r ( u ) ⊆ Y = X r ( δ ). Therefore, u is a right divisor of δ . Similarlyevery right divisor of δ is a left divisor of δ , and δ is balanced. At last, wehave Supp( δ ) = Y . Lemma 4.5.
Div( δ ) = Div(∆) ∩ M Y .Proof. By Lemmas 4.3 and 4.4, δ lies in Div(∆) ∩ M Y , so Div( δ ) is includedin Div(∆) ∩ M Y ( cf. Lemma 4.3(ii)). Conversely, if u lies in Div(∆) ∩ M Y ,then u is the right lcm of X ℓ ( u ), which is a subset of Y by Lemma 4.3(ii).Since δ is the right lcm of Y , the element u belongs to Div( δ ).We can now state and prove the main result of this section: Proposition 4.6.
Under the general hypothesis and notations of this sec-tion, the subgroup G Y , generated by Y , is a standard parabolic subgroup of G .Proof. By Lemma 4.4, δ is a balanced element in Div(∆), with Y = Supp( δ ).Moreover, from Lemma 4.5, Div( δ ) = Div(∆) ∩ M Y . Hence, G Y is a standardparabolic subgroup of G from Definition 3.3.Rump proves that every square-free, non-degenerate and symmetric solu-tion ( X, S ) is decomposable, whenever X is finite [22]. So, from Proposition4.6, the structure group of a square-free solution has standard non-trivialparabolic subgroups. Moreover, in a square-free solution each set { x } with x ∈ X is an invariant subset, since S ( x, x ) = ( x, x ). Square-free solutionsprovide examples of solutions in which there exist invariant subsets Y suchthat X \ Y is not invariant.If we consider that Y , Y ,.., Y k are invariant subsets of ( X, S ) thatcorrespond to indecomposable solutions (in other words that Y , Y ,.., Y k arethe minimal under inclusion among the invariant sets in the decompositionof ( X, S ) to indecomposable solutions), then each set Y , Y ,.., Y k generatesa standard parabolic subgroup. Furthermore, these parabolic subgroups are9-pure Garside, since a solution is indecomposable if and only if its structuregroup is ∆-pure Garside [4, 5.3]. So, we get that G is the crossed productof ∆-pure Garside parabolic subgroups generated by Y , Y ,.., Y k , using [20,Prop.4.5]. As in the previous section, for all this section we assume X is a finite set, and( X, S ) is a non-degenerate symmetric set-theoretical solution of the QYBE.For short, we write G for G ( X, S ). We denote by M the submonoid of G generated by X , and by ∆ the Garside element of M . We recall that wedenote by Div(∆) the set of divisors of ∆. We fix a balanced element δ in Div(∆) such that the subgroup G δ , generated by Div( δ ), is a non-trivialstandard parabolic subgroup of G . We set M δ = M ∩ G δ , which is equal tothe submonoid of M generated by Supp( δ ). Our objective here is to provethat Supp( δ ) is an invariant subset of X . Lemma 4.7.
The balanced element δ is the right lcm and the left lcm of Supp( δ ) . In particular, δ is the Garside element of the Garside monoid M δ .Proof. It is immediate that X ℓ ( δ ) = X r ( δ ) = Supp( δ ) since δ is balanced( cf. see also [18]). Proposition 4.8.
Under the general hypothesis and notations of this sec-tion,
Supp( δ ) is an invariant subset of X .Proof. Since X is finite, from [12] it is enough to show that Y = Supp( δ ) isinvariant, that is S ( Y, Y ) ⊆ ( Y, Y ). Let y, y ′ belong to Y , and assume that S ( y, y ′ ) = ( x, x ′ ), that is x is a left divisor of yy ′ and x ′ is a right divisorof yy ′ . From [18, Prop.2.5], M δ is closed under left and right divisibility, so x, x ′ lie in M δ . As x and x ′ are atoms, they belong to Y . As a consequence,we get S ( Y, Y ) ⊆ ( Y, Y ).Gathering Propositions 4.6 and 4.8, we get Theorem 1.We now extend the result of Proposition 4.8 to parabolic subgroups ingeneral, that is not necessarily standard parabolic subgroups. If gG δ g − isthe conjugate of a standard parabolic subgroup G δ , then gG δ g − is generatedby the set gY g − , where Y generates G δ and Y is an invariant subset of( X, S ) from Proposition 4.8. We recall that two solutions (
X, S ) and ( X ′ , S ′ )are said to be isomorphic if there exists a bijection φ : X → X ′ which maps S to S ′ , that is S ′ ( φ ( x ) , φ ( y )) = ( φ ( S ( x, y )) , φ ( S ( x, y ))).10 roposition 4.9. Consider the general hypothesis and notations of thissection. Let gG δ g − be a non-trivial parabolic subgroup of G , where g belongsto G . Then, g Supp( δ ) g − is an invariant subset of a set-theoretical ( X ′ , S ′ ) which is isomorphic to ( X, S ) .Proof. We define ( X ′ , S ′ ) in the following way: X ′ = gXg − and S ′ ( gx i g − , gx j g − ) = ( gg i ( j ) g − , gf j ( i ) g − ).From the definition, ( X, S ) and ( X ′ , S ′ ) are isomorphic and a direct compu-tation shows that g Supp( δ ) g − is an invariant subset of ( X ′ , S ′ ). If ( X, S )and ( X ′ , S ′ ) are isomorphic, then G and the structure group G ′ of ( X ′ , S ′ )are isomorphic groups (see for example [3]). Moreover, gXg − ⊆ G , so G ′ is a subgroup of G that is isomorphic to G . In the previous section, we investigated the connection between the invariantsubsets of a set-theoretical solution and the standard parabolic subgroups ofits structure group, equipped with is canonical Garside structure. In [18], thesecond author introduced a more general family of subgroups of a Garsidegroup, namely the Garside subgroups. In this section, we study, in the caseof a Garside group that arises as the structure group of a set-theroreticalsolution of the QYBE, how these subgroups can also be associated withset-theoretical solutions. Morover, we extend the notion of a decomposablesolution, and explain why this extention could be useful in order to studythe classification problem.
Parabolic subgroups of Garside groups are
Garside subgroups as introducedin [18]. So, invariant subsets provide Garside subgroups. A question thatarises naturally is whether the converse is true, that is whether a Garsidesubgroup is necessarily a parabolic one. The answer is negative as shown inExample 5.2. Indeed, a Garside subgroup may not be generated by a subsetof atoms.
Definition 5.1.
Let M be a Garside monoid with ∆ as Garside element.Let N be a submonoid of M .(i) [18, Prop. 1.6] We say that N is a Garside submonoid of M if it isgenerated by a non-empty subset D of Div(∆) which is a sublattice of Div(∆)11or left and right divisibility, which is closed by complement for left and rightdivisibility, and such that for every x, y ∈ D , the left gcd and the right gcdof xy and ∆ in M belong to D . In this case, we say that the subgroup of G ( M ) generated by N is a Garside subgroup of G ( M ).(ii) we say that the Garside submonoid N of M is atomic when its atom setis a subset of the atom set of M . Example . Consider the group G ( X, S ) where X = { x , x , x , x } andthe defining relations are x = x ; x x = x x ; x x = x x ; x = x ; x x = x x ; x x = x x . The group G ( X, S ) has no proper standard parabolic subgroup. However,the subgroups generated by the sets { x } , { x } , { x , x } and { x , x } areexamples of Garside subgroups.Since Garside subgroups are not necessarily parabolic, they may notcorrespond to a set-theoretical solution of the QYBE. However, the followingresult proves that under a simple necessary condition, a Garside subgroupis naturally associated to a set-theoretical solution of the QYBE. Proposition 5.3.
Let X be a finite set, and ( X, S ) be a non-degenerate,symmetric set-theoretical solution of the QYBE. Assume H is a Garsidesubgroup of G ( X, S ) such that its atoms set X H is closed under right com-plement in the Garside monoid M ( X, S ) . Then there exists a uniquely well-defined bijective map S H : X H × X H → X H × X H such that the pair ( X H , S H ) is a non-degenerate symmetric set-theoretical solution and G ( X H , S H ) is iso-morphic to H .Proof. We claim that we (uniquely) define a bijective map S H : X H × X H → X H × X H by setting S H ( x, y ) = ( z, t ) for every x, y, z, t in X H with x, z distinct and such that xy = zt = x ∨ z . By assumption, for every x, z distinct in X H , there exists a unique pair y, t of distinct elements in X H such that xy = zt = x ∨ z , that is the map ( x, z ) ( y, t ) is injective. Fromfiniteness, it is also bijective and as a consequence, the set X H is closed byleft complements. Consider the group ˜ H with the generating set X H andthe defining relations: xy = zt whenever xy = zt = x ∨ z . The presentationof ˜ H satisfies the properties (a) and (b) stated in Theorem 4.1. So, thepair ( X H , S H ) is a non-degenerate symmetric set-theoretical solution suchthat G ( X H , S H ) ≃ ˜ H . At last, we show that the surjective morphism ϕ H from ˜ H to H that sends x ∈ X H to itself is an isomorphism. Let w be a word12n X ± H . Then the double reversing process in G ( X, S ) (see [7]) transforms w into a word that represents the same element in ˜ H , since X H is closed byleft and right complements. So, if w represents 1 in H , then the doublereversing process transforms w into the trivial word in ˜ H , that is ϕ H isinjective.As an immediate consequence we get: Corollary 5.4.
Let X be a finite set, and ( X, S ) be a non-degenerate, sym-metric set-theoretical solution of the QYBE. Assume H is an atomic Garsidesubgroup of G ( X, S ) with atoms set X H ⊆ X . Denote by S H the restric-tion of S to X H × X H . The pair ( X H , S H ) is a non-degenerate symmetricset-theoretical solution and G ( X H , S H ) is isomorphic to H .Proof. Using Proposition 5.3, we show that the set X H is closed under rightcomplement. If x, z belong to X H and S ( x, y ) = ( z, t ), then xy = zt = x ∨ z and y, t belong to X . From the definition of a Garside subgroup (see Defn.5.1), H is generated by a lattice closed under right and left complement.So, y, t belong to this generating lattice and also to X , so they belong to X H . Example . Consider Example 5.2. The atomic Garside subgroup of G ( X, S )generated by { x , x } is isomorphic to the structure group defined by thepresentation h x , x | x = x i . The notions of an invariant subset and of a decomposable solution have beenintroduced in [12] as tools to build and classify set-theoretical solutions ofthe QYBE. In this last section, we want to explain how the notion of aGarside subgroup can be used in the same way. Let us first introduce thenotion of a foldable solution . We say that a partition X ∪ · · · ∪ X k of aset X is proper if each X i is not empty and 1 < k < | X | . Definition 5.6.
Let X be a finite set, and ( X, S ) be a non-degenerate,symmetric set-theoretical solution of the QYBE.(i) We say that (
X, S ) is foldable if X has a proper partition X ∪ · · · ∪ X k such that1. Every set X i generates an atomic Garside subgroup of M ( X, S ) withGarside element ∆ i ; 13. The set X ′ = { ∆ , · · · , ∆ k } is closed by right and left complements in M ( X, S ) and is the atom set of a Garside subgroup of G ( X, S ).(ii) In this case, let S ′ : X ′ × X ′ → X ′ × ′ the bijective map induced by S .We say that ( X ′ , S ′ ) is a folding of ( X, S ), or equivalently that G ( X ′ , S ′ ) isa folding of G ( X, S ).(iii) We say that (
X, S ) is strongly foldable when furthermore each X i gen-erates a standard parabolic subgroup of G ( X, S ). In this case, we saythat ( X ′ , S ′ ) is a strong folding of ( X, S ).One can note that the Garside element of M ( X ′ , S ′ ) has to be equal tothe Garside element of M ( X, S ). This notion is very similar to the notion of folding of a Coxeter graph used by Crisp in [6] to study morphisms betweentwo spherical type Artin-Tits groups. Indeed, in this case all the foldingshave to be strong, and it is shown in [18] that such a morphism is character-ized by conditions that are very closed to properties 1 and 2 stated in theabove definition.Foldable solutions exist, since decomposable solutions are foldable as itwill be shown below.
Theorem 5.7.
Let X be a finite set and ( X, S ) be a non-degenerate, sym-metric set-theoretical solution of the QYBE. If ( X, S ) is decomposable then itis strongly foldable, and the folding is a trivial solution to the QYBE. More-over, ( X, S ) is decomposable if and only of it has a strong folding ( X ′ , S ′ ) which is a trivial solution and such that X ′ = 2 . In particular, we get Theorem 2. When proving Theorem 5.7, we shallneed some results proved in [20] and [4], and recalled below.
Proposition 5.8.
Let ( X, S ) be a decomposable non-degenerate, symmetricset-theoretical solution of the QYBE and let M and G be the correspond-ing Garside monoid and group with X as atoms set. For x in X , we set ∆ x = ∨{ b − ( x ∨ b ); b ∈ M } .(i)[20, 19] The relation ∼ on X defined by x ∼ y if ∆ x = ∆ y is an equiva-lence relation. Moreover, if ∆ x = ∆ y then ∆ x ∧ ∆ y = ∆ x ˜ ∧ ∆ y = 1 in M .(ii)[20, 19] Let Y , · · · , Y k be the equivalence classes of ∼ , and set ∆ i = ∆ x for x ∈ Y i . Then the subgroup of G generated by ∆ , · · · , ∆ k is a free abeliangroup with base ∆ , · · · , ∆ k .(iii) [4] Assume X = ∪ i = ki =1 Y i , where Y i are invariant subsets of X and k ≥ , since the solution is decomposable. Then ∆ = ∆ · · · ∆ k , where ∆ isthe Garside element in M and ∆ i is the lcm of Y i . roof of Theorem 5.7. Assume (
X, S ) is decomposable and X = ∪ i = ki =1 Y i ,where Y i are invariant subsets of X and k ≥
2. From Proposition 4.6, thesubgroup generated by Y i is a parabolic subgroup with Garside element ∆ i ,where ∆ i is the lcm of Y i . So, the first condition in the definition of astrongly foldable solution is satisfied. Consider the set D = { ∆ ε · · · ∆ ε k k | ε i = { , }} . Gathering several results from [20, Sec. 2] and Proposition 5.8,we obtain that D is a sublattice of Div(∆) and it is closed by complementand lcm: (∆ ε · · · ∆ ε k k ) ∧ (∆ ε ′ · · · ∆ ε ′ k k ) = ∆ min( ε ,ε ′ )1 · · · ∆ min( ε k ,ε ′ k ) k and (∆ ε · · · ∆ ε k k ) ∨ (∆ ε ′ · · · ∆ ε ′ k k ) = ∆ max( ε ,ε ′ )1 · · · ∆ max( ε k ,ε ′ k ) k . Moreover, we have(∆ ε + ε ′ · · · ∆ ε k + ε ′ k k ) ∧ ∆ = ∆ max( ε ,ε ′ )1 · · · ∆ max( ε k ,ε ′ k ) k Let H be the subgroup generated by D , then H is a Garside subgroup of G ( X, S ), with ∆ as Garside element and atoms set X ′ = { ∆ i ; 1 ≤ i ≤ k } . That is, the second condition in the definition of a strongly foldablesolution is also satisfied, so ( X, S ) is strongly foldable. Furthermore, H isthe structure group of a solution ( X ′ , S ′ ) of the QYBE from Proposition5.3, and it is free abelian (from Proposition 5.8(ii)), so ( X ′ , S ′ ) is the trivialsolution on a set with k elements. Thisa implies that ( X, S ) is stronglyfoldable and that the folding ( X ′ , S ′ ) of ( X, S ) is a trivial solution to theQYBE. It remains to show that it is possible to choose the folding ( X ′ , S ′ )such that X ′ = 2. Let i in { , · · · , k } such that Y i and Z i = ∪ j = i Y i are twoinvariant subsets that generate parabolic subgroups whose Garside elementsare ∆ i and ∆ ˆ i = Q j = i ∆ i , respectively. We have ∆ i ∆ ˆ i = ∆ ˆ i ∆ i = ∆. Thesubgroup H ′ i of G ( X, S ) generated by ∆ i and ∆ ˆ i satisfies H ′ i ∩ Div(∆) = { , ∆ i , ∆ ˆ i , ∆ } . Hence, H ′ i is a strong folding of G ( X, S ) that corresponds to atrivial solution. Conversely, let ( X ′ , S ′ ) be a strong folding which is a trivialsolution and such that X ′ = 2. Then, then by definition, X ′ contains twoelements that are the right (and left) lcm of two invariant subsets of X , Y and Y , that satisfy X = Y ∪ Y . Therefore, X is decomposable.Decomposable solutions of the QYBE are foldable. However, there existfoldable solutions that are not decomposable. Example . Consider the set-theoretical solution ( X ′ , S ′ ) whose structuregroup is defined by the presentation h x, y | x = y i . Then, ( X ′ , S ′ ) is a15olding of the solution ( X, S ) defined in Example 5.2. Indeed, consider thefollowing partition of X : X = { x , x } ∪ { x , x } . The sets { x , x } and { x , x } generate Garside subgroups with Garside elements x and x re-spectively. The set X ′ = { x = x , y = x } is closed under right and leftcomplement and generates a Garside subgroup of G ( X, S ). So, ( X ′ , S ′ ) isa folding of ( X, S ) but this folding is not strong, since the Garside sub-groups generated by { x , x } and { x , x } are not parabolic through theyare atomic.As seen in Theorem 5.7 and Example 5.9, foldable and strongly fold-able solutions do exist. Conversely the notion of folding can be used as atool to build new solutions: starding from a solution ( X ′ , S ′ ) one can tryto substitute each element x of X ′ by the Garside element of another so-lution ( X x , S x ) so that one obtains a new solution ( ∪ x ∈ X ′ X x , S ) which isfoldable, with ( X ′ , S ′ ) as a folding. Clearly, the difficult point is to define abijective map S that is compatible with S ′ and so that one gets a solutionto the QYBE.As a final comment, we raise the question of the existence of a stronglyfoldable solution ( X, S ) that is not decomposable.
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