Skew Braces as Remnants of Co-quasitriangular Hopf Algebras in SupLat
aa r X i v : . [ m a t h . QA ] S e p Skew Braces as Remnants of Co-quasitriangularHopf Algebras in
SupLat
Aryan Ghobadi
Queen Mary University of LondonSchool of Mathematics, Mile End RoadLondon E1 4NS, UKEmail: [email protected]
Abstract
Skew braces have recently attracted attention as a method to study set-theoretical solutions of the Yang-Baxter equation. Here, we present a new ap-proach to these solutions by studying Hopf algebras in the category,
SupLat ,of complete lattices and join-preserving morphisms. We connect the twomethods by showing that any Hopf algebra, H in SupLat , has a correspond-ing group, R ( H ) , which we call its remnant and a co-quasitriangular struc-ture on H induces a YBE solution on R ( H ) , which is compatible with itsgroup structure. Conversely, any group with a compatible YBE solution canbe realised in this way. Additionally, it is well-known that any such grouphas an induced secondary group structure, making it a skew left brace. Byrealising the group as the remnant of a co-quasitriangular Hopf algebra, H ,this secondary group structure appears as the projection of the transmutationof H . Finally, for any YBE solution, we obtain a FRT-type Hopf algebra in SupLat , whose remnant recovers the universal skew brace of the solution.
Mathematics Subject Classification : 16T25, 18M15, 16T99, 17B37
Keywords : braided monoidal category, complete lattice, Hopf algebra, skew braces, transmutation,Yang-Baxter equation
Originally appearing in statistical mechanics [24], the Yang-Baxter equation andits solutions play a fundamental role in the theory of quantum groups, braidedcategories and knot theory. One of the simplest realisations of this equation is oversets: we call a set X and a map r : X × X → X × X , a set-theoretical solution to the Yang-Baxter equation (YBE) if r r r = r r r (1)holds, where r ij : X × X × X → X × X × X are the applications of r to the i and j -th components of X i.e. r = id × r . In [4], Drinfeld proposed theclassification of such solutions as an open problem. Thenceforth, these objectshave garnered large interest due their interactions with combinatorics [5], ring the-ory [22, 21] and their applications to knot theory [3]. More recently, the work ofRump on involutive YBE solutions [19, 18] inspired Guarnieri and Vendramin todevelop of the theory of skew braces , which are sets with two compatible groupstructures [6]. In particular, any set-theoretical YBE solution has a correspondinguniversal skew brace, allowing us to classify set-theoretical YBE solutions, by firstclassifying such algebraic structures. However, when looking at linear YBE solu-tions on vector spaces, there is a well established correspondence between thesesolutions and (co-)quasitriangular Hopf algebras. (Co-)quasitriangular Hopf alge-bras, provide solutions of YBE via their representation theory, and conversly theFadeev-Reshitkhin-Takhtajan (FRT) construction produces such a Hopf algebra,from a given YBE solution. From the latter, we see that skew braces replace Hopfalgebras, in the set-theoretical world. Hence, it would be natural to ask whetherskew braces are related to Hopf algebras in a suitable category related to sets. Ifso, this relation should allow us to (a) apply the usual categorical Hopf algebratechniques to obtain new skew braces, (b) use the FRT construction and recoverthe universal skew brace and (c) explain the nature of the two products on a skewbrace and their interaction, which has been subject to several studies already. Inthis work, we show that the correct category to consider is that of complete latticesand join preserving morphisms, SupLat , and construct skew braces from coquai-triangular Hopf algebras in this category and vice-versa.The Hopf algebra point of view fails when studying set-theoretical YBE solu-tions because of two key reasons:(A) Hopf algebras in the category of sets and functions,
Set , are groups and it iseasy to check that any (co)quasitriangular structure on a group must be trivial.Hence, we can not obtain YBE solutions by looking at (co)modules over agroup in
Set .(B) The key ingredient to the FRT construction is Tannaka-Krein duality, whichrequires the underlying object of the YBE solution to be dualizable, while theonly dualizable object in
Set is the set of one element.The first naive solution is to look at the category of sets and relations,
Rel , whereevery set has itself as a dual, making the category rigid. However,
Rel is not INTRODUCTION 3cocomplete and the colimit needed for Tannaka-Krein reconstruction, 5, will notexist. The second naive solution is to move into the cocompletion of
Rel , namely [Rel op , Set] , via the Yoneda embedding. But this category is rather large and theHopf algebras constructed will not be very intuitive. Instead, we remedy theseissues by embedding the category of sets into the category of complete lattices andjoin-preserving morphisms,
SupLat , via the power-set functor: (Set , × , inc . strong monoidal / / (Rel , × , P ( − )strong monoidal / / (SupLat , ⊗ , P (1)) YBE solutions o o / / YBE solutions o o / / YBE solutionson rigid objects
In particular, all objects of the form P ( X ) , for a set X , are dualizable and converslyYBE solutions on rigid objects of SupLat , provide set-theoretical YBE solutions.The other major benefit of working in
SupLat , is that we can formulate a deeperconnection between co-quasitriangular Hopf algebras in this category and groupswith braiding operators, which are groups with a compatible YBE solutions ontheir underlying sets, see 8 and 9.Our results can be summarised as follows: Given a Hopf algebra structure ona complete lattice H in SupLat , we can form a new Hopf algebra by quotientingout the “kernel” of the counit, Lemma 3.5. The counit of this new Hopf algebrawill send all non-trivial elements to ∈ P (1) and in Lemma 3.6, we show that thiscondition is equivalent to the Hopf algebra being the “group algebra”, see Example3.2, of a group. Hence, this process provides a corresponding group for every Hopfalgebra in SupLat , which we call its remnant and denote by R ( H ) .It is well-known that the multiplication of a co-quasitriangular Hopf algebra isbraided-commutative with respect to a naturally induced braiding, 19, on the Hopfalgebra. Hence, we demonstrate that given a co-quasitriangular structure on H ,the induced braiding of H restricts to a braiding operator on its remnant, Theorem4.2. Additionally, any group with a braiding operator, possesses a secondary groupstructure on the same set, which makes it a skew brace. On the other hand, Majidhas shown that any co-quasitriangular Hopf algebra has an induced secondary mul-tiplication and an antipode which provide it with a braided Hopf algebra structure,called its transmutation , in its category of comodules [15]. A corollary of our workis that the secondary group structure on the remnant agrees with the projection ofthe transmuted multiplication of H , Theorem 4.3. Finally, in Section 5.2, we showthat any skew brace can be recovered as the remnant of a co-quasitriangular Hopfalebra in SupLat .We must point out that similar ideas were discussed in [11, 10], where Hopfalgebras in
Rel are shown to correspond to groups with unique factorisations, G = PRELIMINARIES 4 G + G − , and quasitriangular structures on them are fully described. Although thetheory is presented for finite dimensional positive Hopf algebras, the authors of [11]are aware that the proofs should work for any finite free B -module, where B is theBoolean algebra with two elements. Their work translates into the classification ofHopf algebra structures on free lattices i.e. lattices which are of the form P ( X ) fora set X , in SupLat and the finiteness condition can be completely avoided due tothe rigidity of P ( X ) . We review these results and briefly comment on their proofsin Section 4.1. In [9], the authors describe the properties of the universal groupof a set-theoretical solution by taking inspiration from [11, 10], but do not directlyconnect the works. By providing the correct categorical setting i.e. SupLat (whichgoes beyond the category of B -modules), we are able to present a single machinerywhich captures both constructions, namely by viewing them as remnants of co-quasitriangular Hopf alebras in SupLat .It has been observed that different set-theoretical YBE solutions can have iso-morphic universal skew braces. However, when applying the FRT construction inSection 5, we shall see that the Hopf algebra associated to the solution, remembersa large part of the original solution. It is only when we take the remnant of theHopf algebra, that much of this additional data is lost. Naturally, non-isomorphicco-quasitriangular Hopf alebras can have isomorphic remnants and should providestronger invariants of set-theoretical YBE solutions, while being more difficult towork with. The additional benefit of working in this setting is that one can utiliseclassical Hopf algebraic techniques such as (co-)double bosonasation to producenew examples of skew braces, however, this will be discussed in another place.In Appendix A, we discuss a natural bijection l : X → X , which is inducedwhen the set X is equipped with a YBE solution r . This bijection comes into playwhen we view ( X, r ) as a dualizable object in Rel and appears again in the ourreconstructed Hopf algebra for the solution.
Acknowledgements.
The author would like to thank Shahn Majid, for manyhelpful discussions on the topic
Throughout the article and particularly, in Sections 2.1 and 5, we assume thatthe reader is familiar with the notion of Hopf algebras in symmetric (braided)monoidal categories and refer to Chapter 9 of [17] for any details which we haveleft out. We will however present the structure of Hopf algebras in
SupLat andco-quasitriangular structures on them as definitions, in Section 3, and previousknowledge of Hopf algebras is not essential for the rest of the article.
Notation.
All Hopf algebras considered in this work, will have invertible an- PRELIMINARIES 5tipodes, and as noted later, all YBE solutions considered are assumed to be non-degenerate. We will freely use either m and . to denote the multiplication oper-ation, unless otherwise stated. We will denote the elements of the quotient of aset S , by a ∈ S/ ∼ , for a ∈ S , unless this is clear from context, in this case wewill simply write a . All monoidal categories are assumed to be strict, since theassociator and other structural morphisms will be trivial in the concerned exam-ple. If not stated otherwise a lower case letter such as x , will be an element ofthe set denoted by the upper case lettering, X . The number will both denote theunit element in our constructions and in the right context will denote the set withone element and P (1) = {∅ , } . For an arbitrary monoidal category ( C , ⊗ , ) ,we say an object X in C has a. right dual X ∨ , with duality morphisms ev X : X ⊗ X ∨ → and coev X : → X ∨ ⊗ X if (ev X ⊗ id X )(id X ⊗ coev X ) = id X and (ev X ⊗ id X ∨ )(id X ∨ ⊗ coev X ) = id X ∨ and X ∨ will always denote the rightdual of X . In this section, we review some necessary results about FRT reconstruction. Theseresults are also present in [17] and go back to [14]. However, our presentation ofthe results directly in terms of the coend is closer to [20]. In particular, we adaptthe notation used in [20].Let ˜ B be denote the rigid extension of the category of braids , which is the“smallest” rigid braided monoidal category. Explicitly, ˜ B is the monoidal categorygenerated by two objects, x and y , B and morphisms ev : x ⊗ y → , coev : → y ⊗ x and an invertible morphisms κ a , b : a ⊗ b → b ⊗ a for a , b ∈ { x , y } with the relevant relations [Definition 6.12 [20]], which make κ the braiding of thecategory and y the right dual of x .A pair ( X, r ) is called a braided object or YBE solution , [7], in a monoidalcategory ( C , ⊗ , ) , if X is an object of C and r : X ⊗ X → X ⊗ X an invertiblemorphism satisfying (id X ⊗ r )( r ⊗ id X )(id X ⊗ r ) = ( r ⊗ id X )(id X ⊗ r )( r ⊗ id X ) (2)If X has a right dual X ∨ , with duality morphisms ev X : X ⊗ X ∨ → and coev X : → X ∨ ⊗ X , we say the braided object ( X, r ) is dualizable if themorphisms r ♭ :=(id X ∨ ⊗ X ⊗ ev)(id X ∨ ⊗ r ⊗ id X ∨ )(coev ⊗ id X ⊗ X ∨ ) (3) ( r − ) ♭ :=(id X ∨ ⊗ X ⊗ ev)(id X ∨ ⊗ r − ⊗ id X ∨ )(coev ⊗ id X ⊗ X ∨ ) (4)are invertible. PRELIMINARIES 6Given a dualizable YBE solutions, one can define a functor ω : ˜ B → C by ω ( x ) = X , ω ( y ) = X ∨ and ω ( κ x , x ) = r, ω ( κ x , y ) = ( r − ) ♭ , ω ( κ y , x ) = ( r ♭ ) − , ω ( κ y , y ) = r ∨ ω (coev) = coev X , ω (ev) = ev X Given a YBE solution in a category, it is natural to ask which other braidedobjects are generated by this solution. For example, if ( X, r ) is a braided object,then tensor products of X also have induced braidings e.g. X ⊗ X with the inducedbraiding (id X ⊗ r ⊗ id X )( r ⊗ r )(id X ⊗ r ⊗ id X ) . The direct approach is to generatethe largest category of braided objects generated by ( X, r ) . Explicitly, one startswith ( X, r ) , and adds objects by performing tensor products and possible products,coproducts, etc. Finally, in the spirit of Lyubashenko’s work [12], one can hope torealise this category as comodules over a Hopf algebra.The second point of view, is that objects generated by ( X, r ) will be those,which “braid past, what ( X, r ) braids past”. To make this statement more explicit,we need to recall the definition of the dual of monoidal functors or weak central-izer . The dual of a strong monoidal functor U : D → C is defined as the categorywhose objects are pairs ( X, σ : X ⊗ U ⇒ U ⊗ X ) , where X is an object of C and σ a natural (monoidal) isomorphism satisfying σ = id X and (id M ⊗ τ N )( τ M ⊗ id N ) = ( U ( M, N ) − ⊗ id X ) τ M ⊗ N (id X ⊗ U ( M, N )) where M, N are objects of C . We denote the dual by W ( U ) and the lax left dual,where τ is not assumed to be isomorphisms by W l ( U ) . Note that the left (lax) duallifts the monoidal structure via ( X, τ ) ⊗ ( Y, ρ ) := ( X ⊗ Y, ( τ ⊗ id Y )(id X ⊗ ρ )) with ( C , id X ) as its unit and the strict monoidal functor U : W ( l ) ( U ) → C definedby U ( X, τ ) = X .With the notion of the dual in mind, given a braided object and its correspond-ing functor ω : ˜ B → C , all the braided objects generated by ( X, r ) must braidpast objects of W ( U ) . In particular if U : W ( U ) → C denotes the correspondingforgetful functor, then all the generated braided object will be objects of W l ( U ) .We now recall, the conditions under which this category can be recovered as thecomodule category of a co-quasitriangular Hopf algebra [14].Let D be a small monoidal category and ω : D → C be a strict monoidalfunctor. We consider the functor ω ⊗ ω ∨ : D × D op → C and denote its coend [Chapter IX.6 [13]], if it exists, by H ω := Z a ∈D ω ( a ) ⊗ ω ( a ) ∨ (5) PRELIMINARIES 7Recall that the coend is the colimit of the diagram consisting of objects ω ( a ) ⊗ ω ( b ) ∨ and parallel pairs ω ( f ) ⊗ id ω ( b ) ∨ , id ω ( a ) ⊗ ω ( f ) ∨ corresponding to objects a, b ∈ D and morphisms f : a → b in D , respectively. Theorem 2.1. If ( C , Ψ) is a symmetric monoidal category, and the mentioned co-end exists, it comes equipped with the structure of a bialgebra, such that W l ( U ) ismonoidal equivalent to the the category of left H ω -comodules, H ω C . Additionally,if D is rigid, then H ω admits a bijective antipode, making it a Hopf algebra objectin C . If D is braided, then H ω has an induced co-quasitriangular structure. Here we only recall the induced co-quasitriangular Hopf alebra structure on H ω . The proof of this result can be found in Chapter 9 of [17], where the coendis is described in terms of natural transformations between certain functors or inTheorem 4.3 of [20], which uses the language we will present it as.Because of the simplicity of our examples, we will be assuming that the functor ω is strict monoidal and additionally ω ( x ) ∨ = ω ( x ∨ ) for all x ∈ D . Let µ x : ω ( x ) ⊗ ω ( x ) ∨ → H ω denote the unique natural morphisms, making H ω the colimitof the diagram. The Hopf algebra structure on H ω consists of ( m, η, ∆ , ǫ, S ) whichare the unique morphisms satisfying: m : H ω ⊗ H ω → H ω ; m ( µ x ⊗ µ y ) = µ x ⊗ y (cid:0) id ω ( x ) ⊗ Ψ ω ( x ) ∨ ,ω ( y ) ⊗ id ω ( y ) ∨ (cid:1) η : → H ω ; η = µ ∆ : H ω → H ω ⊗ H ω ; ∆ µ x = ( µ x ⊗ µ x ) (cid:0) id ω ( x ) ⊗ coev ω ( x ) ⊗ id ω ( x ) ∨ (cid:1) ǫ : H ω → ; ǫµ x = ev ω ( x ) S : H ω → H ω ; Sµ x = µ x ∨ (ev ω ( x ) ⊗ id ω ( x ∨ ) ⊗ ω ( x ∨∨ ) )(Ψ ω ( x ∨ ) ⊗ ω ( x ∨∨ ) ,ω ( x ∨ ) )(id ω ( x ) ⊗ ω ( x ) ∨ ⊗ ω (ev x ) ∨ ) where x, y ∈ D . As we will see in our example, Ψ and ev , coev are rather trivialand the expressions will be much simpler to deal with. The key ingredient whichwe need, lies in the co-quasitriangular structure R : H ω ⊗ H ω → induced on H ω , when D is a braided category. If ψ denotes the braiding of D , then R is theunique morphism satisfying R ( µ x ⊗ µ y ) = ev ω ( y ⊗ x ) ( ω ( ψ x,y ) ⊗ id ω ( x ) ∨ ⊗ ω ( y ) ∨ )(id ω ( x ) Ψ ω ( x ) ∨ ,ω ( y ) ⊗ id ω ( y ) ∨ ) for any pair of objects x, y ∈ D . In this section, we review the theory of Skew braces, with [22] as our main refer-ence. PRELIMINARIES 8We have already defined what a set-theoretical YBE solution ( X, r ) is in theintroduction. We will use the notation r ( x, y ) = ( σ x ( y ) , γ y ( x )) , for maps σ x , γ y : X → X corresponding to elements x, y ∈ X . The YBE solution is said to be non-degenerate if σ x , γ y are bijections for all x, y ∈ X . For such a solution,where r is bijective, we adapt the notation r − ( x, y ) = ( τ x ( y ) , ρ y ( x )) , for maps τ x , ρ y : X → X corresponding to elements x, y ∈ X . The solution is called involutive if σ x = τ x and γ x = ρ x for all x ∈ X . Notation.
We will only consider non-degenerate solutions and from here for-ward a set-theoretical YBE solution will refer to a non-degenerate one.We now recall the definition of the a braiding operator on a group and thedefinition of the universal group of a set-theoretical YBE solution from [9].A braiding operator on a group ( G, m, e ) is a map r : G × G → G × G satisfying mr ( a, b ) = a.b (6) r ( e, g ) = ( g, e ) , r ( g, e ) = ( e, g ) (7) r ( g.h, f ) = (id G × m )( r × id G )( g, r ( h, f )) (8) r ( g, h.f ) = ( m × id G )(id G × r )( r ( g, h ) , f ) (9)for any g, h, f ∈ G . It follows from these axioms, that r is invertible and satisfiesthe YBE equation [Corollary 1 of [9]] i.e. ( G, r ) is a braided object in Set . A pair ( G, r ) is sometimes referred to as a braided group . We avoid this term, since inour main reference [17], the term “braided group” is used to discuss braided Hopfalgebras in braided monoidal categories.Given a set-theoretical YBE solution ( X, r ) , we consider the group G ( X, r ) = F g ( X ) / h x.y = σ x ( y ) .γ y ( x ) | x, y ∈ X i , where F g ( X ) denotes the free groupgenerated by the set X . Observe that the braiding on X , extends to a braidingoperator r on G ( X, r ) , defined by r ( x, y ) = (cid:0) σ x ( y ) , γ y ( x ) (cid:1) . Additionally, thenatural map i : X → G ( X, r ) defined by x x commutes with the braidingoperators on both sets. One must keep in mind that i is not necessarily injective,but satisfies a universal property with respect to groups with braiding operators.Explicitly, if ( H, s ) is a group with a braiding operator and f : X → H , a mapwhich commutes with the respective braidings, then f must factorise through i .Now we review the theory of Skew braces with reference to [22], although weadapt the notation of [1, 2], which is compatible with [9].A skew left brace consists of a set B with two group structures ( B, . ) and ( B, ⋆ ) , satisfying a. ( b ⋆ c ) = ( a.b ) ⋆ a ⋆ ⋆ ( a.c ) (10)for all a, b, c ∈ B , where we denote the multiplicative inverse of a with respect to . and ⋆ by a − and a ⋆ , respectively. A skew right brace can be defined accordingly PRELIMINARIES 9[Definition 2.1 [1]]. However, as we will see in Remark 4.4, the theory of rightskew braces is symmetric to that of skew left braces. Hence, in this paper we willonly work with skew left braces and refer to them as skew braces .The reader should be careful when referring to other works related to skewbraces, since many other sources use the notation ◦ and . instead of . and ⋆ , respec-tively.Skew braces are in fact equivalent to groups with braiding operators. This isin fact proved in Theorem 2 of [9], but is stated in the language of skew braces inSection 3 of [22]. Theorem 2.2.
Given a group ( G, . ) , the following additional structures on G areequivalent:(I) A second group structure ( G, ⋆ ) , which makes G a skew brace.(II) A braiding operator r : G × G → G × G on the group G . We briefly recall this correspondence from the mentioned sources. Given askew brace structure as in part (I), the induced braiding operator on G is definedby ( a, b ) (cid:0) a ⋆ ⋆ ( a.b ) , ( a ⋆ ⋆ ( a.b )) − .a.b (cid:1) (11)for a, b ∈ G . Conversely, given a braiding operator r with the usual notation r ( x, y ) = ( σ x ( y ) , γ y ( x )) , the operation x ⋆ y := x.σ − x ( y ) = x.σ x − ( y ) defines acompatible group structure on G . In fact there’s a third equivalent structure namelythe existence of a bijective 1-cocycle, which we will not discuss.For a skew brace ( B, ., ⋆ ) , by axiom 8, the map λ : ( B, . ) → Aut(
B, ⋆ ) defined which sends an element a ∈ B to the map λ a ( b ) = a ⋆ ⋆ ( a.b ) , is a groupmorphisms. Hence, to any skew brace ( B, ., ⋆ ) , one can associate a group structureon the set B × B , namely the cross product ( B, ⋆ ) ⋊ ( B, . ) , with the multiplication ( a, b ) . ( c, d ) = ( a ⋆ λ b ( c ) , b.d ) (12)for a, b, c, d ∈ B . This group is called the crossed group of ( B, ., ⋆ ) [Definition1.10 [22]]. SupLat
We briefly review the monoidal closed structure of
SupLat and its colimits, fromChapter 1 of [8]. The reader can refer to this source for additional details on thestructure of
SupLat .The objects of
SupLat are partially ordered sets ( L , ≤ ) , where any subset S ⊆L , has a least upper bound i.e. an element denote by ∨ S such that if l ∈ L satisfies PRELIMINARIES 10 s ≤ l for all s ∈ S , then ∨ S ≤ l holds. In this case, ∨ S is called the supremum or join of S . If S = { a i | i ∈ I } for some index set I , we use notation ∨ i ∈ I a i insteadof ∨ S . The morphisms of SupLat are join-preserving maps between sets.Notice that a partially ordered set in
SupLat also admits all infima, i.e thegreatest lower bound or meet of a subset S ∈ L will be ∨{ a | a ≤ s , ∀ s ∈ S } and is denoted by ∧ S . Hence, all objects of SupLat are complete lattices , butjoin-preserving morphisms between them, do not necessarily preserve infima. Wedenote the smallest element of every complete lattice by ∅ . Example 2.3.
The simplest example of a complete lattice is the power-set of a set X , denoted by P ( X ) , with ∨ and ∧ being the union and intersection of subsets,respectively. This example is often called the free lattice on set X . Alternatively,we can consider the set of positive integers, div( z ) , which divide a positive integer z . The set div( z ) possesses a partial ordering by division, and ∨ and ∧ are givenby the lowest common multiple and the greatest common divisor, respectively. The category
SupLat , is complete and cocomplete. Limits in
SupLat are easyto construct and follow directly from
Set . The equaliser of to morphisms betweencomplete lattices is exactly the equalizer of the two underlying maps between thetwo underlying sets and the product of complete lattices is the product of the un-derlying sets with coordinate-wise order.The coproduct of complete lattices L i for an index set I with suitable cardi-nality, which will always be the case in our work, will again be the product of thesets L i , with coordinate-wise order Q i ∈ I L i . The lattice Q i ∈ I L i is viewed as acoproduct via the inclusion morphisms L i → Q j ∈ I L j , which for a fixed i ∈ I send an element l ∈ L i to ( a j ) j ∈ I , where a i = l and a j = ∅ for j = i .The coequalizer of a parallel pair f, g : L ⇒ N between lattices is moredifficult to describe. First, we recall the description provided in Proposition 3 of[8]: Let K be the subset of elements in k ∈ N , which satisfy the following prop-erty: ∀ l ∈ L , either f ( l ) ∨ g ( l ) ≤ k or neither f ( l ) ≤ k nor g ( k ) ≤ k hold.The partial order of N restricts to K and the morphism c : N → K , defined by n
7→ ∧{ k ∈ K | n ≤ k in N } , makes K , the coequalizer of the parallel pair f, g .If we compose the morphism c with the inclusion of K into N , we obtain a map , no longer a join-preserving morphism of lattices, c : N → N . It is easy tosee that c is a closure operator i.e satisfies c = c , a ≤ c ( a ) and c ( a ) ≤ c ( b ) for a, b ∈ N , where a ≤ b . Hence, we define K to be the quotient of set N by theequivalence relation n ∼ c ( n ) for all n ∈ N . In other words, K is the coequalizerof the parallel pair id N , c in Set . It follows that K has a partial order defined by a ≤ b iff c ( a ) ≤ c ( b ) and thereby the natural map () : N → K defined by n n is order-preserving. Additionally, we claim that ∨ i ∈ I n i = ∨ i ∈ I n i , making K a PRELIMINARIES 11complete lattice and () a join-preserving morphism. Notice that c ( ∨ i ∈ I n i ) ≥ c ( n i ) holds and if n ≥ n i for all i ∈ I , then c ( n ) ≥ c ( n i ) ≥ n i ∀ i ∈ I ⇒ c ( n ) ≥ ∨ i ∈ I n i ⇒ c ( n ) = c ( n ) ≥ c ( ∨ i ∈ I n i ) which proves our claim. It should be clear to see that () factorises through anisomorphism between K and K . Furthermore, observe that in Set , the map () :
N → K admits a section defined by b c ( b ) . This map is a join-preservingmorphism iff c is join preserving.Now, let us recall the monoidal structure of SupLat . For lattices M , N and L ,a map f : M × N → L is called a bimorphism if it preserves suprema component-wise i.e. f ( ∨ i ∈ I m i , n ) = ∨ i ∈ I f ( m i , n ) and f ( m, ∨ i ∈ I n i ) = ∨ i ∈ I ( m, n i ) . Thereexists a lattice M ⊗ N with the universal property that any bimorphism f , asabove, factorises through a morphism f : M ⊗ N → L . Explicitly,
M ⊗ N can bedescribed as the quotient of the lattice P ( M × N ) by relations { ( ∨ i ∈ I m i , n ) } = ∪ i ∈ I { ( m i , n ) } and { ( m, ∨ i ∈ I n i ) } = ∪ i ∈ I { ( m, n i ) } . Although, we view M ⊗ N as a quotient of P ( M × N ) , we will denote its elements by { ( m, n ) } instead of { ( m, n ) } . Hence, M ⊗ N is “spanned”, via ∪ , by elements of the form { ( m, n ) } and the relations induce component-wise ordering on these elements i.e. if m ≤ m ′ , then { ( m, n ) } ≤ { ( m ′ , n ) } since m ∨ m ′ = m ′ .In fact ⊗ , defines a bifunctor and provides a symmetric monoidal structure on SupLat . The unit of the monoidal structure is given by P (1) , where denotes theset with one element. More generally, for a pair of sets X, Y , P ( X ) ⊗L ∼ = Q x ∈ X L and P ( X ) ⊗ P ( X ) ∼ = P ( X × Y ) [Proposition 2 [8]]. Hence, the natural power-setfunctor P : Set → SupLat is strong monoidal. The symmetric structure of coursefollows from the symmetry of × and viewing M ⊗ N as a quotient of P ( M × N ) .There exists a natural partial ordering on the hom-sets between complete lat-tices defined by f ≤ g iff f ( m ) ≤ g ( m ) , ∀ m ∈ M , for f, g ∈ Hom( M , N ) .By taking point-wise suprema, one can conclude that Hom( M , N ) is also a com-plete lattice. Furthermore, one can prove the familiar hom-tensor adjunction, whichmakes SupLat a closed monoidal category. Next we characterise the rigid or du-alizable objects in SupLat . Definition 2.4.
We say a lattice L has a basis if there exists a set of elements { l i } i ∈ I , such that(A) For any element l ∈ L , there exists a (not necessarily unique) subset J l ⊂ I ,such that l = ∨ i ∈ J l l i (B) For any pair i, j ∈ I , l j ≤ l i iff i = j hold. PRELIMINARIES 12Observe that condition (B), implies that there are no elements under the basiselements i.e. if ∅ 6 = l ≤ l i , then l = l i . Any free lattice P ( X ) admits a basis,consisting of singleton subsets { x } for elements x ∈ X . Additionally, div(6) admits a basis { , } , while div(4) does not. More generally, div( z ) admits a basisiff the power of all primes present in the prime factorisation of z , is one.It’s easy to see that any complete lattice L with basis { l i } i ∈ I , is a rigid objectin SupLat , with itself as its dual. Since L has a basis, then L ⊗ L also admitsa basis with elements { ( l i , l j ) } for pairs i, j ∈ I . Hence, it is straightforward tocheck that morphisms coev : P (1) → M ⊗ L defined by coev(1) = ∪ i ∈ I { ( l i , l i ) } and ev : L ⊗ L → P (1) defined by ev { ( l i , l j ) } = 1 iff i = j which extends to L ⊗ L , by a choice of J l for every l , are well-defined morphisms and satisfy theduality axioms. Notice that although we do not assume that for every l , there be aunique subset J l ⊂ I , we require a choice of such a subset to enable us to definethe evaluation morphism, ev . Lemma 2.5. If L is a rigid object in SupLat , then L has a basis.Proof. Assume M is the right dual of L with duality morphisms ev : L ⊗ M →P (1) and coev : P (1) → M ⊗ L . Let coev(1) = { ( m i , l i ) | i ∈ I } . By theduality axioms e.g. (ev ⊗ id L )(id L ⊗ coev) = id L , we can conclude that for any l ∈ L , there exists a subset J l ⊆ I such that ∨ i ∈ J l l i = l and ev { ( l, m i ) } = 1 iff i ∈ J l . In particular, for a fixed i ∈ I , if j ∈ J l i , then l j ≤ l i and ( l i , m i ) ≥ ( l j , m i ) . Hence, because ev is join-preserving, ev( l i , m i ) = ev( l j , m i ) = 1 forany i ∈ I . Additionally, (id M ⊗ ev)(coev ⊗ id M ) = id M , shows that m i ≤ m j if ev { ( l j , m i ) } = 1 .By the symmetric structure of SupLat , we can deduce that morphisms ev ′ : M⊗L → P (1) and coev ′ : P (1) → L⊗M , defined by coev ′ (1) = { ( l i , m i ) } and ev ′ { ( m, l ) } = ev { ( l, m ) } make M the left dual of L . Hence, if ev ′ { ( m i , l j ) } =1 , the same arguments as above show that m j ≤ m i and l i ≤ l j . Therefore, ev( l j , m i ) = 1 iff i = j .Observe that if for i, j ∈ I , if l j ≤ l i , then { ( l j , m j ) } ≤ { ( l i , m j ) } and thesame line of arguments as above follow demonstrating that i = j . Hence, elements l i form a basis for L .Let us briefly reflect on the fact that YBE solutions on rigid objects of SupLat correspond to set-theoretical YBE solutions. Of course the strong monoidal functor P : Set → SupLat lifts the set-theoretical YBE solutions to YBE solutions on freelattices. Furthermore, an invertible morphism between lattices with bases mustsend basis elements to basis elements. Hence, a YBE solution on a lattice L with abasis is fully determined by a YBE solution on its basis. HOPFALGEBRASIN SUPLAT SupLat
In this section, we introduce the remnant group of a Hopf algebra in
SupLat andshow that any co-quasitriangular Hopf algebra gives rise to a skew brace.As pointed out earlier, for a lattice L , the lattice L ⊗ L can be viewed as aquotient of P ( L × L ) . Hence, any element of can be written as ∪ i ∈ I { ( l i , l ′ i ) } forsome elements l i , l ′ i ∈ L . Therefore, we will use the shorter notation ∨ i ∈ I ( l i , l ′ i ) ,instead of ∪ i ∈ I { ( l i , l ′ i ) } , unless otherwise stated. The category
SupLat has a symmetric monoidal structure via ⊗ , and thereby onecan formulate the notion of a Hopf algebra in SupLat .Let H be a complete lattice in SupLat . An algebra structure on H , consists of ajoin-preserving morphism m : H⊗H → H which is associative i.e. m ( m ⊗ id H ) = m (id H ⊗ m ) and an element denoted by ∈ H , such that m ((1 , h )) = h = m (( h, , for all elements h ∈ H . We will denote the multiplication of pairs ( h, h ′ ) ∈ H ⊗ H , by m ( h, h ′ ) or h.h ′ instead of m (( h, h ′ )) . Remark 3.1.
We should point out that although the multiplication is determinedby its value on pairs of the form ( h, h ′ ) ∈ H ⊗ H , it does not correspond to amonoid structure on the underlying set H , but a stronger structure depending onthe partial ordering, since ∨ i ∈ I h.h i = h. ( ∨ i ∈ I h i ) and ∨ i ∈ I h i .h = ( ∨ i ∈ I h i ) .h must hold. A coalgebra structure on H consists of join-preserving morphisms ∆ : H →H ⊗ H , ǫ : H → P (1) satisfying (∆ ⊗ id H )∆ = (id H ⊗ ∆)∆ , ( ǫ ⊗ id H )∆ = id H = (id H ⊗ ǫ )∆ For h ∈ H , ∆( h ) consists of the image of a subset of H × H in H ⊗ H . Hence,we adapt
Sweedler’s notation from ordinary Hopf algebras and write ∆( h ) = { ( h (1) , h (2) ) } , where ( h (1) , h (2) ) represent all the pairs appearing in ∆( h ) and infact { ( h (1) , h (2) ) } = ∨{ ( a, b ) | ( a, b ) ∈ ∆( h ) } . Consequently, we can write h = ∨{ h (1) | ǫ ( h (2) ) = 1 } = ∨{ h (2) | ǫ ( h (1) ) = 1 } A bialgebra structure on H consists of an algebra and coalgebra structures as abovewith additional compatibility conditions ∆(1) = (1 , , ǫ (1) = 1 , ǫ ( m ( h, h ′ )) = 1 iff ǫ ( h ) = ǫ ( h ′ ) = 1 and { ( h (1) .h ′ (1) , h (2) .h ′ (2) ) } = { (( h.h ′ ) (1) , ( h.h ′ ) (2) ) } (13) HOPFALGEBRASIN SUPLAT H a Hopf algebra , if there exists an invertible join-preserving morphism S : H → H , such that ∨ h (1) .S ( h (2) ) = ∨ S ( h (1) ) .h (2) is equal to iff ǫ ( h ) = 1 and equal to ∅ otherwise. Example 3.2.
Since the power-set functor P ( − ) : Set → SupLat is strongmonoidal, the image of any Hopf algebra in
Set i.e. P ( G ) for any group G , willhave an induced Hopf algebra structure. Here, we refer to P ( G ) as the group al-gebra of G . The induced Hopf algebra structure is defined by { g } . { h } = { gh } , S ( { g } ) = { g − } , { e } and ǫ ( { g } ) = 1 , where g, h ∈ G and e ∈ G is theidentity element. Since the singleton sets { g } , where g ∈ G , form a basis of P ( G ) ,it is sufficient to define the structure on the basis elements. Example 3.3.
As demonstrated in Lemma 2.5, free lattices are rigid objects andin fact self-dual. Hence, we can provide a dual Hopf algebra structure on P ( G ) for a group G . We will refer to this structure as the function algebra , which as forordinary Hopf algebras, is defined by { g } . { h } = { g } iff g = h , S ( { g } ) = { g − } , ∆( { g } ) = { ( h, l ) | h.l = g } , { g | g ∈ G } and ǫ ( { g } ) = 1 iff g = 1 . Example 3.4.
Any complete lattice L which satisfies the following distributivityconditions ∨ i ∈ I ( a i ∧ b ) = ( ∨ i ∈ I a i ) ∧ b and ∨ i ∈ I ( b ∧ a i ) = b ∧ ( ∨ i ∈ I a i ) , for a i , b ∈L , obtains a natural Hopf algebra structure defined by ∨L , m ( a, b ) = a ∧ b , S = id L , ǫ ( a ) = 1 iff a = ∨L and ∆( a ) = { ( a, } ∨ { (1 , a ) } . Let H be a Hopf algebra in SupLat . We first observe that ǫ − ( ∅ ) is a completesublattice of L . Hence, we can consider the parallel pair ∅ , inc . : ǫ − ( ∅ ) ⇒ H ,where ∅ denotes the morphisms which sends all elements to ∅ ∈ H and inc . denotesthe natural inclusion morphism. We denote the coequalizer of this pair by π : H →Q . Recall from the description of colimits in Section 2.3, that as a set Q is thequotient of H by relation h = ∨ (cid:0) { h } ∪ ǫ − ( ∅ ) (cid:1) , and the morphism π is defined by π ( h ) = h . If we denote D := ∨ ǫ − ( ∅ ) , then ∨ (cid:0) { h } ∪ ǫ − ( ∅ ) (cid:1) = h ∨ D . Since theclosure operator on H , defined by h h ∨ D is join-preserving, it follows that π admits a well-defined join-preserving section ι : Q → H , defined by ι ( h ) = h ∨ D . Lemma 3.5.
The quotient lattice, Q of a Hopf algebra H , as described above hasan induced Hopf algebra structure, such that π becomes a Hopf algebra morphismi.e. π commutes with all structural morphisms.Proof. We define the Hopf algebra structure on Q denoted by ( m ′ , , ∆ ′ , ǫ ′ , S ′ ) via m ′ = πm ( ι ⊗ ι ) , ∆ ′ = ( π ⊗ π )∆ ι, ǫ ′ = ǫι, S ′ = πSι Since ιπ = id Q , then we only need to show that the defined morphisms induce aHopf algebra structure. Checking that m ′ ( h, g ) = πm ( h ∨ D, g ∨ D ) is associative, HOPFALGEBRASIN SUPLAT π ( h. ( g ∨ D )) = π ( h.g ) = π (( h ∨ D ) .g ) , for any h, g ∈ H .This follows from the fact that h.D ∈ ǫ − ( ∅ ) and π ( h. ( g ∨ D )) = π ( h.g ) ∨ π ( h.D ) .Consequently, π ((1 ∨ D ) . ( h ∨ D )) = h = π (( h ∨ D ) . (1 ∨ D )) and m ′ , form anassociative algebra structure on Q .Similarly, showing that ∆( h ) = { ( π ( h (1) ) , π ( h (2) )) } is coassociative dependson showing that ( π ⊗ π )∆( h ∨ D ) = ( π ⊗ π )∆( h ) . This fact follows from theobservation that ∆( h ∨ D ) = { ( h (1) , h (2) ) } ∨ { ( D (1) , D (2) ) } and { ( D (1) , D (2) ) } ∈ Im(
H ⊗ ǫ − ( ∅ ) ∪ ǫ − ( ∅ ) ⊗ H ) ⊂ H ⊗ H , since ǫ = ( ǫ ⊗ ǫ )∆ . Note that ǫ ′ ( h ) = ǫ ( h ) and it follows immediately that ∆ ′ ( h ) = { ( h (1) , h (2) ) } defines a coalgebrastructure on Q .The bialgebra axioms follow directly from our observations in the paragraphabove, that π ( h. ( g ∨ D )) = π ( h.g ) = π (( h ∨ D ) .g ) and ( π ⊗ π )∆( h ∨ D ) =( π ⊗ π )∆( h ) and is left to the reader. Note that S ′ ( h ) = S ( h ) , since ǫS = ǫ andthereby S ( D ) ∈ ǫ − ( ∅ ) . The Hopf algebra axioms then follow directly from thisobservation.As we noted in the proof of Lemma 3.5, ǫ ′ ( h ) = ǫ ( h ) and thereby ǫ ′ ( h ) = ∅ iff h = ∅ . We can further characterise such Hopf algebras. In what follows we willcall an element q ∈ Q group-like , if ∆( q ) = { ( q, q ) } . Lemma 3.6.
Given a Hopf algebra structure on a complete lattice Q , TFAE(I) For any ∅ 6 = q ∈ Q , ǫ ( q ) = 1 .(II) There exists a set X , such that Q ∼ = P ( X ) as a lattice and a group structureon X inducing a group algebra structure on Q .Proof. ( I ) ⇒ ( II ) We will prove this statement in several steps.Assume that for any ∅ 6 = q ∈ Q , ǫ ( q ) = 1 . First, note that for any ∅ 6 = q ∈ Q , the coalgebra structure implies that ∨{ q (1) } = q = ∨{ q (2) } , so that forany pair ( q (1) , q (2) ) ∈ ∆( q ) , we have q (1) , q (2) ≤ q . Secondly, we observe that ∨{ q (1) .S ( q (2) ) } = 1 and consequently ≤ q.S ( q ) . Claim 1. If ∅ 6 = q ≤ , then q = 1 .Since ≤ q.S ( q ) ≤ q.S (1) ≤ .S (1) = 1 and S (1) = 1 , then q = 1 . Claim 2.
For any ∅ 6 = q ∈ Q , there exists at least one pair pair ( a, b ) ∈{ ( q (1) , q (2) ) } , such that a.S ( b ) = 1 = S ( b ) .a .Since ∨ q (1) .S ( q (2) ) = 1 , then by Claim 1 it follows immediately that thereexists a pair ( a, b ) ∈ { ( q (1) , q (2) ) } such that a.S ( b ) = 1 . Consequently, by as-sociativity we conclude that p.a = ∅ iff p = ∅ . Since ( b, a ) ∈ { ( q (2) , q (1) ) } ,thereby S − ( b ) .a ≤ . Hence, S − ( b ) .a = 1 and by associativity it follows that S − ( b ) = S ( b ) . A symmetric argument also shows that S ( a ) .b = 1 = b.S ( a ) and S − ( a ) = S ( a ) . CO-QUASITRIANGULARHOPFALGEBRASANDSKEWBRACES 16 Claim 3. If a.b = 1 = b.a , then { ( a (2) , a (1) ) } = { ( a, a ) } .The bialgebra axiom, 13, implies that { ( a (1) .b (1) , a (2) .b (2) ) } = { (1 , } . Hence,for any ( a ′ , b ′ ) ∈ { ( a (1) , b (1) ) } , we have a ′ .b ′ = 1 or ∅ . On the other hand, Claim2 shows that there exists a pair ( a ′ , a ′′ ) ∈ { ( a (1) , a (2) ) } such that S ( a ′′ ) .a ′ = 1 = S ( a ′ ) .a ′′ . Therefore, for any ( b ′ , b ′′ ) ∈ { ( b (1) , b (2) ) } , ( a ′ .b ′ , a ′′ .b ′′ ) = (1 , and { ( b (1) , b (2) ) } = { ( S ( a ′′ ) , S ( a ′ )) } . Since b = ∨{ b (1) } = ∨{ b (2) } , then b = S ( a ′ ) and a ′ = a ′′ . By a symmetric argument, we see that ∆( a ) = { ( a ′ , a ′ ) } and a ′ = a = a ′′ .So far we have shown that for every element ∅ 6 = q ∈ Q , there exists a group-like element a , such that ( a, a ) ∈ { ( q (1) , q (2) ) } . Additionally, the multiplication on Q ⊗ Q restricts to the set of group-like elements,
Grp( Q ) , in Q . Claim 4. If ∅ 6 = a ∈ Grp( Q ) is group-like and ∅ 6 = q ≤ a , then q = a .The same argument as Claim 1, can be used ≤ q.S ( q ) ≤ q.S ( a ) ≤ a.S ( a ) =1 . By symmetric arguments, we conclude that q is the inverse of S ( a ) , and byassociativity a = q . Claim 5. If ∅ 6 = q, p ∈ Q , and q.p = ∅ .By Claim 2, we see that there exist group-like elements a, b ∈ Q such that a ≤ q and b ≤ p . Hence ∅ (cid:12) a.b ≤ q.p .Since for an arbitrary ∅ 6 = q ∈ Q , and ( a, b ) ∈ { ( q (1) , q (2) ) } a.S ( b ) = 1 or ∅ .By Claims 5 and 2, we conclude that ∆( q ) = ∨ i ∈ I { ( a i , a i ) } for a set of group-likeelements a i such that q = ∨ i ∈ I a i . Furthermore, the set of group-like elementsforms a basis for Q . What remains to show is that the factorisation q = ∨ i ∈ I a i is unique. If b is a group-like element such that b ≤ q , then ∆( b ) = ( b, b ) ≤∨ i ∈ I { ( a i , a i ) } . Since the group-like elements are minimal, by the lattice structureof Q ⊗ Q = P ( Q × Q ) / ∼ , we can conclude that the statement only holds if thereexists an i ∈ I such that b = a i . Hence, Q ∼ = P (Grp( Q )) . ( II ) ⇒ ( I ) is true by the definition of the Hopf algebra structure on the groupalgebra P ( X ) . Corollary 3.7.
Any Hopf algebra H in SupLat , we has a corresponding group R ( H ) such that the quotient Hopf algebra H /ǫ − ( ∅ ) is isomorphic to the inducedgroup algebra of R ( H ) . We call R ( H ) the remnant of H . In this section, we show that the remnant of a co-quasitriangular Hopf algebra hasan induced braiding operator. Consequently, we show that the secondary product ⋆ on a such a group agrees with the restriction of the transmuted product of the Hopfalgebra to its remnant. CO-QUASITRIANGULARHOPFALGEBRASANDSKEWBRACES 17Before we prove our results, we review the definitions of co-quasitriangularstructures on Hopf algebras in SupLat and present Majid’s transmutation theory inthis case. Recall from the last section, that we are simply translating result on Hopfalgebras in symmetric categories to the setting of
SupLat . But in doing so, ourequations will look essentially the same as the ordinary Hopf algebra case, whereinstead of a implicit sum a (1) ⊗ a (2) ∈ H ⊗ H , we have an implicit supremum { ( a (1) , a (2) ) } ∈ H ⊗ H . In particular, for an arbitrary morphism f : H ⊗ H → L ,by f ( a (1) , a (2) ) , we always mean ∨{ f ( a (1) , a (2) ) | ( a (1) , a (2) ) ∈ ∆( a ) } .We call a join-preserving morphism R : H ⊗ H → P (1) a co-quasitriangular structure on a Hopf algebra H , if the below conditions hold:(A) R is convolution-invertible i.e., there exists a join-preserving morphism R − : H ⊗ H → P (1) satisfying R − ( a (1) , b (1) ) . R ( a (2) , b (2) ) = ǫ ( a ) .ǫ ( b ) = R ( a (1) , b (1) ) . R − ( a (2) , b (2) ) (14)for any pair of elements a, b ∈ H , where . here signifies the natural monoidstructure on P (1) , defined by . . , . and . .(B) For any a, b, c ∈ H , the following equalities hold: R ( a.b, c ) = R ( b, c (1) ) . R ( a, c (2) ) (15) R ( a, b.c ) = R ( a (1) , b ) . R ( a (2) , c ) (16) R ( b (1) , a (1) ) a (2) .b (2) = b (1) .a (1) R ( b (2) , a (2) ) (17)where in 17, we are using the natural action of P (1) on all lattices P (1) ⊗ L ∼ = L , defined by .l = 0 and .l = l .For any Hopf algebra, we can consider its category of left comodules , denoteby H SupLat , which has pairs ( L , δ : L → H ⊗ L ) , satisfying ( ǫ ⊗ id L ) δ =id L and (∆ ⊗ id L ) δ = (id H ⊗ δ ) δ , as objects and join-preserving morphismswhich commute with the coactions , δ , appropriately as its morphisms. If we utilisenotation δ ( l ) = { ( l (0) , l (1) ) } , for elements l ∈ H , so that l (0) ∈ L and l (1) ∈ L ,then the monoidal structure of SupLat , lifts to the category of H -comodules andthe coaction on L ⊗ M for arbitrary comodules L and M is defined by { (( l, m ) (0) , ( l, m ) (1) ) } := { ( l .m (0) , ( l (1) , m (1) )) } Consequently, when given a co-quasitriangular structure on H , H SupLat is pro-vided with an induce braiding ψ , defined by ψ L , M ( l, m ) = R ( l (0) , m (0) ) . ( m (1) , l (1) ) (18)for arbitrary comodules L and M . CO-QUASITRIANGULARHOPFALGEBRASANDSKEWBRACES 18 Remark 4.1.
The equations, 15, 16, 17, described in the above definition are mir-rors to the usual definition of co-quasitriangular Hopf algebras, as presented inChapter 2 of [17]. Notice, that this is because the definition we are presenting isused to provide a braiding on the category of left H -comodules, while the descrip-tion in [17] is used for right comodules. Nevertheless, they are equivalent, since R − in our case will satisfy the axioms presented in [17]. It is known that any co-quasitriangular Hopf algebra has a well-defined braid-ing operator
Γ :
H ⊗ H → H ⊗ H , defined by ( a, b )
7→ R ( a (1) , b (1) ) . ( b (2) , a (2) ) . R − ( a (3) , b (3) ) (19)for elements a, b ∈ H , which make ( H , Γ) a braided object in SupLat . Addition-ally, it follows from 17, that H is braided-commutative with respect to this braidingi.e. m Γ = m holds. It follows directly from 15 and 16 that Γ( a.b, c ) = (id H ⊗ m )(Γ ⊗ id H )( a, Γ( b, c )) (20) Γ( a, b.c ) = ( m ⊗ id H )(id H ⊗ Γ)(Γ( a, b ) , c ) (21)hold for a, b, c ∈ H and it should also be clear that Γ(1 , a ) = ( a, and Γ( a,
1) =(1 , a ) . Consequently, we can show that Γ restricts to a braiding operator on R ( H ) . Notation.
Since R ( H ) is a subset of the quotient Hopf algebra, Q , constructedin Lemma 3.5, we denote its elements by a for some a ∈ H , where ι ( a ) = a ∨ D ,with D = ∨ ǫ − ( ∅ ) . Theorem 4.2. If H is a Hopf algebra in SupLat and R is a co-quasitriangularstructure on H , then the induced braiding, Γ , on H restricts to a braiding operator r on the group R ( H ) .Proof. Let D , Q , π and ι be as in Section 3.1 and define ξ : Q ⊗ Q → Q ⊗ Q by ξ = ( π ⊗ π )Γ( ι ⊗ ι ) . Observe that by 14, we can conclude that ( ǫ ⊗ ǫ )Γ = ǫ ⊗ ǫ .Consequently, for any a, b ∈ H , the equality ( π ⊗ π )Γ( a, b ∨ D ) = ( π ⊗ π )Γ( a, b ) = ( π ⊗ π )Γ( a, b ∨ D ) (22)holds. A symmetric argument proves the exact same statements where Γ is replacedby Γ − . Hence, ( π ⊗ π )Γ − ( ι ⊗ ι )( π ⊗ π )Γ( ι ⊗ ι ) = ( π ⊗ π )Γ − Γ( ι ⊗ ι ) =id Q⊗Q . Thereby, ξ is invertible and since Q = P ( R ( H )) as a lattice, by Lemma3.6, ξ must send basis elements to basis elements, and restricts to a bijective map r : R ( H ) × R ( H ) → R ( H ) × R ( H ) , where r ( g, h ) = ξ ( { g } , { h } ) .By the properties of Γ described before (20, 21), 22 and π ( h. ( g ∨ D )) = π ( h.g ) = π (( h ∨ D ) .g ) from the proof of Lemma 3.5, we can immediately con-clude that ξ satisfies the same properties and r is a braiding operator on R ( H ) . CO-QUASITRIANGULARHOPFALGEBRASANDSKEWBRACES 19We will prove 8 as an example and leave the other properties to the reader. Let a, b, c ∈ Q , then (id Q ⊗ m ′ )( ξ ⊗ id Q )( a, ξ ( b, c )) = (id Q ⊗ πm ( ι ⊗ ι ))(( π ⊗ π )Γ) ⊗ id Q )(id H ⊗ ιπ ⊗ π )(id H ⊗ Γ)( a ∨ D, b ∨ D, c ∨ D )=(id Q ⊗ πm ( ι ⊗ ι ))( π ⊗ π ⊗ π )(Γ ⊗ id H )( a ∨ D, Γ( b ∨ D, c ∨ D ))=( π ⊗ πm )(Γ ⊗ id H )( a ∨ D, Γ( b ∨ D, c ∨ D ))=( π ⊗ π )Γ( m ( a ∨ D, b ∨ D ) , c ∨ D ) = ξ ( m ′ ( a, b ) , c ) Hence, the remnant of every co-quasitriangular Hopf alebra in
SupLat , be-comes a group with a braiding operator. As described in Theorem 2.2, any suchgroup carries a secondary group structure, denoted by ⋆ , which makes it a skewbrace. We now show that the source of this second multiplication is transmutation ,[15].We recall the theory of transmutation of co-quasitriangular Hopf alebras fromChapter 9 of [17]. Any Hopf algebra H can be viewed as an object of its category ofleft comodules via the left coadjoint coaction defined by a
7→ { ( a (1) .S ( a (3) ) , a (2) ) } .In fact, ( H , ∆ , ǫ ) becomes a comonoid in H SupLat i.e. ∆ and ǫ commute withthe appropriate coactions. If additionally H is equipped with a co-quasitriangularstructure, one can defined a secondary multiplication and antipode, denoted by ⋆ and S ⋆ , respectively. a ⋆ b = R (cid:0) S ( a (2) ) ⊗ b (1) S ( b (3) ) (cid:1) a (1) .b (2) (23) S ⋆ ( a ) = R (cid:0) a (1) ⊗ S ( a (4) ) S ( a (2) ) (cid:1) S ( a (3) ) (24)In this case, ( H , ⋆, , S ⋆ , ∆ , ǫ ) becomes a braided Hopf algebra in the braidedmonoidal category H SupLat , Example 9.4.10 [17]. A braided Hopf algebra in thiscase only differs from a Hopf algebra, in the statement of the bialgebra condition,13, where the braiding of the category comes, into play, but we will not utilise thisin what follows.
Theorem 4.3.
Let ( H , R ) and r be as in Theorem 4.2. The transmuted product on H , 23, restricts to a secondary group structure on X and agrees with the induced ⋆ multiplication of ( R ( H ) , r ) , from Theorem 2.2.Proof. First, we must consider the induced braiding operator r more carefully. Let a, b ∈ Q , since ( π ⊗ π )Γ( a ∨ D, b ∨ D ) = ( π ⊗ π )Γ( a, b ) then r ( a, b ) = R ( a (1) , b (1) ) . (cid:0) π ( b (2) ) , π ( a (2) ) (cid:1) . R − ( a (3) , b (3) ) We again adapt the notation r ( a, b ) = (cid:0) σ a ( b ) , γ b ( a ) (cid:1) of Section 2.2. It is a straight-forward consequence of the definition of a a co-quasitriangular structure R , that it CO-QUASITRIANGULARHOPFALGEBRASANDSKEWBRACES 20must satisfy R ( a, S ( b )) = R − ( a, b ) for any a, b ∈ H [Lemma 2.2.2 [17]]. Since ǫ ′ π = ǫ , and ǫ ′ is non-trivial on all elements other than ∅ , then { σ a ( b ) } = (id Q ⊗ ǫ ′ ) ξ ( { a } , ξ { b } )= R ( a (1) , b (1) ) . R − ( a (3) , b (3) ) .π ( b (2) ) .ǫ ( a (2) ) (25) = R ( a (1) , b (1) ) . R ( a (2) , S ( b (3) )) .π ( b (2) ) = R (cid:0) a ⊗ b (1) S ( b (3) ) (cid:1) π ( b (2) ) Since, ǫ ⊗ ǫ = ǫ⋆ , we can proceed as in the proof of Lemma 3.5, and observethat ⋆ , restricts to an associative product, ⋆ ′ , on Q . More importantly, π ( a ∨ D ⋆b ∨ D ) = π ( a ⋆ b ) . Additionally, we note that for any a ∈ R ( H ) and for any q ∈ { q | ∃ b ∈ H , ( b, q ) ∈ ∆( a ) } , the value of π ( q ) is either ∅ or a . Thereby, ( π ⊗ id H )∆( a ) = ∨ { b | ( q,b ) ∈ ∆( a ) , ǫ ( q )=1 } ( a, b ) = ( a, a ) ∈ Q ⊗ H . Hence, a ⋆ b = R (cid:0) S ( a (2) ) ⊗ b (1) S ( b (3) ) (cid:1) .π ( a (1) ) .π ( b (2) )= R (cid:0) S ( a ) ⊗ b (1) S ( b (3) ) (cid:1) .a.π ( b (2) )= R (cid:0) S ( a ) ⊗ b (1) S ( b (3) ) (cid:1) .a.π ( b (2) ) = a.σ S ( a ) ( b ) = a.σ a − ( b ) Remark 4.4.
As mentioned in Section 2.2, there is also a notion of skew rightbraces , which we do not discuss here. Although, we do not present the details,the author believes the secondary group structure for skew right braces, shouldarise by applying the right-handed transmutation of the Hopf algebra i.e. the mul-tiplication which makes ( H , R ) a braided Hopf algebra in the category of right H -comodules. As mentioned in 2.2, to any skew brace ( B, ., ⋆ ) , one can associate a crossedgroup ( B, ⋆ ) ⋊ ( B, . ) . On the other hand, in the theory of co-quasitriangular Hopfalgebras, there is a well-known construction for new Hopf algebras called the crossproduct or bosonisation , using the original Hopf algebra and a braided Hopf al-gebra in its braided category of comodules [16]. In particular, if we denote thetransmutation of H by H ad , we can form the cross product H ad ⋊ H Hopf algebraon the underlying object H ad ⊗ H [Corollary 4.6 [16]]. For the general theoryof bosonisation as we will use here, we refer to Theorem 9.4.12 of [17] and thedescription of the dual statement afterwards. Corollary 4.5. If ( B, ., ⋆ ) is the skew brace arising as the remnant of a co-quasitriangularHopf alebra ( H , R ) , then the crossed group ( B, ⋆ ) ⋊ ( B, . ) is the remnant of theHopf algebra H ad ⋊ H .Proof. First, we note that the induced Hopf algebra H ad ⋊ H is an induced structureon the object H ad ⊗ H = H ⊗ H and its counit takes the form ǫ ( a, b ) = ǫ ( a ) .ǫ ( b ) for a, b ∈ H [Proposition 1.6.18 [17]]. It follows directly, that the quotient Hopf CO-QUASITRIANGULARHOPFALGEBRASANDSKEWBRACES 21algebra of H ad ⋊ H , in the sense of Lemma 3.5, must be of the form P ( R ( H )) ⊗P ( R ( H )) = P ( R ( H ) × R ( H )) .Hence, we know that as a set R ( H ad ⋊ H ) = B × B . Now, we only need torecall the multiplication on H ad ⋊ H from Equation (9.51) of [17]. Recall that H ad is an object of H SupLat , via its left coadjoint action. Hence, for a, b, c, d ∈ H , wehave the following induced multiplication ( a, b ) . ( c, d ) = R ( b (1) , c (1) .S ( c (3) )) . ( a.c (2) , b (2) .d ) As in the proof of Theorem 4.2, we observe that the induced multiplication on B × B for elements a, b, c, d ∈ B , will have the form ( a, b ) . ( c, d ) = R ( b (1) , c (1) .S ( c (3) )) . (cid:0) π ( a ) .π ( c (2) ) , π ( b (2) ) .π ( d ) (cid:1) And from the observations in Theorem 4.2 and in particular equation 25, it followsthat the multiplication agrees with the multiplication of the crossed group, 12.
Hopf algebras in
Rel were classified in [LYZ1] and quasitriangular structures onthem were described in [LYZ2]. Consequently, we can classify co-quasitriangularHopf alebra structures on free lattices in
SupLat . Here we briefly review the proofsof these results and our interpretation of these results in terms of the remnant.If G is a group and G + , G − are subgroups of G such that for any element g ∈ G , there exists a unique pair g + ∈ G + and g − ∈ G − satisfying g = g + .g − , G is said to have a unique factorisation denoted by G = G + .G − . By applyinginverses, we see that G also factorises as G = G − .G + , with notation g = g − .g + .Hence, one can define the following actions of G + and G − on each other: g g − + = g + , g + g − = g − , g g + − = g − , g − g + = g + For more details about the properties which these actions satisfy, we refer the readerto [10, 11]. For a group G with a unique factorisation, P ( G ) admits a Hopf algebrastructure defined by g.h = ( g.h − = g − .h = g − .h + h − iff h + = g + ∅ otherwise ∨ { g + | g + ∈ G + } , S ( g ) = g − ∆( g ) = ∨ h + ∈ G + { ( g + h − ( h + g − ) , h + g − ) } ǫ ( g ) = ( iff g ∈ G − ∅ otherwise CO-QUASITRIANGULARHOPFALGEBRASANDSKEWBRACES 22We should note that this structure arises as the bicrossproduct [Example 6.2.11[17]] of the group algebra P ( G − ) and function algebra P ( G + ) , from Examples3.2 and 3.3, respectively. Theorem 4.6. [10, 11] For a set G , any Hopf algebra structure on the power-set P ( G ) corresponds to a group structure on G with a unique factorisation G = G + .G − , with the resulting Hopf algebra structure described above on P ( G ) . The proof of this statement is presented in [11], for finite-dimensional Hopfalgebras with positive basis and is said to follow for free modules over the Booleanalgebra with 2 elements i.e. free lattices. The positive basis assumption is replacingthe fact that every element in P ( G ) , must be the join of a unique set of basiselements. However, in this case things are much simpler, since there are no scalarsto worry about. It is not difficult to follow the proof and check that all argumentsdo hold as already mentioned in the article. The first step of this proof is to identify G + = { g ∈ G | g ∈ } and observe that P ( G + ) is a commutative Hopf subalgebraof P ( G ) . The finiteness condition comes in use when the authors apply the sameargument to dual Hopf algebra of P ( G ) . However, we know that any free lattice isa dualizable object and finiteness is no longer an issue for defining the dual Hopfalgebra structure on P ( G ) . Recall that the dual Hopf algebra structure on P ( G ) , isdefined by (∆ ∨ , ǫ ∨ , m ∨ , η ∨ , S ∨ ) , where η : P (1) → P ( G ) is the morphism whichsends to the designated unit element ∈ P ( G ) . In this way G − = { g ∈ G | ǫ ( g ) = 1 } and P ( G − ) is shown to be a cocommutative Hopf subalgebra of P ( G ) .The authors also use the classification of finite-dimensional cocommutative Hopfalgebras, to show that P ( G − ) is a group algebra. While for us, this follows fromthe properties of the counit on P ( G − ) and Lemma 3.6.Notice that by definition, the quotient algebra, Q , of Lemma 3.5, will be iso-morphic to P ( G − ) . We must emphasise that in general the quotient constructed inLemma 3.5, will not be isomorphic to the sublattice ǫ − (1) ∪ ∅ . It is only in thissimple case, that they agree, and even here, the map ι is different to the natural in-clusion of P ( G − ) as a subalgebra. Nevertheless, in the next proof, this differencewill be ineffective, due to the application of π .In [10], positive quasitriangular structure of such Hopf algebras were classified.Again this statement holds for free lattices. It should be clear that the dual of the P ( G ) , will again be P ( G ) , but the dual Hopf algebra structure will reverse thefactorisation and use G = G − .G + . Using this technique, we can also clasify allpossible co-quasitriangular structures on P ( G ) . Co-quasitriangular structure on P ( G + .G − ) correspond to a pair of group morphisms η, ξ : G − → G + satisfying v ξ ( u ) = ξ (cid:0) η ( v ) u (cid:1) η ( v ) u = η (cid:0) v ξ ( u ) (cid:1) uv = (cid:0) η ( u ) v (cid:1)(cid:0) u ξ ( u ) (cid:1) u xη ( u x ) = ξ ( u ) x u xξ ( u x ) = η ( u ) x CO-QUASITRIANGULARHOPFALGEBRASANDSKEWBRACES 23for u, v ∈ G − and x ∈ G + [Proposition 1 and Theorem 1 [10]]. When providedwith such a pair, we can define R : P ( G × G ) → P (1) by R ( g, h ) = ( iff there exists a pair u, v ∈ G − s.t g = vξ ( u ) , h = u ( η ( v ) u ) − ∅ otherwise Theorem 4.7.
Let H = P ( G ) be the resulting Hopf algebra of a group G = G + G − with unique factorisation, and η, ξ : G − → G + provide a co-quasitriangularstructure on H . The remnant of H is isomorphic to the group G − and its inducedbraiding operator is defined by ( g − , h − ) (cid:16) η ( g − ) h − , g ξ ( h − ) − (cid:17) (26) for g − , h − ∈ G − .Proof. We have already mentioned why the remnant of H is the group G − . Wemust describe the induced braiding operator r ( a, b ) = R ( a (1) , b (1) ) . (cid:0) π ( b (2) ) , π ( a (2) ) (cid:1) . R − ( a (3) , b (3) ) in this case, where a, b ∈ G − . For a ∈ G − ⊂ G , { ( a (1) , a (2) , a (3) ) } = { (cid:0) l − ( l + a ) , l + k − k + a, k + a (cid:1) | l + , k + ∈ G + } Since π ( g + g − ) = g − iff g + = e and ∅ otherwise, then r ( a, b ) = ∨ l + ,k + ∈ G + R (cid:0) l − ( l + a ) , k − ( k + b ) (cid:1) . (cid:0) k + b, l + a (cid:1) . R − ( l + a, k + b ) We note that l − ( l + a ) = a (cid:0) a − l − (cid:1) , by equations (1) of [10]. The first term R (cid:0) l − ( l + a ) , k − ( k + b ) (cid:1) takes the value if and only if ξ ( b ) = a − l − and b − k − = (cid:0) η ( a ) b (cid:1) − . The first equation is resolved by l − = a ξ ( b ) and for the second equa-tion we recall that b − k − = ( k b + ) − , from equations (2) of [10]. Hence, k + = η ( a ) . It is easy to check that R − ( l + a, k + b ) is also only non-trivial for the samevalues of l + , k + ∈ G + . By (1) of [10], we conclude that l + a = ξ ( b − ) a − a = a ξ ( b ) ,thereby demonstrating that r ( a, b ) = (cid:0) η ( a ) b, a ξ ( b ) (cid:1) . Example 4.8. [LYZ2] Given a group with a unique factorisation G = G + G − ,there is a natural braiding operator on G itself, sometimes referred to as Weinsteinand Xu’s solution [23]. In Section 7 of [LYZ2], it was pointed out that the relevantbraiding operator on G appears from the Drinfeld double of P ( G ) . FRTRECONSTRUCTION 24
In this section, we use the FRT reconstruction as formulated in Section 2.1, firstto recover the universal group of a set-theoretical YBE solution, Theorem 5.1, andsecondly, to obtain a co-quasitriangular Hopf alebra for every group with a braidingoperator, whose remnant recovers the group and the operator, Theorem 5.2.
Let ( X, r ) be set-theoretical YBE solution with the notation presented in Section2.2 i.e. r ( x, y ) = ( σ x ( y ) , γ y ( x )) and r − ( x, y ) = ( τ x ( y ) , ρ y ( x )) . First let ustake the solution into the category of sets and relations Rel , via the natural faithfulfunctor inc . : Set → Rel .It should be easy to see that a bijective YBE solution, is a braided object in
Set . Additionally, our assumption for non-degeneracy of the solution becomesequivalent to ( X, r ) being dualizable in Rel :Any set X is dualizable in (Rel , × , ) with itself as its dual and the evaluationand coevaluation morphisms given by ev = { (( x, x ); 1) | ∀ x ∈ X } ⊂ X × X × (27) coev = { (1; ( x, x )) | ∀ x ∈ X } ⊂ × X × X (28)and the image of these relations under P ( − ) : Rel → SupLat , become exactly theduality morphisms for P ( X ) . Notice that morphisms in Rel are invertible, if andonly if they describe bijective maps between the sets. If we denote the set X whenregarded as its own dual by X ∨ , we observe that non-degeneracy is necessary since r ♭ ( x, y ) = ( γ − x ( y ) , ρ − y ( x )) and ( r − ) ♭ ( x, y ) = ( ρ − x ( y ) , γ − y ( x )) .Hence, with reference to Section 2.1, we have a strict monoidal functor ω :˜ B →
Rel , which sends x to X , y to X ∨ and κ x , x to r . The image of ˜ B forms arigid braided monoidal subcategory of Rel . Notice that included in this subcategoryare morphisms ω ( κ x , y ) , ω ( κ y , y ) and ω ( κ y , y ) which define the braidings between X and X ∨ , X ∨ and X and X ∨ with itself, respectively. Using the laws of a braidedcategory e.g. ω ( κ x , y ) = ω (cid:0) (id x ⊗ y ⊗ ev)(id y ⊗ κ − x ⊗ x ⊗ id y )(coev ⊗ id x ⊗ y ) (cid:1) , wecan calculate these morphisms directly: ω ( κ x , y ) : X × X ∨ → X ∨ × X ; ( x, y ) ( ρ − x ( y ) , γ − y ( x )) ω ( κ y , y ) : X ∨ × X → X × X ∨ ; ( x, y ) ( σ − x ( y ) , τ − y ( x )) ω ( κ y , y ) : X ∨ × X ∨ → X ∨ × X ∨ ; ( x, y ) ( ρ x ( y ) , τ y ( x )) Let w denote a finite sequence of values from {− , ∨} , so that we can denote theobject X × X ∨ × X ∨ by X w for the sequence w = ( − , ∨ , ∨ ) . We denote the set FRTRECONSTRUCTION 25of such sequences by W and define the inverse of a sequence w = ( w , . . . w n ) by the sequence w − = ( w − n , . . . w − ) , where − − = ∨ and ∨ − = − . Hence,by the braiding principle of κ a , b ⊗ c = (id b ⊗ κ a , c )( κ a , b ⊗ id c ) , we can calculatethe induced braidings between any pair of objects X v and X w , in the image of ω .We denote this braiding by r v,w : X v × X w → X w × X v and extend our notationfor r , so that for a pair of words x ∈ X v and y ∈ X w , we write r v,w ( x, y ) =( σ x ( y ) , γ y ( x )) and similarly for r − .Now we apply the power-set functor to the above constructions and denote thestrong monoidal functor P ( ω ) : ˜ B →
SupLat , by ω . As mentioned in Section 2.3, SupLat is cocomplete and we can construct the corresponding coend, 5, for FRTreconstruction on ω .Observe that H ω = Z a ∈ ˜ B ω ( a ) ⊗ ω ( a ) ∨ = Z w ∈W P ( X w × X w − )= a w ∈W P ( X w × X w − ) (cid:14) { Relations } where we are using the fact that the braidings on X ∨∨ agree with X and X w − =( X w ) ∨ . By the symmetric structure of the category we can reorganise the elementsof X w × X w − = X ( w ,...,w n ,w − n ,...,w − ) into X ( w ,w − ,...,w n ,w − n ) , as long as werecall this change when deriving the mentioned relations and Hopf algebra struc-ture. In this way H ω along with its induced multiplication takes a simpler formsince a w ∈W P (cid:16) X ( w ,w − ,...,w n ,w − n ) (cid:17) = P (cid:0) F ( X × X ∨ ⊔ X ∨ × X ) (cid:1) where F ( S ) , for a set S , denotes the free monoid on the set S . If we look carefullyat the induced multiplication on H ω , we can see that it must agree with the inducedmultiplication of the free monoid, hence this form of the coend is more desirableto work with. For the set S = X × X ∨ ⊔ X ∨ × X , we will denote, elements of X × X ∨ and X ∨ × X by ( x, y ) and ( x, y ) , respectively.The category ˜ B is generated by two types of morphisms, namely braidings κ and the duality morphisms ev , coev . Hence, there are two types of relations whichmust be quotiented out from P ( F ( S )) to obtain H ω .First we resolve the relations coming from the braidings. Let F , be the quo-tient of the monoid F ( S ) as above, by the two-sided ideal generated by the follow-ing relations κ x , x ; ( x, y ) . ( a, b ) = ( σ x ( a ) , σ y ( b )) . ( γ a ( x ) , γ b ( y )) FRTRECONSTRUCTION 26 κ x , y ; ( x, y ) . ( a, b ) = ( ρ − x ( a ) , ρ − y ( b )) . ( γ − a ( x ) , γ − b ( y )) κ y , x ; ( x, y ) . ( a, b ) = ( σ − x ( a ) , σ − y ( b )) . ( τ − a ( x ) , τ − b ( y )) κ y , y ; ( x, y ) . ( a, b ) = ( ρ x ( a ) , ρ y ( b )) . ( τ a ( x ) , τ b ( y )) for all x, y, a, b ∈ X . Now we must resolve the relations which arise from the eval-uation and coevaluation morphisms. This is the step which would not be possibleif we intended to construct this coend in Rel . We apply a further adjustment to F .Consider the two sided ideal generated by elements ( x, a ) . ( y, a ) , ( a, x ) . ( a, y ) (29)for all x, y, a ∈ X , such that x = y , and denote it by J . Let F = F \ J and observe that it no longer has a monoid structure since, the multiplication ofcertain pairs of elements is not defined, while P ( F ) continues to carry an algebrastructure in SupLat , where the pairs whose multiplication is undefined multiply to ∅ . The above relations come from the fact that the evaluation morphism, 27, send-ing non-equal pairs to ∅ . The lattice H ω , will be the quotient of P ( F ) by thefollowing relations { f. ( x, a ) . ( x, a ) .h | a ∈ X } = { f.h } = { f. ( a, x ) . ( a, x ) .h | a ∈ X } (30)for all x ∈ X and f, h ∈ F . Hence, for any word w ∈ W , we have morphisms µ w : P ( X w × X w − ) → H ω defined by the composition of the natural inclusions ̺ w : P ( X w × X w − ) → P ( F ( S )) and the induced projection ς : P ( F ( S )) → H ω . It is straightforward to see from the relations imposed that H ω along withmorphisms µ w , becomes the coend of the mentioned diagram.Observe that in the last step we are constructing a quotient of the lattice struc-ture, as formulated in Section 2.3, the relation automatically implies that for anypair x, a ∈ X and y ∈ F , { ( a, x ) . ( a, x ) .y } ≤ { y } in H ω . Hence, notice thatthis lattice is far from admitting a basis since we have infinitely ordered chainswhere the order is strict e.g for any pair x, a ∈ X and y ∈ F , we have a chain · · · < { ( a, x ) . ( a, x ) . ( a, x ) . ( a, x ) .y } < { ( a, x ) . ( a, x ) .y } < { y } Hence, the image of no element in H ω is minimal i.e for every element ∅ 6 = h ∈ H ω , there exists an element ∅ 6 = g ∈ H ω , such that g < h and g = h .Now we can describe the induced Hopf algebra structure on H ω , by Theorem2.1. As mentioned earlier the multiplication, is exactly the image of the multipli-cation for P ( F ) , where certain pairs of elements multiply to give ∅ due to the FRTRECONSTRUCTION 27reduction in the structure of F . Additionally, the image of the unit of F ( S ) , de-noted by { } , acts as the unit of H ω .Let us denote arbitrary elements { ( x , y ) i . . . . . ( x n , y n ) i n } ∈ H ω by ( x, y ) i ,where x = ( x , . . . x n ) , y = ( y , . . . y n ) and i = ( i , . . . i n ) . The counit in thiscase is defined by ǫ (( x, y ) i ) = 1 if and only if x i j = y i j for all ≤ j ≤ n . Noticethat ǫ is well-defined since it is invariant under the imposed relations on F ( S ) ,which define H ω . The coalgebra structure is defined by ∆(( x, y ) i ) = ∨{ (cid:0) ( x, l ) i , ( l, y ) i (cid:1) | ∀ l ∈ X n } Moreover, H ω admits an involutive antipode defined by S (( x, y ) i ) = ( y f , x f ) i − ,where x f denotes the sequence x being flipped i.e. x f = ( x n , . . . , x ) and i − de-notes the sequence being flipped as well as and being switched e.g. (1 , , − =(1 , , .As described in Section 2.1, H ω will have an induced co-quasitriangular struc-ture, R : H ω ⊗ H ω → P (1) defined by R (( x, y ) i , ( a, b ) j ) = 1 iff y = σ x ( a ) and b = γ a ( x ) (31)where σ, γ denote the extensions of the braiding to arbitrary X w and X v . Theorem 5.1.
Given a set-theoretical YBE solution ( X, r ) , the remnant of theHopf algebra H ω recovers the universal group G ( X, r ) of the solution along withits braiding operator.Proof. We must form the quotient Q of H ω , as done in Lemma 3.5. First observethat by the definition of the counit ǫ , we can write Q as a quotient of P ( F ( X ⊔ X ∨ )) , where we view ( x, y ) i with x = y as the word x i in F ( X ⊔ X ∨ ) . Noticethat thereby the deleted elements 29, are ineffective and would be sent to ∅ anywayin Q . Furthermore, Q should be written as a quotient of P ( F ( X ⊔ X ∨ ) / I ) , where I is the two sided ideal generated by the set of relations x.a = σ x ( a ) .γ a ( x ) , x.b = ρ − x ( b ) .γ − b ( x ) b.x = σ − b ( x ) .τ − x ( b ) , y.b = ρ y ( b ) .τ b ( y ) where x, a ∈ X and b, y ∈ X ∨ . Lastly, if D = ∨ ǫ − ( ∅ ) , we note that by relation30, for any x ∈ X and f, h ∈ F , we have { f. ( x, x ) . ( x, x ) .h } ∨ D = { f.h } ∨ D = { f. ( x, x ) . ( x, x ) .h } ∨ D (32)Hence, the remaining relations imposed on Q , will be that { x } and { x } , are mul-tiplicative inverses for any x ∈ X . In other words, Q is isomorphic to P ( G ) , where FRTRECONSTRUCTION 28 G is the group obtained by quotienting the free group generated by X , F g ( X ) , bythe mentioned braiding relations, where elements x ∈ X ∨ are now written as x − .It is straightforward to see that the latter three braiding relations then follow fromthe first namely x.a = σ x ( a ) .γ a ( x ) for a, x ∈ X , and the inverse laws. Hence, R ( H ω ) ∼ = G ( X, r ) as groups.It remains to show that the induced braiding on R ( H ω ) agrees with that of theuniversal group of ( X, r ) . Let g, h ∈ G ( X, r ) , then we recall the structure of theinduced braiding on R ( H ) from Theorem 4.3 and observe that r ( g, h ) = R (( g, g ) (1) , ( h, h ) (1) ) . (cid:0) π (( g, g ) (2) ) , π (( h, h ) (2) ) (cid:1) . R − (( g, g ) (3) , ( h, h ) (3) )= ∨ l,k,m,n R (( g, l ) , ( h, m )) . (cid:0) π (( l, k )) , π (( m, n )) (cid:1) . R − (( k, g ) (3) , ( n, h ))= (cid:0) π (( σ g ( h ) , σ g ( h )) , π (( γ h ( g ) , γ h ( g )) (cid:1) = (cid:0) σ g ( h ) , γ h ( g ) (cid:1) where if g ∈ X w , then l, k are take values in all elements of X w and similarly for m, n .In the proof of Theorem 4.3, we used the structural properties of the remnant,to deduce the projection of the transmuted product on the remnant. In the case of H ω , one can directly compute the multiplication and antipode of the transmutationof H ω by 23 and 24. We present these structures for the interested reader and omittheir verification: ( x.y ) i ⋆ ( a.b ) j = (cid:16) x.σ − y ( a ) , τ − b ( τ a ( y )) .ρ τ − a ( y ) ( d ) (cid:17) i.j S ⋆ (( x.y ) i ) = (cid:16) ρ − x ( x ) f , l ( σ − y ( x )) f (cid:17) i − where l is the unique isomorphism induced on the set X and its powers, which wediscuss in Appendix A. Observe that when x = y and a = b , the multiplicationtakes exactly the required form of ⋆ on the skew brace. Let G be a group with a braiding operator r : G × G → G × G . In this section, weconstruct a co-quasitriangular Hopf alebra in SupLat , which recovers ( G, r ) as itsremnant. To do this we apply the FRT construction as in last section, while replac-ing ˜ B , with ˜ B m , which is the smallest rigid braided monoidal category generatedby a commutative monoid.We define the category ˜ B m , by adding two new generating morphisms to ˜ B ,namely m : x ⊗ x → x and u : → x , and imposing additional relations m ( u ⊗ id x ) = id x = m (id x ⊗ u ) , m ( m ⊗ id x ) = m (id x ⊗ m ) FRTRECONSTRUCTION 29 mκ x , x = m, κ x , x ( u ⊗ id x ) = id x ⊗ u, κ x , x (id x ⊗ u ) = u ⊗ id x which make x a commutative monoid in ˜ B m .Given a group G with a braiding operator r , we can define a functor ω m :˜ B m → SupLat , as before by sending x to P ( G ) , morphisms m , u and κ to P ( . ) , P ( u ) and r , respectively. This of course defines a strong monoidal functor andsince ˜ B m is a rigid braided category, the Hopf algebra H ω m constructed from thefunctor ω m will have an induced co-quasitriangular structure. Theorem 5.2.
Given a group G with a braiding operator r , the remnant of thereconstructed co-quasitriangular Hopf alebra, H ω m , described above, recovers ( G, r ) .Proof. We must again first construct the coend H ω m . As all the previous mor-phisms appear, H ω m will of course be a quotient of H ω , where ω : ˜ B → SupLat is the relevant functor for the underlying set-theoretical YBE solution ( G, r ) . Theadditional relations arise from the presence of the morphisms m and u in ˜ B m . Thefirst relation comes from the parallel pair µ x ( e ⊗ id P (1) ) , µ (id P (1) ⊗ e ∨ ) : P ( X × ⇒ H ω and imposess { } = { ( e, e ) } . A symmetric relation coming from e ∨ imposes that { } = { ( e, e ) } . The second set of relations arise from the parallel pair µ x ( P ( . ) ⊗ id P ( X ∨ ) ) , µ x ⊗ x (id P ( X × X ) ⊗ P ( . ) ∨ ) : P ( X × X × X ∨ ) ⇒ H ω m Consequently, for any a, b, c ∈ G and f, h ∈ F (where F is as in the proof ofTheorem 5.1), we have the following relation { f. ( a.b, c ) .h } = { f. ( a, d ) . ( b, e ) .h | ∀ d, e ∈ G satisfying d.e = c } There is also a symmetric relation, where ( , ) is replaced by ( , ) . From these rela-tion it should be clear that once we construct the remnant of H ω m , the last relation,will imply that the image of any word x = x x . . . x n ∈ G n in F g ( G ) is identifiedwith the multiplication of the sequence in G . The additional braiding relations in H ω m , are assumed to commute with the multiplication since r is a braiding oper-ator and hence, do not affect the computation. Consquently, R ( H ω m ) ∼ = G and r becomes the induced braiding on G , with exactly the same arguments as in Theo-rem 5.1. APPENDIX:ANINDUCEDBIJECTION 30 A Appendix: An Induced Bijection
In this section, we investigate a bijection l : X → X , which is induced for anyset-theoretical YBE solution ( X, r ) . We will first define l and show it is indeedbijective, by explicit computation, before commenting on its categorical origin atthe end of the section.For this section, we adapt a graphical notation, in the same vain as [9]. If ( a, b ) = r ( x, y ) holds, for x, y, a, b ∈ X , we draw x y a ba b x y Figure 1: Graphical notationIf we read these diagrams from up to down, considering the values in the topline as the entries and the bottom line as outputs, it is well-known that the Yang-Baxter equation is equivalent to the following diagrams having the same output: x y z x y z. . z x . .. . c a . .a b c a b c
Figure 2: Yang-Baxter Equationwhere a straight line, denotes the identity map on X .Since for any x ∈ X , γ x is bijective, there exists a unique a ∈ X such that γ x ( a ) = x . We can in fact show that for any a ∈ X , there exists a unique x ∈ X satisfying γ x ( a ) = x : Lemma A.1.
Let ( X, r ) be a set-theoretical YBE solution, with notation as before,for any a ∈ X , there exists a unique x ∈ X such that γ x ( a ) = x .Proof. We first prove the existence of such an x ∈ X . Let a ∈ X , and pick p, y ∈ X , such that σ a ( p ) = a and ρ a ( y ) = a , and observe that for some l ∈ X ,the following diagrams APPENDIX:ANINDUCEDBIJECTION 31 τ y ( a ) a p τ y ( a ) a py a p τ y ( a ) a γ p ( a ) y a γ p ( a ) y a γ p ( a )? l γ p ( a ) y l ?? Figure 3: Proof of existancehold. Hence, ? = y and ?? = γ p ( a ) , so that γ γ p ( a ) ( a ) = γ p ( a ) .Now, we assume there exists a b ∈ X such that γ x ( b ) = x and γ y ( b ) = y for x = y . Hence, since, σ x is bijective, pick l so that σ x ( l ) = y and observe that forsome p, q, m, t ∈ X , the following diagrams b x l b x lp x l b y mp y m q y mq t m q t ? Figure 4: Proof of uniquenesshold. Hence, ? = m and we have that γ m ( y ) = m . Now we pick j ∈ X suchthat σ m ( j ) = m and observe that for some t ∈ X , the figures x l j x l jy m j x ? jy m γ j ( m ) t m jt m γ j ( m ) t m γ j ( m ) Figure 5: Proof of uniquenesshold. Hence ? = m and y = τ t ( m ) = x .In the situation of Lemma A.1, if γ x ( a ) = x , we will denote σ a ( x ) by l ( a ) .Observe that a completely symmetric argument shows that for any a ∈ X , thereexists a unique y ∈ X such that τ y ( a ) = y . In this case, we denote ρ a ( y ) by r ( a ) . From Figure 3, we can say that for any a ∈ X , there exist unique elementsEFERENCES 32 x, y, l ∈ X such that r ( a, x ) = ( l, x ) and r ( y, a ) = ( y, l ) . Hence, l ( a ) = l and r ( l ) = a , making the maps l and r inverses, and consequently, bijections. In ourdiagrammatic notation, we have that the following diagrams hold. a x y l ( a ) l ( a ) x y a Figure 6: Unique pair a, l ( a ) In fact, in the proof of Lemma A.1, we have explicitly given x, y, l ( a ) in termsof a ∈ X . In the paragraph before Figure 3, we chose x = τ − a ( a ) , y = ρ − a ( a ) and from Figure 3, we observe that l ( a ) = σ a ( τ − a ( a )) = γ a ( ρ − a ( a )) .As mentioned throughout this work, a set-theoretical YBE solution ( X, r ) pro-vides a functor ω : ˜ B →
Rel . From this statement, it follows that the morphisms coev ′ := ω ( κ − y , x )coev : → X × X ∨ and ev ′ := ev ω ( κ y , x ) : X ∨ × X → ,must also satisfy the duality axioms. When written explicitly, the two morphisms,take the forms ev ′ = { (( x, l ( a )); 1) | ∀ x ∈ X } ⊂ X × X × coev ′ = { (1; ( x, r ( x )) | ∀ x ∈ X } ⊂ × X × X which as we have demonstrated above, are well-defined and satisfy the dualityaxioms. References [1] David Bachiller. Solutions of the Yang-Baxter equation associated to skewleft braces, with applications to racks.
Journal of Knot Theory and Its Rami-fications , 27(08):1850055, 2018.[2] David Bachiller P´erez.
Study of the algebraic structure of left braces andthe Yang-Baxter equation . PhD thesis, Universitat Aut`onoma de Barcelona,2016.[3] J Carter, Mohamed Elhamdadi, and Masahico Saito. Homology theory for theset-theoretic Yang-Baxter equation and knot invariants from generalizationsof quandles.
Fundamenta Mathematicae , 184(1):31–54, 2004.[4] VG Drinfeld. On some unsolved problems in quantum group theory. In
Quantum groups , pages 1–8. Springer, 1992.EFERENCES 33[5] Tatiana Gateva-Ivanova. A combinatorial approach to the set-theoretic so-lutions of the Yang–Baxter equation.
Journal of mathematical physics ,45(10):3828–3858, 2004.[6] Leandro Guarnieri and Leandro Vendramin. Skew braces and the Yang-Baxter equation.
Mathematics of Computation , 86(307):2519–2534, 2017.[7] Andr´e Joyal and Ross Street. Braided tensor categories.
Advances in Mathe-matics , 102(1):20–78, 1993.[8] Andr´e Joyal and Myles Tierney.
An extension of the Galois theory ofGrothendieck , volume 309. American Mathematical Soc., 1984.[9] Jiang-Hua Lu, Min Yan, Yong-Chang Zhu, et al. On the set-theoretical Yang-Baxter equation.
Duke Mathematical Journal , 104(1):1–18, 2000.[10] Jiang-Hua Lu, Min Yan, and Yongchang Zhu. Quasi-triangular structureson Hopf algebras with positive. In
New Trends in Hopf Algebra Theory:Proceedings of the Colloquium on Quantum Groups and Hopf Algebras, LaFalda, Sierras de C´ordoba, Argentina, August 9-13, 1999 , volume 267, page339. American Mathematical Soc., 2000.[11] Jiang-Hua Lu, Min Yan, and Yongchang Zhu. On Hopf algebras with positivebases.
Journal of Algebra , 237(2):421–445, 2001.[12] Volodymyr Vasyliovych Lyubashenko. Hopf algebras and vector symmetries.
Russian Mathematical Surveys , 41(5):153, 1986.[13] Saunders Mac Lane.
Categories for the working mathematician , volume 5.Springer Science, 2013.[14] Shahn Majid. Braided groups.
Journal of pure and applied algebra ,86(2):187–221, 1993.[15] Shahn Majid. Transmutation theory and rank for quantum braided groups. In
Mathematical proceedings of the Cambridge Philosophical Society , volume113, pages 45–70. Cambridge University Press, 1993.[16] Shahn Majid. Cross products by braided groups and bosonization.
Journal ofalgebra , 163(1):165–190, 1994.[17] Shahn Majid.
Foundations of quantum group theory . Cambridge universitypress, 2000.EFERENCES 34[18] Wolfgang Rump. Modules over braces.
Algebra and Discrete Mathematics ,2006.[19] Wolfgang Rump. Braces, radical rings, and the quantum Yang-Baxter equa-tion.
Journal of Algebra , 307(1):153–170, 2007.[20] Kenichi Shimizu. Tannaka theory and the FRT construction over non-commutative algebras. arXiv preprint arXiv:1912.13160 , 2019.[21] Agata Smoktunowicz. On Engel groups, nilpotent groups, rings, braces andthe Yang-Baxter equation.
Transactions of the American Mathematical Soci-ety , 370(9):6535–6564, 2018.[22] Agata Smoktunowicz and Leandro Vendramin. On skew braces (with an ap-pendix by n. Byott and l. Vendramin).
Journal of Combinatorial Algebra ,2(1):47–86, 2018.[23] Alan Weinstein and Ping Xu. Classical solutions of the quantum Yang-Baxterequation.
Communications in mathematical physics , 148(2):309–343, 1992.[24] Chen-Ning Yang. Some exact results for the many-body problem in one di-mension with repulsive delta-function interaction.