Reflection equation as a tool for studying solutions to the Yang-Baxter equation
aa r X i v : . [ m a t h . QA ] A ug REFLECTION EQUATION AS A TOOL FOR STUDYINGSOLUTIONS TO THE YANG–BAXTER EQUATION
VICTORIA LEBED AND LEANDRO VENDRAMIN
Abstract.
Given a right-non-degenerate set-theoretic solution (
X, r ) to theYang–Baxter equation, we construct a whole family of YBE solutions r ( k ) on X indexed by its reflections k (i.e., solutions to the reflection equation for r ).This family includes the original solution and the classical derived solution.All these solutions induce isomorphic actions of the braid group/monoid on X n . The structure monoids of r and r ( k ) are related by an explicit bijective1-cocycle-like map. We thus turn reflections into a tool for studying YBEsolutions, rather than a side object of study. In a different direction, we studythe reflection equation for non-degenerate involutive YBE solutions, show itto be equivalent to (any of the) three simpler relations, and deduce from thelatter systematic ways of constructing new reflections. To the memory of Patrick Dehornoy,a connoisseur of strange structures
Introduction
Let X be a set. A map r : X × X → X × X will be called a solution if it is a(set-theoretic) solution to the Yang–Baxter equation (YBE) on X : r r r = r r r . (1)Here and below the notation φ i means the map φ applied starting from the i thposition of some X m . A map k : X → X will be called a reflection for ( X, r ) if itsatisfies the reflection equation (RE) on X : rk rk = k rk r. (2)The linear versions of these equations (that is, when X is a vector space, thedirect products × are replaced with the tensor products ⊗ , and all maps are re-quired to be linear) appeared in physics. In particular, in the study of n particlescattering on a half-line, a solution r represents a collision between two particles,and a reflection k represents a collision between a particle and the wall delimitingthe half-line. See Figs. 1–2 for an illustration.Since then the YBE has made its way into many areas of mathematics. The REis far less familiar to mathematicians. An exception is the study of braids with afrozen strand, or, equivalently, the study of the motion of n particles in an annulus(whereas usual braids describe the motion of n particles in a disk). Figs. 3–4 shouldmake this connection clear. See [Cho48, Sos92, Sch94] for more detail. Mathematics Subject Classification.
Key words and phrases.
Yang–Baxter equation, reflection equation, structure monoid, struc-ture shelf, braid group. time wallhalf-line =
Figure 1.
The Yang–Baxter equation governing collisions be-tween particles moving on a half-line=
Figure 2.
The reflection equation governing collisions betweenparticles and the wallThe idea of studying set-theoretic rather than linear solutions goes back toDrinfel ′ d. His goal was to restrict attention to this more tractable, but still ex-tremely rich class of solutions. This class is currently an object of active mathe-matical research; see [SS18, GI18, JKVA19, MBBER19, Rum20, CCS20, CJO20a]and references therein for some recent advances. The interest in set-theoretic so-lutions to the RE is on the contrary rather new. The only articles going in thisdirection we are aware of are [DC19, KO19, SVW20, Kat19, DS20]. The objectiveof all these contributions is to systematically construct and study reflections fordifferent types of solutions.In the present paper, we are looking at reflections from a completely differentangle. For us they are tools for studying the corresponding solution. Concretely,with any solution r on X , written explicitly as r ( a, b ) = ( λ a ( b ) , ρ b ( a )) , (3)come the following invariants: • the structure monoid of r , given by the presentation M ( X, r ) = h a ∈ X | ab = λ a ( b ) ρ b ( a ) , for all a, b ∈ X i ;(4) • the structure shelf of r : a ⊳ b = ρ b λ ρ − a ( b ) ( a ); • the derived monoid A ( X, r ) of r , which is the structure monoid of thefollowing derived solution on X : r ⊳ : ( a, b ) ( b ⊳ a, a );(5) • the structure group G ( X, r ) and the derived group of r , given by the samepresentation as the monoids M ( X, r ) and A ( X, r ) respectively.The constructions of structure shelf and derived monoid/group are valid only for right-non-degenerate (RND) solutions r , that is, their right components ρ b arebijective on X for all b ∈ X . Similarly, a solution is called left-non-degenerate(LND) if all the λ a ’s are bijective, and non-degenerate if it is both RND and LND. In [LV17] we used the term
LND instead; here we stick to the more popular convention.
Recall that a shelf is a set X with a binary self-distributive operation ⊳ , in thesense of ( a ⊳ b ) ⊳ c = ( a ⊳ c ) ⊳ ( b ⊳ c )for all a, b, c ∈ X . For any shelf, the map (5) is a solution.Note that all these constructions are not only useful invariants of solutions, butalso a rich source of monoids and groups with remarkable properties.The structure and the derived monoids are related by a bijective monoid 1-cocycle J : M ( X, r ) → A ( X, r ) . Similarly, the structure and the derived groups are related by a bijective group 1-cocycle. As a result, one can deduce various properties of the structure monoid andgroup of a solution from those of its structure shelf. This was used to understandthe possible values of the size of the degree 2 component of M ( X, r ) in [CJO20b];to construct a remarkable finite quotient of G ( X, r ) in [LV19], generalising theCoxeter-like quotients from [Deh15]; and to understand when G ( X, r ) has torsionin [JKVAV20]. Also, J can be used to transport the product of A ( X, r ) to M ( X, r ).Combining it with the original product of M ( X, r ), one gets a ring-like structurecalled skew-brace (see [GV17] and numerous recent references thereto). This allowsfor fruitful transport of techniques and intuitions from group and ring theories tothe study of YBE solutions.In this paper, we show that r ⊳ is actually a part of a whole family of solutions r ( k ) indexed by the reflections k of r . We recover r ⊳ as r (Id) , and we get r itselfby adding an element ∗ to X and taking as k the projection onto ∗ (Example 1.3).In general, there are many more solutions (of variable complexity) in this family,which we illustrate with examples. All these solutions induce isomorphic actionsof braid monoids/groups, and share many properties (invertibility, involutiveness,idempotence etc.). They have the same structure shelves. The structure monoidsof these solutions are related by bijective maps J ( k ) : M ( X, r ) → M ( X, r ( k ) )generalising J . These maps are not monoid 1-cocycles in general, but satisfy amore involved property (11). We expect that, for a wide class of solutions, similarbijections relate the structure groups of solutions from this family. We have acandidate for such bijections, but cannot prove that it does the job. Finally, everymonoid M ( X, r ( k ) ) left-acts on X ; this generalises the left action of the derivedmonoid, given on the generators by a · b = b ⊳ a .In the second part of the paper, we take a closer look at the reflections for aninvolutive solution. We show that to check the reflection equation for an LND/RNDsolution, it is enough to compare the first/second coordinates of (2) only. As aconsequence, a map commuting with all the λ a ’s is a reflection for an LND solution,and a map k satisfying ρ k ( a ) = ρ a for all a ∈ X is a reflection for an RND solution.The LND statements appeared in [SVW20], while the RND ones are new. We alsosuggest two ways (a stronger and a weaker) of dividing reflections for an involutivesolution into equivalence classes, making the study of the (generally huge) set ofreflections more organised. We show that two reflections equivalent in the strongersense yield isomorphic solutions r ( k ) . On the other hand, non-equivalent reflectionsmight give the same solutions r ( k ) . VICTORIA LEBED AND LEANDRO VENDRAMIN
Finally, given an LND involutive solution and a map k : X → X , we construct abinary operation b ∗ a = λ b kλ − b ( a ). We show that k is a reflection if and only if ∗ satisfies the strange property ( a ∗ b ) ∗ a = b ∗ a for all a, b ∈ X . Moreover, k can be uniquely reconstructed from ∗ .The constructions, results and proofs in this paper are given using a mixture oftwo languages: algebraic formulas, and the graphical calculus developed in [LV17](which we use freely with a minimum of explanations; the reader is referred to[LV17] for more detail). The formulas we get are often difficult to digest; theirpictorial analogues are on the contrary intuitive, and explain their origin.1. Reflections as parameters for generalised derived solutions
Let (
X, r ) be a right-non-degenerate (RND) set-theoretic solution to the Yang–Baxter equation (further simply called solution ). Let k be a reflection for ( X, r ).We will use k to construct another solution r ( k ) on X . For k = Id X we recover thederived solution. We then describe bijective maps J n ; k : X n → X n intertwining thebraid monoid/group actions on X n coming from r and r ( k ) respectively; the case k = Id X yields the guitar map from [Sol00, LYZ00, LV17]. Thus with any RNDsolution comes a whole family of solutions parametrised by its reflections. Thesesolutions vary in complexity, but all yield isomorphic braid monoid/group actions,the same structure shelf, and, as we shall see in the next section, intimately relatedstructure monoids.Now let us turn to concrete statements. Definition 1.1.
The generalised derived solution associated to a reflection k foran RND solution ( X, r ), or simply a k - derived solution , is the map r ( k ) : X × X −→ X × X, ( a, b ) ( b ′ = ρ k ( a ′ ) λ ρ − k ( b ) ( a ) ( b ) , a ′ = ρ b ρ − k ( b ) ( a )) . In Corollary 1.9 we will show that r ( k ) is indeed a solution. Example 1.2.
The trivial reflection k = Id X yields the classical derived solution( a, b ) ( ρ a λ ρ − b ( a ) ( b ) , a ). Example 1.3.
Extend the solution (
X, r ) to X ∗ = X ⊔ {∗} by imposing λ ∗ = ρ ∗ = Id X ∗ , and λ a ( ∗ ) = ρ a ( ∗ ) = ∗ for all a ∈ X . Consider the projection onto {∗} : k ( a ) = ∗ for all a ∈ X ∗ . It is a reflection, since both sides of (2) yield ( ∗ , ∗ ) forall a, b ∈ X ∗ . This reflection yields r ( k ) ( a, b ) = ( λ a ( b ) , ρ b ( a )), which is simply theoriginal solution extended to X ∗ .In the next example, on the contrary, the generalised derived solution is verydifferent from the original one: Example 1.4.
Consider the solution X = { , , } , λ = ρ = (12), and itsreflection k ( a ) = 3 for all a . Then for r ( k ) we have λ = ρ = λ = ρ = (12). Hereand below the omitted λ ’s and ρ ’s are all identities.In the last example, we see that even small solutions may have many non-isomorphic generalised derived solutions of very different nature. Example 1.5.
Consider the solution X = { , , , } , λ a = (132) for all a , ρ =(13) , ρ = (12) , ρ = (23) , ρ = (123). To see that it is a solution, consider the map r ( a, b ) = ( b − , − a − b −
1) on { , , } . One checks that r r r ( a, b, c ) = ( c + 1 , − b − c + 1 , a + b − c ) = r r r ( a, b, c ) . In fact it is sufficient to compute the first two coordinates only (which are thesimplest ones), since r preserves the signed sum of its entries:( b − − ( − a − b −
1) = a − b. Further, the map σ = (132) is compatible with this map r , in the sense of r ( σ × σ ) =( σ × σ ) r . Thus one can extend r to X = { , , , } by putting λ = σ and ρ = σ − .This is precisely the solution described above.One checks that the maps of the following type are reflections for our solution: • projections p = aaaa for all a ; • maps p = aaa a ; • maps p = 444 a for all a .Here and below when writing k = abcd we mean k (1) = a, k (2) = b , k (3) = c , k (4) = d . Together with k = Id, this sums up to eleven reflections. A computer-aided verification shows that this is the complete list.The reflection k = Id produces the derived solution given by ρ a = Id for all a, λ a = ( a − , a + 1) for all a = 4 , λ = Id . Here and below a ± k = 4444 produces the solution given by ρ a = ( a − , a + 1) for all a = 4 , ρ = Id , λ a = Id for all a. This is the mirror variant of the preceding solution.The projection k = 1111 produces the solution given by ρ = Id , ρ = (132) , ρ = (123) , ρ = (23) , λ a = (23) for all a. The reflection k = 1114 produces the solution given by ρ = ρ = Id , ρ = (132) , ρ = (123) , λ a = (23) for all a = 4 , λ = Id . It differs from the preceding solution by the values of ρ and λ .These four solutions are clearly non-isomorphic. In fact, according to our com-puter, this is a complete list of generalised derived solutions in this case, up toisomorphism.Already our central Definition 1.1 is difficult to grasp in the algebraic form. Letus give its pictorial interpretation.Graphically, the solution r and the reflection k are represented by a crossing anda bead respectively. The YBE then becomes the Reidemeister III move (Fig. 3),in the sense that, putting the same colors at the bottom of the two diagrams, onepropagates them upwards to get the same colors at the top. In other words, theReidemeister III move has only a local effect on colorings. Here and below we markin green the “starting” colors of a diagram, which uniquely determine the colors ofall the remaining diagram parts. The reflection equation is depicted (in the samesense as the YBE) on Fig. 4. It does not seem natural drawn this way, but itbecomes so if one thinks of the beads as the twists around a straight pole, as shown VICTORIA LEBED AND LEANDRO VENDRAMIN in the right part of Fig. 4. In what follows we will use the beads rather than thepole in order to make our diagrams lighter. a bλ a ( b ) ρ b ( a ) ak ( a ) Y BE = Figure 3.
Graphical depiction of a solution, its reflexion, and theYang–Baxter equation RE = RE = Figure 4.
A short-hand and a detailed graphical depictions of thereflection equationNow, in the left diagram of Fig. 5, the (green) colors a and b put just below thetwo beads uniquely determine all the remaining colors, in particular the bottom leftcolor c = ρ − k ( b ) ( a ). Then in the right diagram, the colors c and b put at the bottomuniquely determine all the remaining colors, in particular the colors a ′ = ρ b ( c ) and b ′ = ρ k ( a ′ ) λ c ( b ) just below the beads. The map r ( k ) from Definition 1.1 works withthe colors just below the beads: it sends ( a, b ) to ( b ′ , a ′ ). For future reference, notethat the top colors of the two diagrams of Fig. 5 coincide due to the reflectionequation. ba k ( b ) c = ρ − k ( b ) ( a ) c b λ c ( b ) a ′ = ρ b ( c ) b ′ = ρ k ( a ′ ) λ c ( b ) Figure 5.
The generalised derived solution ( a, b ) ( b ′ , a ′ ).Now, we have to prove that our maps r ( k ) are indeed YBE solutions. This fact,as well as the fundamental properties of these solutions, are based on the followingconstruction. Definition 1.6.
Given a solution (
X, r ) and its reflection k , the k -Garside maps are the following operators on X n :∆ n ; k = k n ( r n − · · · r r ) k n · · · k n ( r n − r n − ) k n r n − k n . The k -guitar maps are then defined by J n ; k : X n −→ X n , ( a , a , . . . , a n ) ( ρ ∆ n − k ( a ,...,a n ) ( a ) , ρ ∆ n − k ( a ,...,a n ) ( a ) , . . . ,ρ ∆ k ( a n − ,a n ) ( a n − ) , ρ k ( a n ) ( a n − ) , a n ) . Here we extended the maps ρ to(6) ρ ( b ,b ,...,b m ) = ρ b m · · · ρ b ρ b . The maps ∆ n ; k resemble the action of the classical Garside elements of thepositive braid monoids, except for the reflection k repeatedly applied to rightmostelements, hence the name.The maps ∆ n ; k and J n ; k are graphically interpreted in Fig. 6. As with thegeneralised derived solution r ( k ) , the important colors to look at are the ones justbelow the beads. a a · · · a n − a n a ′ a ′ ... a ′ n − a ′ n J n ; k ( a , a , . . . , a n − , a n )= ( a ′ , a ′ , . . . , a ′ n − , a ′ n )∆ n ; k ( a , a , . . . , a n − , a n ) Figure 6.
The k -Garside map ∆ n ; k and the k -guitar map J n ; k . Lemma 1.7.
The maps J n ; k are bijective if r is RND.Proof. If all the ρ a ’s are bijective, then from J n ; k ( a , a , . . . , a n ) = ( a ′ , a ′ , . . . , a ′ n )one can reconstruct a n = a ′ n , then a n − = ρ − k ( a n ) ( a ′ n − ), and so on until a = ρ − n − k ( a ,...,a n ) ( a ′ ) . (cid:3) Theorem 1.8.
Let ( X, r ) be an RND solution, and k its reflection. Then the k -guitar maps intertwine the solution r and the k -derived solution r ( k ) . That is, forall n ∈ N and all ≤ i < n , one has J n ; k r i = r ( k ) i J n ; k . (7) VICTORIA LEBED AND LEANDRO VENDRAMIN
As for the k -Garside map, it is compatible with r in the following sense: ∆ n ; k r i = r n − i ∆ n ; k . (8)This theorem directly yields the following fundamental properties of generalisedderived solutions: Corollary 1.9.
Let ( X, r ) be an RND solution, and k its reflection. Then:(1) r ( k ) is an RND YBE solution;(2) r and r ( k ) induce isomorphic braid monoid/group actions;(3) the relation r s = r t for some s > t ≥ is equivalent to ( r ( k ) ) s = ( r ( k ) ) t ; inparticular, r is invertible/idempotent if and only if r ( k ) is so;(4) r is invertible if and only if r ( k ) is so.Proof of Theorem 1.8. A graphical proof is given in Fig. 7. r J a a a a J ( r ( a )) J ( r ( a )) J ( r ( a )) J ( r ( a )) YBE J a a a a J ( a ) J ( a ) J ( a ) J ( a ) YBE a a a a J ( r ( a ))= J ( a ) J ( r ( a ))= J ( a ) ′ J ( r ( a ))= J ( a ) ′ J ( r ( a ))= J ( a ) a a a a J ( a ) J ( a ) J ( a ) J ( a )definition of r ( k ) and RE Figure 7.
The entwining relation J n ; k r i = r ( k ) i J n ; k (here n =4, i = 2) is established by comparing the colors just below thebeads in the bottom left diagram as calculated from the upper left(blue labels) and the two right diagrams (red labels). The relation∆ n ; k r i = r n − i ∆ n ; k is established by comparing the top colors ofthe two upper diagrams.We start from the upper left diagram representing J n ; k r i . The bottom colors a are fixed, and uniquely determine all the remaining colors. We then pull thecrossing corresponding to r i to the right, and get the lower left diagram. Due tothe YBE, this does not alter the (blue) colors just below the beads. Then we makeone bead slide around its neighbouring bead in the grey zone, and obtain the lower right diagram. This preserves the colors outside the grey zone because of the RE.And for the colors just below the beads, this operation boils down to applying r ( k ) (directed from right to left). Now, one can pull the beadless crossing from the greyzone upward, and get the upper right diagram. Once again, the YBE guaranteesthat the the (red) colors just below the beads remain unchanged. But from theupper right diagram, one sees that these colors are precisely J n ; k ( a ).The argument above shows that the colors at the top of the two upper diagramscoincide. But these are precisely the left and the right sides of (8). (cid:3) We have thus learnt that any RND solution comes with a whole family of closelyrelated solutions parametrised by its reflections. It is natural to ask what happensif we iterate this construction, and look at the generalised derived solutions ofgeneralised derived solutions. Computer experiments show that new iterations mayproduce new solutions, but the situation tends to stabilise fast. One of the reasonsis the following result:
Theorem 1.10.
Let ( X, r ) be an RND solution, and k its reflection. Then thestructure shelf operations ⊳ and ⊳ k for r and r ( k ) respectively coincide. If moreover k and r are both invertible, then one has k ( b ⊲ k a ) = k ( a ) ⊳ k ( b ) for all a, b ∈ X . Here ⊲ k is the left structure shelf operations for r ( k ) . Recall that the left structure shelf operations for an LND solution is given bythe formula b ⊲ a = λ b ρ λ − a ( b ) ( a ) . In particular, under the assumptions of the proposition, the solution r ( k ) is auto-matically LND. Note that, according to [LV19], the right and left structure shelvesof an invertible non-degenerate solution are isomorphic. Proof.
In Fig. 8 we showed a double application of r ( k ) . ba c k ( a ′′ ) k ( b ′ ) k ( a ) r ( k ) : ( a, b ) b ′ k ( b ′ ) a ′ c b k ( a ′′ ) ( b ′ , a ′ ) a ′ b ′′ a ′′ k ( a ′′ ) c b ( a ′′ , b ′′ ) Figure 8.
Comparing structure shelves for r and r ( k ) : the di-agrams yield b ⊳ k a ′ = b ′′ = b ⊳ a ′ , b ′ ⊲ k a = a ′′ and k ( a ) ⊳ k ( b ′ ) = k ( a ′′ ).As usual, the starting colors in each diagram are in green: a and b for the leftone, and c and b for the remaining ones. The definition of r ( k ) yields r ( k ) : ( a, b ) ( b ′ , a ′ ) ( a ′′ , b ′′ ) . By the definition of structure shelves, this implies b ⊳ k a ′ = b ′′ and b ′ ⊲ k a = a ′′ . On the other hand, by the RE, the coloring changes due to a sliding bead are local.Hence the upper right colors are the same on the three diagrams. From the rightdiagram we see that it is k ( a ′′ ). In the same way, the color k ( b ′ ) can be transportedfrom the middle to the left diagram, and the color a ′ from the middle to the rightdiagram. Finally, from the right diagram and the definition of the right structureshelf we conclude b ⊳ a ′ = b ′′ = b ⊳ k a ′ . It remains to prove that this identity holds for all b, a ′ ∈ X . For this one needs todeduce the starting color a from a ′ and b . But this is easy: a = ρ k ( b ) ( c ) = ρ k ( b ) ρ − b ( a ′ ) . Similarly, from the left diagram follows k ( a ) ⊳ k ( b ′ ) = k ( a ′′ ) = k ( b ′ ⊲ k a ) . The starting color b can be deduced from a and b ′ only when k and r are invertible.Indeed, the colors k ( a ) and k ( b ′ ) on the left diagram can be propagated everywhere,and uniquely determine the remaining colors, in particular the bottom right color b . (cid:3) Structure monoids for generalised derived solutions
The classical guitar maps yield a bijective monoid 1-cocycle M ( X, r ) → A ( X, r ) = M ( X, r (Id) ) . In this section, we will see that the k -guitar maps induce a bijection M ( X, r ) → M ( X, r ( k ) ), satisfying a more general cocycle-like property (11). Moreover, eachstructure monoid M ( X, r ( k ) ) acts on the set X .Recall the definition (4) of the structure monoid of ( X, r ). It can be reinterpretedas the quotient of the free monoid on X by the relations r i ( a , . . . , a n ) = ( a , . . . , a n )for all n , all i < n , and all a j ∈ X . Thus the entwining relations J n ; k r i = r ( k ) i J n ; k and ∆ n ; k r i = r n − i ∆ n ; k from Theorem 1.8, and the invertibility of J n ; k , imply Proposition 2.1.
Let ( X, r ) be an RND solution, and k its reflection. The k -guitarmaps J n ; k induce a bijection of Z ≥ -graded monoids J ( k ) : M ( X, r ) → M ( X, r ( k ) ) . Further, the k -Garside maps ∆ n ; k induce a bijection of Z ≥ -graded monoids ∆ ( k ) : M ( X, r ) → M ( X, r ) . The map J ( k ) is not necessarily a monoid 1-cocycle. It interacts with the prod-ucts in M ( X, r ) and in M ( X, r ( k ) ) in a more intricate way: Proposition 2.2.
Let ( X, r ) be a solution, and k its reflection. Take p, q ∈ N , a ∈ X p , b ∈ X q . One has: J p + q ; k ( ab ) = J p ; k ( ρ ∆ q ; k ( b ) ( a )) J q ; k ( b ) , (9) ∆ p + q ; k ( ab ) = λ a (∆ q ; k ( b )) ∆ p ; k ( ρ ∆ q ; k ( b ) ( a )) . (10) Without the superscripts, the formulas take a more readable form: J ( ab ) = J ( ρ ∆( b ) ( a )) J ( b ) , (11) ∆( ab ) = λ a (∆( b )) ∆( ρ ∆( b ) ( a )) . (12)Here λ and ρ are extended to the powers of X as shown in Fig 9. Note that thedefinition (6) is a particular case of this extension. a ∈ X × p b ∈ X × q ρ b ( a ) ∈ X × p λ a ( b ) ∈ X × q Figure 9.
The solution r extended to T ( X ). Remark . The YBE for r implies that these extended λ and ρ induce left andright actions of M ( X, r ) on itself respectively. Thus formulas (11)–(12) are validfor all a, b ∈ M ( X, r ), and the induced maps J and ∆. Proof of Proposition 2.2.
The proof is given on Fig. 10. The dotted lines mean thatwe are simply reading the colors of the strands we intersect, without modifyingthem. a a p · · · b b q · · · J p ; k ( ρ ∆ q ; k ( b ) ( a )) J q ; k ( b ) λ a (∆ q ; k ( b )) ∆ p ; k ( ρ ∆ q ; k ( b ) ( a )) ∆ q ; k ( b ) ρ ∆ q ; k ( b ) ( a ) Figure 10.
The k -Garside map ∆ n ; k and the k -guitar map J n ; k evaluated on products. (cid:3) Remark . In the classical case k = Id, the YBE implies λ ∆ n ;Id ( c ) = λ c and ρ ∆ n ;Id ( c ) = ρ c for all c ∈ X n , so (9) yields J p + q ; k ( ab ) = J p ; k ( ρ b ( a )) J q ; k ( b ) . To turn it into a 1-cocycle condition, one needs to rewrite J ( ρ b ( a )) as J ( a ) ↼ b forsome right action of M ( X, r ) on M ( X, r (Id) ). We omitted the superscripts for themap J for simplicity. This action thus has to be defined by a ↼ b = J ( ρ b ( J − ( a ))) . One easily checks that it is an action of M ( X, r ) on M ( X, r (Id) ) by monoid mor-phisms. If one naively transposes this procedure to k -guitar maps, the operationobtained is no longer a monoid action.We have seen that the structure shelves of a solution and its generalised derivedsolutions coincide. This renders the structure shelves uninteresting invariants ofgeneralised derived solutions. The k -versions of structure shelves (which need notto be shelves) are on the contrary rather useful. Definition 2.5.
Let (
X, r ) be an RND solution, and k its reflection. Define thefollowing binary operation on X : a ◭ k b = ρ k ( b ) λ ρ − a ( b ) ( a ) . If (
X, r ) is LND instead, define the following binary operation on X : b ◮ k a = λ b kρ λ − a ( b ) ( a ) . Graphical versions of these operations in Fig. 11 are very intuitive. When k = Id,one recovers the right and the left structure shelf operations respectively. ab a ◭ k b ab a ◮ k b Figure 11.
The k -versions of structure shelf operations.We will now study the operation ◭ k . The second operation ◮ k will appear laterin Remark 3.12. Note that, contrary to the two structure shelf operations, thesetwo operations play asymmetric roles. Lemma 2.6.
Let ( X, r ) be an RND solution, and k its reflection. For all a, b, c ∈ X , one has ( a ◭ k b ) ◭ k c = ( a ◭ k c ′ ) ◭ k b ′ , where ( b ′ , c ′ ) = r ( k ) ( c, b ) .Proof. A graphical proof is given on Fig. 12. (cid:3)
This lemma directly implies
Proposition 2.7.
Let ( X, r ) be an RND solution, and k its reflection. Then theright translations ◭ k b extend to a left action of the structure monoid M ( X, r ( k ) ) on X . a ◭ k b ) ◭ k c cba YBE cba RE c ′ b ′ a YBE b ′ c ′ a ( a ◭ k c ′ ) ◭ k b ′ Figure 12.
The relation ( a ◭ k b ) ◭ k c = ( a ◭ k c ′ ) ◭ k b ′ , where( b ′ , c ′ ) = r ( k ) ( c, b ), is established by comparing the upper colors.3. Reflections for involutive solutions
In this section (
X, r ) is an involutive set-theoretic YBE solution.First, we will show that in this case in order to check the reflection equation (2)on X , it is often sufficient to look at one of the two coordinates of X only.To make this statement concrete, put t ( a, b ) = λ λ a ( b ) kρ b ( a ) , u ( a, b ) = ρ kρ b ( a ) λ a ( b ) , where a, b ∈ X . With these notations, 2 can be rewritten as( t ( a, k ( b )) , u ( a, k ( b ))) = ( t ( a, b ) , k ( u ( a, b ))) . (13) Theorem 3.1. (1) [SVW20]
Let ( X, r ) be an LND involutive solution. A map k : X → X is a reflection for ( X, r ) if and only if the left part t ( a, k ( b )) = t ( a, b )(14) of (13) holds for all a, b ∈ X .(2) Let ( X, r ) be an RND involutive solution. A map k : X → X is a reflectionfor ( X, r ) if and only if the right part u ( a, k ( b )) = k ( u ( a, b ))(15) of (13) holds for all a, b ∈ X . Relations (14) and (15) can be explicitly written as follows: λ λ a ( b ) kρ b ( a ) = λ λ a ( k ( b )) kρ k ( b ) ( a ) , (16) ρ kρ k ( b ) ( a ) λ a k ( b ) = kρ kρ b ( a ) λ a ( b ) . (17) Proof.
We will prove only Point 2. Point 1 can be shown using similar arguments;an alternative approach can be found in [SVW20].Our aim is to deduce (14) from (15). Assume (15). Take any a, b ∈ X . Put( c, d ) = rk ( a, b ) , ( e, f ) = rk ( c, d ) . Since r is involutive, this yields r ( e, f ) = k ( c, d ) = ( c, k ( d )) . (18)By (15), we have k rk r ( a, b ) = ( e ′ , f ) for some e ′ ∈ X . We need to prove that e ′ = e . Put( g, h ) = r ( e ′ , f ) = rk rk r ( a, b ) . By (15) again, h is the same as the second component of k rk rr ( a, b ) = k rk ( a, b ) = k ( c, d ) = ( c, k ( d )) , that is, h = k ( d ). This yields ( g, k ( d )) = r ( e ′ , f ) . Comparing with (18) and using the right non-degeneracy, one concludes that e ′ = e ,as desired. (cid:3) This is used in the following construction of reflections:
Theorem 3.2. (1) [SVW20]
Let ( X, r ) be an LND involutive solution. A map k : X → X commuting with all the λ a ’s, in the sense of (19) kλ a = λ a k for all a ∈ X, is a reflection for ( X, r ) .(2) Let ( X, r ) be an RND involutive solution. A map k : X → X satisfying (20) ρ k ( a ) = ρ a for all a ∈ X is a reflection for ( X, r ) .Proof. (1) According to Theorem 3.1, it is sufficient to check the relation (16)for all a, b ∈ X . We have λ λ a ( b ) kρ b ( a ) = kλ λ a ( b ) ρ b ( a ) = k ( a ) . The first equality follows from (19), and the second one from the involutivityof r . Similarly, λ λ a ( k ( b )) kρ k ( b ) ( a ) = kλ λ a ( k ( b )) ρ k ( b ) ( a ) = k ( a ) . (2) According to Theorem 3.1, it is sufficient to check the relation (17) for all a, b ∈ X . We have ρ kρ k ( b ) ( a ) λ a k ( b ) = ρ ρ b ( a ) λ a k ( b ) = k ( b ) . The first equality follows from (20), and the second one from the involutivityof r . Similarly, kρ kρ b ( a ) λ a ( b ) = kρ ρ b ( a ) λ a ( b ) = k ( b ) . (cid:3) Example 3.3.
Consider a permutation solution r ( a, b ) = ( f ( b ) , f − ( a )), where f is a permutation on X . It is an involutive non-degenerate solution. Point 1 of thetheorem shows that any k commuting with f is a reflection, while Point 2 assertsthat all maps k : X → X are reflections. In this case, all the generalised derivedsolutions are simply the flips: r ( k ) ( a, b ) = ( b, a ). Example 3.4.
Let us resume Example 1.4. It is an involutive non-degeneratesolution. Point 1 of the theorem yields three reflections: 123, 213 and 333. Point 2yields four reflections: all the maps respecting the decomposition X = { , } ⊔ { } ,that is, 123, 213, 113, 223. In total one gets five reflections, which is the completelist for this solution. For all these reflections except for 333, we get ρ b = ρ k ( b ) for all b , hence the k -derived solution is of the form r ( k ) ( a, b ) = ( b ′ , a ). But it is involutive since the original solution is so, thus r ( k ) is a flip. The solution r (333) was describedin Example 1.4. Proposition 3.5.
Let k be a reflection for an involutive solution ( X, r ) , and taketwo maps ϕ, ψ : X → X satisfying (19) – (20) . Then the map ϕkψ is a reflection.Proof. Relations (19)–(20) can be assembled into rϕ = ϕ r . But for involutive r ,this implies rϕ = ϕ r , since ϕ r = rrϕ r = rϕ rr = rϕ . The same holds for ψ . This yields rϕ k ψ rϕ k ψ = ( ϕ × ϕ ) rk rk ( ψ × ψ ) = ( ϕ × ϕ ) k rk r ( ψ × ψ )= ϕ k ψ rϕ k ψ r. (cid:3) This result suggests how to divide reflections for involutive solutions into equiv-alence classes.
Example 3.6.
For a trivial solution r ( a, b ) = ( b, a ), all maps X → X are reflec-tions, and they form a single equivalence class. Example 3.7.
For a general permutation solution r ( a, b ) = ( f ( b ) , f − ( a )) fromExample 3.3, all maps X → X are still reflections, but they may fall into severalequivalence classes. For example, for X = { , } and f = (12), there are twomaps satisfying (19)–(20): ϕ = Id and ϕ = (12). Then the reflections fall into twoclasses: { , } and { , } . On the other hand, if f has a fixed point p , then anyreflection can be postcomposed with the projection ϕ ( a ) = p , hence there is onlyone equivalence class. Example 3.8.
Let us resume Example 3.4. There are two maps satisfying (19)–(20): ϕ = Id and ϕ = (12). Pre- and post-composition with the latter divides thefive solutions into three classes: { , } , { , } , and { } .In the last two examples, the equivalence class turns out to be an invariantstrictly refining the generalised derived solution. We will now show that in generaltwo equivalent reflections yield isomorphic generalised derived solutions wheneverthe ψ ’s used in the equivalence relation are bijections: Proposition 3.9.
Let k be a reflection for an RND involutive solution ( X, r ) , andtake two maps ϕ, ψ : X → X satisfying (19) – (20) . One has r ( k ) ( ψ × ψ ) = ( ψ × ψ ) r ( ϕkψ ) . Proof.
In the proof of Propositoin 3.5, we showed that (19)–(20) for ψ imply rψ = ψ r , hence ψρ a = ρ a ψ and λ ψ ( a ) = λ a for all a ∈ X . Now, recalling the definitionof r ( k ) , one has r ( k ) ( ψ × ψ ) : ( a, b ) ( b ′ = ρ k ( a ′ ) λ ρ − kψ ( b ) ψ ( a ) ψ ( b ) , a ′ = ρ ψ ( b ) ρ − kψ ( b ) ψ ( a ))= ( b ′ = ψρ k ( a ′ ) λ ψρ − kψ ( b ) ( a ) ( b ) , a ′ = ψρ b ρ − kψ ( b ) ( a ))= ( b ′ = ψρ k ( a ′ ) λ ρ − kψ ( b ) ( a ) ( b ) , a ′ = ψρ b ρ − kψ ( b ) ( a ))= ( b ′ = ψρ ϕkψ ( a ′′ ) λ ρ − ϕkψ ( b ) ( a ) ( b ) , a ′ = ψ ( a ′′ ))where a ′′ = ρ b ρ − ϕkψ ( b ) ( a )= ( ψ × ψ ) r ( ϕkψ ) ( a, b ) . (cid:3) Finally, we unveil an unexpected relation between reflections and another alge-braic structure. Let Refl ( X,r ) be the set of all reflections for ( X, r ). Let Strange X bethe set of all strange binary operations ∗ on a set X , that is, operations satisfying( a ∗ b ) ∗ a = b ∗ a for all a, b ∈ X. Theorem 3.10.
For any LND involutive YBE solution ( X, r ) , one has an injection Refl ( X,r ) ֒ → Strange X ,k ( b ◮ k a = λ b kλ − b ( a )) . Moreover, k is a reflection if and only if the operation ◮ k is strange. The operation ◮ k above is a particular case of that from Definition 2.5.Thus to check if k is a reflection, it suffices to check the relation λ λ a kλ − a ( b ) kλ − λ a kλ − a ( b ) ( a ) = λ b kλ − b ( a ) . Contrary to the alternative one-relation criteria (16) and (17) from Theorem 3.1,it involves only the λ ’s, and no ρ ’s.The remarkable feature of our injection is the independence of the set Strange X from the map r . We identify reflections with strange operations ∗ such that themaps λ − b l b λ b do not depend on b . Here l b is the left translation for ∗ : l b ( a ) = b ∗ a .Using constraint programming, we computed the size of Strange n for n ≤ n = 5, the calculation took about 15 minutes. The size of Strange is expected to be huge. n | Strange n | Table 1.
Number of strange binary operations.
Proof of Theorem 3.10.
The map k can be uniquely reconstructed from ◮ k (whichin this proof we will write as ∗ for simplicity) and the λ b ’s, hence the injectivity. Itremains to show that the proposed operation ∗ is indeed strange.Due to the RE and the involutivity of r , one has the following equality of oper-ators on X : k rk r = rk rk = rk rk rr. Let us now apply these operators to ( a, λ − a ( b )). Since we are interested in whathappens to the first component of X only, we will write • in the second position.Applying k rk r , one obtains( a, λ − a ( b )) r ( b, • ) k ( b, • ) r ( b ∗ a, • ) k ( b ∗ a, • ) . Applying rk rk rr , one gets( a, λ − a ( b )) r ( b, • ) r ( a, • ) k ( a, • ) r ( a ∗ b, • ) rk (( a ∗ b ) ∗ a, • ) . Hence ( a ∗ b ) ∗ a = b ∗ a , as desired.To prove the last statement, assume that ◮ k is strange. The computationsabove show that the maps k rk r and rk rk = rk rk rr then have the same firstcomponent. But then, according to Theorem 3.1, these maps have to coincide. (cid:3) Example 3.11.
Let us resume Example 1.4. Recall that we have an involutivenon-degenerate solution with 5 reflections. The associated strange operations are b ◮ k a = ( k ( a ) if k = 123 ,
213 or 333 ,λ b k ( a ) if k = 113 or 223 . In other words, b ◮ k a is always k ( a ) except when k = 113 or 223, b = 3, and a = 3. Remark . The proof of Theorem 3.10 can be adapted to show that for anyinvertible LND solution, the reflection equation is equivalent to the following twoconditions for all a, b ∈ X : ( ( b ◮ k a ) ◮ k b = ( b ⊲ a ) ◮ k b, (( b ◮ k a ) ◮ k b ) ⊲ ( b ◮ k a ) = (( b ⊲ a ) ◮ k b ) ◮ k ( b ⊲ a ) . Here ⊲ is the left structure shelf operation for ( X, r ). Using notations b ⇀ a = b ◮ k ( b e ⊲ a ) = λ b kλ − b ( a ) ,b ⇁ a = b e ⊲ ( b ◮ k a ) = ρ − λ − a ( b ) kρ λ − a ( b ) ( a ) , where e ⊲ is the left-inverse of ⊲ , in the sense of b ⊲ ( b e ⊲ a ) = b e ⊲ ( b ⊲ a ) = a, the above system can be rewritten in a more compact form: ( ( b ⇀ a ) ◮ k b = a ◮ k b, ( a ◮ k b ) ⇁ a = b ⇀ a. For an involutive solution, the left structure shelf is trivial: a ⊲ b = b . Hence theoperations ⇁ , ⇀ , and ◮ k coincide, and both relations in the above system coincidewith the strange axiom. References [CCS20] Francesco Catino, Ilaria Colazzo, and Paola Stefanelli. The matched product ofset-theoretical solutions of the Yang–Baxter equation.
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LMNO, Universit´e de Caen–Normandie, BP 5186, 14032 Caen Cedex, France
E-mail address : [email protected] Departamento de Matem´atica – FCEN, Universidad de Buenos Aires, Pab. I – CiudadUniversitaria (1428), Buenos Aires, Argentina
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