Coxeter-like groups for groups of set-theoretic solutions of the Yang--Baxter equation
aa r X i v : . [ m a t h . G R ] M a y COXETER-LIKE GROUPS FOR GROUPS OF SET-THEORETICSOLUTIONS OF THE YANG–BAXTER EQUATION
PATRICK DEHORNOY
Abstract.
We attach with every finite, involutive, nondegenerate set-theoreticsolution of the Yang–Baxter equation a finite group that plays for the associ-ated structure group the role that a finite Coxeter group plays for the associ-ated Artin–Tits group. Introduction A set-theoretic solution of the Yang–Baxter equation (YBE) is a pair ( X, R )where R is a bijection from X to itself satisfying R R R = R R R , inwhich R ij : X → X corresponds to R acting in positions i and j . Set-theoreticsolutions of YBE provide particular solutions of the (quantum) Yang–Baxter equa-tion, and received some attention in recent years.A set-theoretic solution ( X, R ) of YBE is called involutive for R = id, and nondegenerate if, writing R ( x, y ) = ( R ( x, y ) , R ( x, y )), the maps y R ( x, y )and x R ( x, y ) are one-to-one. In this case, the group ( resp . monoid) presentedby h X | { xy = z, t | x, y, z, t ∈ X and R ( x, y ) = ( z, t ) }i is called the structure group ( resp . structure monoid ) of ( X, R ) [5].Such structure groups happen to admit a number of alternative definitions andmake an interesting family. Among others, every structure group is a
Garsidegroup [1], meaning that there exists a pair ( M, ∆) such that M is a cancellativemonoid in which left-divisibility—defined by g h ⇔ ∃ h ′ ∈ M ( h = gh ′ )—is a lattice,∆ is a Garside element in M —meaning that the left- and right-divisors coincide,are finite in number, and generate M —and G is a group of fractions for M [3].In the case of Artin’s braid group B n , the seminal example of a Garside group,the Garside structure ( B + n , ∆ n ) is connected with the exact sequence 1 → P n → B n → S n →
1, where P n is the pure braid group: B + n is the monoid of positivebraids, the lattice made by the divisors of ∆ n in B + n is isomorphic to the weakorder on S n . A presentation of S n is obtained by adding n − σ i = 1 tothe standard presentation of B n , and the germ derived from S n and the transposi-tions σ i , meaning the substructure of S n where multiplication is restricted to thecases when lengths add, generates B + n [4] and its Cayley graph is the Hasse diagramof the divisors of ∆ n . The results extend to all types in the Cartan classification,connecting spherical Artin–Tits groups with the associated finite Coxeter group.As Garside groups extend spherical type Artin–Tits groups in many respects, itis natural to ask: Question 1.1.
Assume that G is a Garside group, with Garside structure ( M, ∆) .Does there exist a finite quotient W of G such that W provides a Garside germ for M and the Cayley graph of W with respect to the atoms of M is isomorphic tothe lattice of divisors of ∆ in M ? In other words, does every Garside group admit some Coxeter-like group? Thegeneral question remains open. The aim of this note is to establish a positive answerfor structure groups of set-theoretic solutions of YBE. We attach with every suchsolution a number called its class and establish:
Theorem 1.2.
Assume that G (resp. M ) is the structure group (resp. monoid) ofan involutive, nondegenerate solution ( X, R ) of YBE with X of size n and class p .Then there exist a Garside element ∆ in M and a finite group W of order p n entering a short exact sequence → Z n → G → W → such that ( W, X ) providesa germ for M whose Cayley graph is the Hasse diagram of the divisors of ∆ in M .A presentation of W is obtained by adding n relations w x = 1 to that of G , with w x an explicit length p word beginning with x . Theorem 1.2 extends the results of [2], in which solutions of class 2 are addressedby a different method. Our approach relies on the connection with the right-cycliclaw of [8] and on the existence of an I -structure [6] [7] which enables one to carryto arbitrary structure monoids results that are trivial in the case of Z n .2. The class of a finite RC-quasigroup
The first step consists in switching from solutions of YBE to RC-quasigroups.
Definition
RC-system is a pair (
X, ⋆ ) with ⋆ a binary operation on X that obeys the RC-law ( x⋆y ) ⋆ ( x⋆ z ) = ( y ⋆x ) ⋆ ( y ⋆z ). An RC-quasigroup is an RC-system in which the maps y x⋆y are bijections. An RC-quasigroup is bijective if the map ( x, y ) ( x⋆y, y ⋆x ) from X to X is bijective. The associated group ( resp . monoid ) is presented by h X | { x ( x⋆y ) = y ( y ⋆x ) | x, y ∈ X }i .As proved in [8], if ( X, R ) is an involutive, nondegenerate set-theoretic solutionof YBE, then defining x⋆y to be the (unique) z satisfying R ( x, z ) = y makes X into a bijective RC-quasigroup and the group and monoid associated with ( X, ⋆ )coincide with those of (
X, R ). Conversely, every bijective RC-quasigroup (
X, ⋆ )comes associated with a set-theoretic solution of YBE. Thus investigating structuregroups of set-theoretic solutions of YBE and groups of bijective RC-quasigroups areequivalent tasks.
Definition ( x ) = x and(1) Π n ( x , ..., x n ) = Π n − ( x , ..., x n − ) ⋆ Π n − ( x , ..., x n − , x n ) . An RC-quasigroup (
X, ⋆ ) is said to be of class p if Π p +1 ( x, ..., x, y ) = y holds forall x, y in X . Lemma 2.3.
Every finite RC-quasigroup is of class p for some p .Proof. Let (
X, ⋆ ) be a finite RC-quasigroup. First, (
X, ⋆ ) must be bijective, thatis, the map Ψ : ( x, y ) ( x⋆ y, y ⋆x ) is bijective on X [8] (or [7] for a differentargument). Now, consider the map Φ : ( x, y ) ( x⋆ x, x⋆y ) on X . Assume( x, y ) = ( x ′ , y ′ ). For x = x ′ , Ψ( x, x ) = Ψ( x ′ , x ′ ) implies x⋆x = x ′ ⋆x ′ ; for x = x ′ ,we have y = y ′ , whence x⋆ y = x⋆y ′ since left-translations are injective; so Φ( x, y ) =Φ( x ′ , y ′ ) always holds. So Φ is injective, hence bijective on X , and Φ p +1 = id holdsfor some p ≥
1. An induction gives Φ r ( x, y ) = (Π r ( x, ..., x, x ) , Π r ( x, ..., x, y )). SoΦ p +1 = id implies Π p +1 ( x, ..., x, y ) = y for all x, y , that is, ( X, ⋆ ) is of class p . (cid:3) OXETER-LIKE GROUPS FOR GROUPS OF YANG–BAXTER EQUATION 3 Using the I -structure From now on, assume that M ( resp . G ) is the structure group of some finiteRC-quasigroup ( X, ⋆ ) of size n and class p . The form of the defining relations of M implies that the Cayley graph of M with respect to X is an n -dimensional lattice.It was proved in [6] that M admits a (right) I -structure , defined to be a bijection ν : N n → M satisfying ν (1) = 1 and { ν ( ux ) | x ∈ X } = { ν ( u ) x | x ∈ X } for every u in N n , that is, equivalently, ν ( ux ) = ν ( u ) π ( u )( x ) for some permutation π ( u ) of X .The monoid M is then called of right- I -type . Our point is that the I -structure(which is unique) is connected with ⋆ . Without loss of generality, we shall assumethat X is the standard basis of N n and that ν ( x ) = x holds for x in X . Lemma 3.1.
For all x , ..., x r in X , we have ν ( x ··· x r ) = Σ r ( x , ..., x r ) , with Σ r inductively defined by Σ ( x ) = x and Σ r ( x , ..., x r ) = Σ r − ( x , , ..., x r − ) · Π r ( x , ..., x r ) . Proof.
The result can be established directly by developing a convenient RC-calculusand proving that Σ r ( x , ..., x r ) satisfies all properties required for an I -structure. Ashorter proof is to start from the existence of the I -structure ν and just connect itwith the values of Σ r . As established in [6] (see also [7, Chapter 8, Lemma 8.2.2]),the following inductive relations are satisfied for all u, v in N n :(2) ν ( uv ) = ν ( u ) ν ( π ( u )[ v ]) and π ( uv ) = π ( π ( u )[ v ]) ◦ π ( u )where π [ u ] is the result of applying π to u componentwise.We then use induction on r . For r = 1, the result is obvious. Assume r = 2 and x = x . By definition, we have ν ( x x ) = x π ( x )( x ) = ν ( x x ) = x π ( x )( x ).This shows that ν ( x x ) must be the right-lcm (least common right-multiple) of x and x in M . On the other hand, x ( x ⋆x ) = x ( x ⋆x ) holds in M by definition,and this must also represent the right-lcm of x and x . By uniqueness of the right-lcm and left-cancellativity, we deduce π ( x )( x ) = x ⋆x . Next, for x = x , thevalue of π ( x )( x ), as well as that of x ⋆x , must be the unique element of X that is not of the form π ( x )( x ) or x ⋆x with x = x , respectively. This forces π ( x )( x ) = x ⋆x in this case as well, implying ν ( x x ) = x ( x ⋆x ) = Σ ( x , x )in every case. Assume now r ≥
3. We find ν ( x ··· x r ) = x ν ( π ( x )[ x ··· x r ]) = x ν (( x ⋆x ) ··· ( x ⋆x r ))= x Σ r − ( x ⋆x , ..., x ⋆x r ) = Σ r ( x , x , ..., x r ) , the first equality by (2), the second by the case r = 2, the third by the inductionhypothesis, and the last one by expanding the terms. (cid:3) Lemma 3.2.
For x ∈ X and r ≥ , let x [ r ] = ν ( x r ) . For all x ∈ X and u ∈ N n ,we have ν ( x p u ) = x [ p ] ν ( u ) in M . In particular, we have π ( x p ) = id and, for all x, y in X , the elements x [ p ] and y [ p ] commute in M .Proof. Let y ··· y q be a decomposition of u in terms of elements of X . By Lemma 3.1,we have ν ( x p u ) = Σ p + q ( x, ..., x, y , ..., y q )= Σ p ( x, ..., x )Σ q (Π p +1 ( x, ..., x, y ) , ..., Π p +1 ( x, , ..., x, y q ))= Σ p ( x, ..., x )Σ q ( y , ..., y q ) = ν ( x p ) ν ( y ··· y q ) = x [ p ] ν ( u ) , PATRICK DEHORNOY in which the second equality comes from expanding the terms and the third onefrom the assumption that M is of class p . Applying with u = y in X and mergingwith ν ( x p y ) = ν ( x p ) π ( x p )( y ), we deduce π ( x p ) = id. On the other hand, applyingwith u = y [ p ] , we find x [ p ] y [ p ] = ν ( x p y p ) = ν ( y p x p ) = y [ p ] x [ p ] . (cid:3) Lemma 3.3.
Assume p ≥ and define ∆ = ν ( Q x ∈ X x p − ) . Then ∆ is a Garsideelement in M , and its family of divisors is ν ( { , ..., p − } n ) , which has p n elements.Moreover ∆ p is central in M .Proof. The map ν is compatible with : for all u, v in N n , we have u v in N n if and only if ν ( u ) ν ( v ) holds in M . Indeed, by (2), v = ux with x in X implies ν ( v ) = ν ( u ) π ( u )( x ), whence ν ( u ) ν ( v ) in M . Conversely, for ν ( v ) = ν ( u ) x with x in X , as π ( u ) is bijective, we have π ( u )( y ) = x for some y in X , whence ν ( uy ) = ν ( u ) π ( u )( y ) = ν ( u ) x = ν ( v ), and v = uy since ν is injective, that is, u v in N n . Hence the left-divisors of ∆ in M are the image under ν of the p n divisorsof δ p − in N n , with δ = Q x ∈ X x . For right-divisors, the maps π ( u ) are bijective, soevery right-divisor of ∆ must be a left-divisor of ∆. Then the duality map g h for gh = ∆ is a bijection from the left- to the right-divisors of ∆. So the left- andright-divisors of ∆ coincide, and they are p n in number. Since every element of X divides ∆, the latter is a Garside element in M . Finally, by Lemma 3.1, ∆ p isthe product of the elements x [ p ] repeated p − σ [ δ ] = δ holds for everypermutation σ , we deduce x ∆ p = ∆ p x for every x . (cid:3) For u ∈ N n and x ∈ X , write | u | x for the (well-defined) number of x in an X -decomposition of u . Lemma 3.4.
For u, u ′ in N n , say that u ≡ p u ′ holds if, for every x in X , we have | u | x = | u ′ | x mod p , and, for g, g ′ in M , say that g ≡ g ′ holds for ν − ( g ) ≡ p ν − ( g ′ ) .Then ≡ is an equivalence relation on M that is compatible with left- and right-multiplication.Proof. As ν is bijective, carrying the equivalence relation ≡ p of N n to M yieldsan equivalence relation. Assume ν ( u ) ≡ ν ( u ′ ). Without loss of generality, wemay assume u ′ = ux p = x p u with x in X . Applying (2) and Lemma 3.2, wededuce π ( u ) = π ( u ′ ) and, therefore, ν ( u ) π ( u )( y ) = ν ( uy ) ≡ ν ( u ′ y ) = ν ( u ′ ) π ( u )( y ).As π ( u )( y ) takes every value in X when y varies, ≡ is compatible with right-multiplication by X . On the other hand, u ≡ p u ′ implies σ [ u ] ≡ p σ [ u ′ ] for everypermutation σ in S X , so we obtain yν ( u ) = ν ( yπ ( y ) − [ u ]) ≡ ν ( yπ ( y ) − [ u ′ ]) = yν ( u ′ ), and ≡ is compatible with left-multiplication by X . (cid:3) Lemma 3.5.
For g = ∆ pe h , g ′ = ∆ pe ′ h ′ in G with h, h ′ ∈ M , say that g ≡ g ′ holdsif h ≡ h ′ does. Then ≡ is a congruence on G with p n classes, and the kernel of G → G/ ≡ is the Abelian subgroup of G generated by the elements x [ p ] with x ∈ X .Proof. As ∆ is a Garside element in M , every element of G admits a (non-unique)expression ∆ pe h with e ∈ Z and h ∈ M . Assume g = ∆ pe h = ∆ pe h with e > e .As M is left-cancellative, we find h = ∆ p ( e − e ) h , whence h ≡ h . So, for every h ′ in M , we have h ≡ h ′ ⇔ h ≡ h ′ and ≡ is well-defined on G . That ≡ is compatiblewith multiplication on G follows from the compatibility on M and the fact that ∆ p lies in the centre of G . Next, by definition, every element of G is ≡ -equivalent tosome element of M , so the number of ≡ -classes in G equals the number of ≡ -classesin M , hence the number p n of ≡ p -classes in N n . OXETER-LIKE GROUPS FOR GROUPS OF YANG–BAXTER EQUATION 5
Finally, u ≡ p x p u holds for all x in X and u in N n . This, together withLemma 3.1, implies x [ p ] ≡
1. Conversely, assume g ≡ M . By definition, ν − ( g ) lies in the ≡ p -class of 1, hence one can go from ν − ( g ) to 1 by multiplyingor dividing by elements x p with x ∈ X . By Lemma 3.1 again, this means that onecan go from g to 1 by multiplying or dividing by elements x [ p ] with x ∈ X . In otherwords, the latter elements generate the kernel of the projection of G to G/ ≡ . (cid:3) Now Theorem 1.2 readily follows. Indeed, define W to be the finite quotient-group G/ ≡ . We saw that the kernel of the projection of G onto W is the freeAbelian group generated by the n elements x [ p ] with x ∈ X , thus giving an ex-act sequence 1 → Z n → G → W →
1. A presentation of W is obtained byadding to the presentation of G in Definition 2.1 the n relations x [ p ] = 1, thatis, x ( x⋆ x )(( x⋆ x ) ⋆ ( x⋆ x )) ... = 1. By construction, the Hasse diagram of the lat-tice made of the p n divisors of ∆ is the image under ν of the sublattice of N n made of the p n divisors of δ in N n , whereas the Cayley graph of the germ derivedfrom ( W, X )—that is, W equipped with the partial product obtained by restrictingto the cases when the X -lengths add—is the image under ν of the Cayley graphof the germ derived from the quotient-group Z n / ≡ p : the (obvious) equality in thecase of N n implies the equality in the case of M .4. An example
For an RC-quasigroup of class 1, that is, satisfying x⋆ y = y for all x, y , thegroup G is a free Abelian group, the group W is trivial, and the short exact sequenceof Theorem 1.2 reduces to 1 → Z n → G → x⋆x ) ⋆ ( x⋆y ) = y holds for all x, y , is addressed in [2](with no connection with RC-quasigroups). The element ∆ is the right-lcm of X ,it has 2 n divisors which are the right-lcms of subsets of X , and the group W is theorder 2 n quotient of G obtained by adding the relations x ( x⋆ x ) = 1.For one example in class 3, consider { a , b , c } with x⋆y = f ( y ), f : a b c a . The associated presentation is h a , b , c | ac = b , ba = c , cb = a i . The smallestGarside element is a , but, here, in class 3, we consider the next one, namely ∆ = a .Adding to the above presentation of G the three relations x ( x⋆ x )(( x⋆ x ) ⋆ ( x⋆ x )) =1, namely abc = bca = cab = 1, here reducing to abc = 1, one obtains for W a b cab b bc a c caac b a ab a ba a b ca b a a b a c c c b b a c ∆ Figure 1.
An example in class 3: here W has 3 = 27 elements,and its Cayley graph is a cube with edges of length 3 − PATRICK DEHORNOY the presentation h a , b , c | ac = b , ba = c , cb = a , abc = 1 i . The lattice Div(∆)has 27 elements, its diagram is the cube shown on the right. The latter is also theCayley graph of the germ derived from ( W, X ). References [1] F. Chouraqui,
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Semigroups of I -type , J. of Algebra 206 (1998)97-112.[7] E. Jespers and J. Okninski, Noetherian semigroup algebras, Algebra and Applications vol. 7,Springer-Verlag (2007).[8] W. Rump, A decomposition theorem for square-free unitary solutions of the quantum Yang–Baxter equation , Adv. in Math. 193 (2005) 4055.
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