Idempotent solutions of the Yang-Baxter equation and twisted group division
aa r X i v : . [ m a t h . QA ] F e b IDEMPOTENT SOLUTIONS OF THE YANG-BAXTER EQUATION ANDTWISTED GROUP DIVISION
DAVID STANOVSK ´Y AND PETR VOJTˇECHOVSK ´Y
Abstract.
Idempotent left nondegenerate solutions of the Yang-Baxter equation are in one-to-onecorrespondence with twisted Ward left quasigroups, which are left quasigroups satisfying the identity( x ∗ y ) ∗ ( x ∗ z ) = ( y ∗ y ) ∗ ( y ∗ z ). Using combinatorial properties of the Cayley kernel and the squaringmapping, we prove that a twisted Ward left quasigroup of prime order is either permutational ora quasigroup. Up to isomorphism, all twisted Ward quasigroups ( X, ∗ ) are obtained by twistingthe left division operation in groups (that is, they are of the form x ∗ y = ψ ( x − y ) for a group( X, · ) and its automorphism ψ ), and they correspond to idempotent latin solutions. We solve theisomorphism problem for idempotent latin solutions. Introduction
We continue our program of studying left nondegenerate set-theoretic solutions of the Yang-Baxter equation from an algebraic perspective, taking advantage of the associated left quasigroups[2, 12].It is well-known that the algebraic counterpart to derived left nondegenerate solutions of theYang-Baxter equation are racks , which are left quasigroups satisfying the identity( x ∗ y ) ∗ ( x ∗ z ) = x ∗ ( y ∗ z ) . (1.1)There is vast literature on racks, on idempotent racks (i.e., quandles ), and on their usage in knottheory [1, 7, 9, 15, 21].Jones [11] and Turaev [22] showed that Yang-Baxter operators r : V ⊗ V → V ⊗ V that satisfya quadratic equation r = ar + b give rise to polynomial invariants of knots. In the realm of set-theoretic solutions, the quadratic equation reduces to either r = 1 (the involutive case) or r = r (the idempotent case).Rump showed in [20] that the algebraic counterpart to involutive left nondegenerate solutionsare cycle sets , which are left quasigroups satisfying the identity( x ∗ y ) ∗ ( x ∗ z ) = ( y ∗ x ) ∗ ( y ∗ z ) . (1.2)We have renamed cycle sets Rump left quasigroups in [2] and we continue to use this terminologyhere.In this paper we focus on idempotent left nondegenerate solutions. In Section 2 we provide auniform treatment to the correspondences for derived, involutive and idempotent left nondegeneratesolutions. We recover the above correspondences for derived and involutive left nondegenerate
Date : February 10, 2020.1991
Mathematics Subject Classification.
Key words and phrases.
Quantum Yang-Baxter equation, set-theoretical solution to Yang-Baxter equation, braid-ing, idempotent braiding, twisted Ward quasigroup, Ward quasigroup.The paper is written within the framework of the cooperation grant LTAUSA19070. David Stanovsk´y partiallysupported by the GA ˇCR grant 18-20123S. Petr Vojtˇechovsk´y partially supported by the 2019 PROF grant of theUniversity of Denver. olutions, and we prove that the algebraic counterpart to idempotent left nondegenerate solutionsare left quasigroups satisfying the identity( x ∗ y ) ∗ ( x ∗ z ) = ( y ∗ y ) ∗ ( y ∗ z ) , (tW)which we call twisted Ward left quasigroups for reasons explained below.In Section 3 we completely classify twisted Ward quasigroups and hence idempotent latin solu-tions of the Yang-Baxter equation, cf. Theorem 3.10. First we show that, unlike in the case of latinracks and Rump quasigroups, every twisted Ward quasigroup is isotopic to a group. In fact, everytwisted Ward quasigroup has the form ( X, ∗ ) with x ∗ y = cψ ( x − y ), where ( X, · ) is a group, ψ is anautomorphism of ( X, · ) and c ∈ X . Classifying these quasigroups up to isomorphism, it suffices toconsider groups ( X, · ) up to isomorphism, automorphisms ψ up to conjugation in the automorphismgroup of ( X, · ), and c = 1. Summarizing, idempotent latin solutions of the Yang-Baxter equationare in one-to-one correspondence with conjugacy classes of automorphisms of groups .The above representation theorem is the reason why we have chosen the terminology “twistedWard quasigroup.” Quasigroups ( X, ∗ ) defined over groups ( X, · ) via x ∗ y = x − y were firstinvestigated by Ward in [23] and later became known as Ward quasigroups . They are precisely thequasigroups satisfying the identity ( x ∗ y ) ∗ ( x ∗ z ) = y ∗ z and they can be though of as groups axiomatized by left division instead of multiplication, cf.[10, 18, 19, 23].In Section 4 we classify twisted Ward left quasigroups of prime order. We consider two equivalencerelations on a twisted Ward left quasigroup, namely the Cayley kernel and the kernel of the squaringmap, and we show that, in the finite case, each of the equivalences has blocks of uniform size.Consequently, a twisted Ward left quasigroup of prime order is either permutational or a quasigroup.This complements the result of Etingof, Soloviev and Guralnick [8] who classified all indecomposablenondegenerate solutions to the Yang-Baxter equation with a prime number of elements.2. Three classes of left nondegenerate braidings
A binary algebraic structure ( X, ∗ ) is a left quasigroup if all left translations L x : X → X , y x ∗ y are bijections of X . In a left quasigroup ( X, ∗ ) we can define the left division operationby x \ ∗ y = L − x ( y ) and observe that, obviously, x \ ∗ ( x ∗ y ) = y = x ∗ ( x \ ∗ y ) (2.1)holds for all x, y ∈ X . Conversely, if ( X, ∗ , \ ∗ ) is a set equipped with two binary operations satisfyingthe identity (2.1), then ( X, ∗ ) is a left quasigroup with left division \ ∗ . Note that if ( X, ∗ ) is a leftquasigroup with left division \ ∗ , then ( X, \ ∗ ) is a left quasigroup with left division ∗ [17].Dually, ( X, ∗ ) is a right quasigroup if all right translations R x : X → X , y y ∗ x are bijectionsof X . The right division operation is then defined by x/ ∗ y = R − y ( x ). A left quasigroup that is alsoa right quasigroup is called a quasigroup .A set-theoretic solution of the Yang-Baxter equation( r × × r )( r ×
1) = (1 × r )( r × × r ) (YB)is a mapping r : X × X → X × X such that (YB) holds as an equality of mappings X × X × X → X × X × X under composition [5]. Set-theoretic solutions of (YB) are also known as braidings [13].For any mapping (not necessarily a braiding) r : X × X → X × X we write r ( x, y ) = ( x ◦ y, x • y ) or suitable binary operations ◦ and • on X . Straightforward calculation then shows that r is abraiding if and only if the following three identities hold: x ◦ ( y ◦ z ) = ( x ◦ y ) ◦ (( x • y ) ◦ z ) , (YB1)( x ◦ y ) • (( x • y ) ◦ z ) = ( x • ( y ◦ z )) ◦ ( y • z ) , (YB2)( x • y ) • z = ( x • ( y ◦ z )) • ( y • z ) . (YB3)A mapping r : X × X → X × X is said to be left nondegenerate if ( X, ◦ ) is a left quasigroup; nondegenerate if ( X, ◦ ) is a left quasigroup and ( X, • ) is a right quasigroup; latin if ( X, ◦ ) is aquasigroup; derived if x • y = x for every x, y ∈ X ; involutive if r = 1 [20]; and idempotent if r = r [13].We are mostly interested in braidings where one of the two operations ◦ , • is either trivial orcan be reconstructed from the other one. The following result gives three classes of mappings withsuch a property. Lemma 2.1.
Let r : X × X → X × X be a mapping, r ( x, y ) = ( x ◦ y, x • y ) . Then: (i) r is derived if and only if x • y = x . (ii) r is involutive and left nondegenerate if and only if ( X, ◦ ) is a left quasigroup and x • y =( x ◦ y ) \ ◦ x . (iii) r is idempotent and left nondegenerate if and only if ( X, ◦ ) is a left quasigroup and x • y =( x ◦ y ) \ ◦ ( x ◦ y ) .Proof. Part (i) holds by definition. For (ii), note that r is involutive if and only if the identities( x ◦ y ) ◦ ( x • y ) = x, ( x ◦ y ) • ( x • y ) = y hold. If r is also left nondegenerate then x • y = ( x ◦ y ) \ ◦ x is equivalent to the first identity, andthe second identity becomes (( x ◦ y ) ◦ (( x ◦ y ) \ ◦ x )) \ ◦ ( x ◦ y ) = y, which holds in any left quasigroup ( X, ◦ ). Finally, for (iii), note that r is idempotent if and only ifthe identities ( x ◦ y ) ◦ ( x • y ) = x ◦ y, ( x ◦ y ) • ( x • y ) = x • y hold. If r is also left nondegenerate then x • y = ( x ◦ y ) \ ◦ ( x ◦ y ) is equivalent to the first identity,and the second identity becomes ( z ◦ ( z \ ◦ z )) \ ◦ ( z ◦ ( z \ ◦ z )) = z \ ◦ z with z = x ◦ y , which holds in any left quasigroup ( X, ◦ ). (cid:3) The proof of the following result is somewhat involved. Nevertheless it is purely equational andcan be verified in a fraction of a second by an automated theorem prover, such as
Prover9 [14].
Proposition 2.2.
Let r : X × X → X × X be a mapping. Then: (i) r is a derived braiding if and only if x • y = x and x ◦ ( y ◦ z ) = ( x ◦ y ) ◦ ( x ◦ z ) . (2.2)(ii) r is an involutive left nondegenerate braiding if and only if ( X, ◦ ) is a left quasigroup, x • y = ( x ◦ y ) \ ◦ x and x ◦ ( y ◦ z ) = ( x ◦ y ) ◦ ((( x ◦ y ) \ ◦ x ) ◦ z ) . (2.3)(iii) r is an idempotent left nondegenerate braiding if and only if ( X, ◦ ) is a left quasigroup, x • y = ( x ◦ y ) \ ◦ ( x ◦ y ) and x ◦ ( y ◦ z ) = ( x ◦ y ) ◦ ((( x ◦ y ) \ ◦ ( x ◦ y )) ◦ z ) . (2.4) roof. (i) By Lemma 2.1, r is a derived mapping if and only if x • y = x holds. Then (YB1) isequivalent to (2.2), (YB2) is equivalent to the trivial identity x ◦ y = x ◦ y , and (YB3) is equivalentto the trivial identity x = x .For the rest of the proof, let us write xy instead of x ◦ y , x \ y instead of x \ ◦ y , and [ x, y ] insteadof x • y to save space and improve legibility. The identities (YB1)–(YB3) then become x ( yz ) = ( xy )([ x, y ] z ) , [ xy, [ x, y ] z ] = [ x, yz ][ y, z ] , [[ x, y ] , z ] = [[ x, yz ] , [ y, z ]] . (ii) By Lemma 2.1, r is an involutive left nondegenerate mapping if and only if ( X, · ) is a leftquasigroup and [ x, y ] = ( xy ) \ x . Hence (YB1) holds if and only if (2.3) holds. Suppose that (YB1)holds and let us rewrite it as ( xy ) \ ( x ( yz )) = [ x, y ] z . Upon substituting y \ z for z , we obtain( xy ) \ ( xz ) = [ x, y ]( y \ z ) . (2.5)Using (YB1), the left hand side of (YB2) can be written as [ xy, [ x, y ] z ] = (( xy )([ x, y ] z )) \ ( xy ) =( x ( yz )) \ ( xy ). Upon substituting y \ z for z into (YB2), we therefore obtain the identity( xz ) \ ( xy ) = [ x, z ][ y, y \ z ] = [ x, z ]( z \ y ) , which is (2.5) with y and z interchanged. Hence (YB2) holds. To show that (YB3) also holds,first note that (2.3) can be written as L x L y = L xy L ( xy ) \ x . Substituting x \ y for y , we obtain L x L x \ y = L y L y \ x , and taking inverses yields L − x \ y L − x = L − y \ x L − y . (2.6)We will return to (2.6) shortly. By (YB1), the left hand side of (YB3) is equal to[[ x, y ] , z ] = ([ x, y ] z ) \ [ x, y ] = (( xy ) \ ( x ( yz ))) \ (( xy ) \ x ) . By (YB2) and (YB1), the right hand side of (YB3) is equal to[[ x, yz ] , [ y, z ]] = ([ x, yz ][ y, z ]) \ [ x, yz ] = [ xy, [ x, y ] z ] \ [ x, yz ]= [ xy, ( xy ) \ ( x ( yz ))] \ [ x, yz ] = (( x ( yz )) \ ( xy )) \ (( x ( yz )) \ x ) . Upon substituting y \ z for z , we see that (YB3) is then equivalent to(( xy ) \ ( xz )) \ (( xy ) \ x ) = (( xz ) \ ( xy )) \ (( xz ) \ x ) , which is further equivalent, upon substitution of x \ y for y and x \ z for z , to( y \ z ) \ ( y \ x ) = ( z \ y ) \ ( z \ x ) . But this says L − y \ z L − y = L − z \ y L − z , which is (2.6) with the variables renamed.(iii) By Lemma 2.1, r is an idempotent left nondegenerate mapping if and only if ( X, · ) is a leftquasigroup and [ x, y ] = ( xy ) \ ( xy ). Hence (YB1) holds if and only if (2.4) holds. Suppose that(YB1) holds. As in (ii), we obtain the equivalent identity (2.5). Usinq (YB1), the left hand side of(YB2) can be written as (( xy )([ x, y ] z )) \ (( xy )([ x, y ] z )) = ( x ( yz )) \ ( x ( yz )). Upon substituting y \ z for z into (YB2), we therefore obtain the identity( xz ) \ ( xz ) = [ x, z ][ y, y \ z ] = [ x, z ]( z \ z ) , which is an instance of (2.5) with y = z . Hence (YB2) holds. To see that (YB3) also holds, notethat (2.4) is equivalent to L x L y = L xy L ( xy ) \ ( xy ) , which is the same as L x L x \ y = L y L y \ y and hence L − x \ y L − x = L − y \ y L − y . (2.7)Following the same series of steps as in (ii), we can rewrite (YB3) as(( xy ) \ ( x ( yz ))) \ (( xy ) \ ( x ( yz ))) = (( x ( yz )) \ ( x ( yz ))) \ (( x ( yz )) \ ( x ( yz ))) , hich is equivalent, upon substitution of y \ z for z , to(( xy ) \ ( xz )) \ (( xy ) \ ( xz )) = (( xz ) \ ( xz )) \ (( xz ) \ ( xz )) , and hence to ( y \ z ) \ ( y \ z ) = ( z \ z ) \ ( z \ z ) . But this is implied by L − y \ z L − y = L − z \ z L − z , which is (2.7) with the variables renamed. (cid:3) As has become clear from the proof of Proposition 2.2, the identities (2.3) and (2.4) are somewhatinconvenient to work with. We will therefore employ the following syntactic trick due to Rump[20] to arrive at simpler left quasigroup identities that correspond to the same braidings as inProposition 2.2 (but not necessarily to the same left quasigroups).Given a left nondegenerate mapping r : X × X → X × X , the trick is to write r ( x, y ) = ( x \ ◦ y, x • y )and express the equations (YB1)–(YB3) in terms of the operations ◦ and • , rather than \ ◦ and • . Proposition 2.3.
Let r : X × X → X × X be a left nondegenerate mapping written as r ( x, y ) =( x \ ◦ y, x • y ) . Then: (i) r is a derived left nondegenerate braiding if and only if ( X, ◦ ) is a left quasigroup, x • y = x and ( x ◦ y ) ◦ ( x ◦ z ) = x ◦ ( y ◦ z ) . (2.8)(ii) r is an involutive left nondegenerate braiding if and only if ( X, ◦ ) is a left quasigroup, x • y = ( x \ ◦ y ) ◦ x and ( x ◦ y ) ◦ ( x ◦ z ) = ( y ◦ x ) ◦ ( y ◦ z ) . (2.9)(iii) r is an idempotent left nondegenerate braiding if and only if ( X, ◦ ) is a left quasigroup, x • y = ( x \ ◦ y ) ◦ ( x \ ◦ y ) and ( x ◦ y ) ◦ ( x ◦ z ) = ( y ◦ y ) ◦ ( y ◦ z ) . (2.10) Proof. (i) By Proposition 2.2, r is a derived left nondegenerate braiding if and only if ( X, \ ◦ ) is aleft quasigroup (equivalently, ( X, ◦ ) is a left quasigroup), x • y = x and x \ ◦ ( y \ ◦ z ) = ( x \ ◦ y ) \ ◦ ( x \ ◦ z ) . In terms of the left translations in ( X, ◦ ), the last identity is equivalent to L − x L − y = L − x \ ◦ y L − x .Taking inverses on both sides, we obtain the equivalent identity L y L x = L x L x \ ◦ y . Substituting x ◦ y for y , we obtain L x ◦ y L x = L x L y , which is (2.8).(ii) By Proposition 2.2, r is an involutive left nondegenerate braiding if and only if ( X, ◦ ) is aleft quasigroup, x • y = ( x \ ◦ y ) ◦ x and x \ ◦ ( y \ ◦ z ) = ( x \ ◦ y ) \ ◦ ((( x \ ◦ y ) ◦ x ) \ ◦ z ) . This says L − x L − y = L − x \ ◦ y L − x \ ◦ y ) ◦ x , which is equivalent to L x ◦ y L x = L y ◦ x L y , i.e., to (2.9).(iii) By Proposition 2.2, r is an idempotent left nondegenerate braiding if and only if ( X, ◦ ) is aleft quasigroup, x • y = ( x \ ◦ y ) ◦ ( x \ ◦ y ) and x \ ◦ ( y \ ◦ z ) = ( x \ ◦ y ) \ ◦ ((( x \ ◦ y ) ◦ ( x \ ◦ y )) \ ◦ z ) . This says L − x L − y = L − x \ ◦ y L − x \ ◦ y ) ◦ ( x \ ◦ y ) , which is equivalent to L x ◦ y L x = L y ◦ y L y , i.e., to (2.10). (cid:3) Recall from the introduction that a left quasigroup ( X, ∗ ) is a rack (resp. Rump left quasigroup ,resp. twisted Ward left quasigroup ) if it satisfies (1.1) (resp. (1.2), resp. (tW)).Part (i) of Theorem 2.4 is well-known and part (ii) can be found in [20, Proposition 1].
Theorem 2.4.
Let X be a set. Denote a typical braiding on X by r ( x, y ) = ( x ◦ y, x • y ) . Then: i) There is a one-to-one correspondence between derived left nondegenerate braidings on X and racks on X , given by r ( X, ∗ ) , x ∗ y = x \ ◦ y, ( X, ∗ ) r, r ( x, y ) = ( x \ ∗ y, x ) . (ii) There is a one-to-one correspondence between involutive left nondegenerate braidings on X and Rump left quasigroups on X , given by r ( X, ∗ ) , x ∗ y = x \ ◦ y, ( X, ∗ ) r, r ( x, y ) = ( x \ ∗ y, ( x \ ∗ y ) ∗ x ) . (iii) There is a one-to-one correspondence between idempotent left nondegenerate braidings on X and twisted Ward left quasigroups on X , given by r ( X, ∗ ) , x ∗ y = x \ ◦ y, ( X, ∗ ) r, r ( x, y ) = ( x \ ∗ y, ( x \ ∗ y ) ∗ ( x \ ∗ y )) . Proof.
It is clear that in each case the two mappings are mutually inverse. The rest follows fromProposition 2.3. (cid:3)
Remark 2.5. (a) As we have shown, the identities x ◦ ( y ◦ z ) = ( x ◦ y ) ◦ ( x ◦ z ) and x \ ◦ ( y \ ◦ z ) =( x \ ◦ y ) \ ◦ ( x \ ◦ z ) are equivalent in the variety of left quasigroups. It is therefore customary to replacethe correspondence from Theorem 2.4(i) with the correspondence r ( X, ∗ ) , x ∗ y = x ◦ y, ( X, ∗ ) r, r ( x, y ) = ( x ∗ y, x ) . Our version of the correspondence for racks fits better with the uniform approach employed here.(b) Neither of the identities (2.3) and (2.9) implies the other in the variety of left quasigroups.Likewise, neither of the identities (2.4) and (2.10) implies the other in the variety of left quasigroups.Therefore, in both cases, there are two varieties of left quasigroups that can be chosen to correspondto the braidings in question.
Corollary 2.6.
Let X be a set, | X | ≥ . Then every idempotent braiding on X is degenerate.Proof. Let r ( x, y ) = ( x ◦ y, x • y ) be an idempotent nondegenerate braiding and let ( X, ∗ ) bethe corresponding twisted Ward left quasigroup, i.e., x ∗ y = x \ ◦ y . For every y ∈ X , the righttranslation x x • y = ( x \ ∗ y ) ∗ ( x \ ∗ y ) is a bijection of X , which immediately implies that thesquaring mapping σ : x x ∗ x is onto X . We will prove that σ is also injective. First observe that x • ( x ∗ y ) = ( x \ ∗ ( x ∗ y )) ∗ ( x \ ∗ ( x ∗ y )) = y ∗ y , hence x = ( y ∗ y ) / • ( x ∗ y ) is independent of y , and thus x = ( x ∗ x ) / • ( x ∗ x ). Consequently, if σ ( u ) = σ ( v ), we have u = ( u ∗ u ) / • ( u ∗ u ) = ( v ∗ v ) / • ( v ∗ v ) = v .Now, σ ( x ∗ y ) = ( x ∗ y ) ∗ ( x ∗ y ) = ( y ∗ y ) ∗ ( y ∗ y ) = σ ( y ∗ y ) is an instance of (tW), and since σ is bijective, we have x ∗ y = σ ( y ) for every x, y ∈ X . But then x • y = ( x \ ∗ y ) ∗ ( x \ ∗ y ) = σ ( x \ ∗ y ) = σ ( σ − ( y )) = y , hence r is right degenerate, a contradiction. (cid:3) The following are examples of twisted Ward left quasigroups.
Example 2.7.
Let ( X, ∗ ) be an elementary abelian 2-group. Then ( x ∗ y ) ∗ ( x ∗ z ) = y ∗ z =( y ∗ y ) ∗ ( y ∗ z ), so ( X, ∗ ) is a twisted Ward (left) quasigroup. Example 2.8.
Let ( X, +) be an abelian group, ϕ ∈ End( X, +), ψ ∈ Aut( X, +) and c ∈ X . Definea binary operation ∗ on X by x ∗ y = ϕ ( x ) + ψ ( y ) + c. It is easy to check that the resulting left quasigroup ( X, ∗ ) satisfies (tW) if and only if ϕψ = ψϕ and ϕ + ϕψ = 0 . Example 2.9.
Let x ∗ y = f ( y ) for some bijection f of X . Then ( X, ∗ ) is clearly a left quasigroup,usually called a permutational left quasigroup . Every permutational left quasigroup satisfies (tW). Lemma 2.10.
Let ( X, ∗ ) be a twisted Ward left quasigroup. Then the following conditions areequivalent: i) ( X, ∗ ) is a rack, (ii) ( X, ∗ ) is a Rump left quasigroup, (iii) ( X, ∗ ) is permutational.Proof. If ( X, ∗ ) is permutational then it satisfies both (1.1) and (1.2). If ( X, ∗ ) satisfies (1.1) then x ∗ ( y ∗ z ) = ( x ∗ y ) ∗ ( x ∗ z ) = ( y ∗ y ) ∗ ( y ∗ z ) and substituting y \ ∗ z for z yields x ∗ z = ( y ∗ y ) ∗ z ,which means that all left translations are the same and ( X, ∗ ) is permutational. If ( X, ∗ ) satisfies(1.2) then ( y ∗ x ) ∗ ( y ∗ z ) = ( x ∗ y ) ∗ ( x ∗ z ) = ( y ∗ y ) ∗ ( y ∗ z ) and substituting y \ ∗ z for z and y \ ∗ x for x again yields x ∗ z = ( y ∗ y ) ∗ z . (cid:3) We will return to twisted Ward left quasigroups in Section 4.3.
Twisted Ward quasigroups
A quasigroup ( X, ∗ ) is a twisted Ward quasigroup if it satisfies the identity (tW).Note that in Example 2.8, ( X, ∗ ) is a quasigroup if and only if ϕ ∈ Aut( X, +), in which case( X, ∗ ) satisfies (tW) if and only if ϕ = − ψ . This motivates the following construction. Example 3.1.
Let ( X, · ) be a group, ψ ∈ Aut( X, · ) and c ∈ X . Then ( X, ∗ ) = tWq( X, · , ψ, c )defined by x ∗ y = cψ ( x − y )is a twisted Ward quasigroup. Indeed, to verify (tW), we compute( x ∗ y ) ∗ ( x ∗ z ) = ( cψ ( x − y )) ∗ ( cψ ( x − z )) = cψ (( cψ ( x − y )) − cψ ( x − z )) = cψ ( ψ ( y − z )) , which is independent of x and therefore equal to ( y ∗ y ) ∗ ( y ∗ z ).As we shall see in Theorem 3.7, all twisted Ward quasigroups are of the form tWq( X, · , ψ, c ). Westart by proving in two ways that every twisted Ward quasigroup is isotopic to a group. The firstproof uses the quadrangle criterion known from the theory of latin squares and the second proof isbased on the structure of the displacement group.Recall that two quasigroups ( X, · ), ( Y, ∗ ) are isotopic if there are bijections α, β, γ : X → Y suchthat α ( x ) ∗ β ( y ) = γ ( x · y ) for every x, y ∈ X . Replacing ( Y, ∗ ) with an isomorphic copy, we canassume that γ = 1 in an isotopism [17, Theorem III.1.4]. Proposition 3.2 ([3, p. 18] or [6, Theorem 2.2]) . A quasigroup ( X, ∗ ) is isotopic to a group ifand only if it satisfies the quadrangle criterion , i.e., for every a i , b i , c i , d i ∈ X , if a ∗ c = a ∗ c , a ∗ d = a ∗ d and b ∗ c = b ∗ c then b ∗ d = b ∗ d . Lemma 3.3.
Let ( X, ∗ ) be a twisted Ward quasigroup and a , a , c , c ∈ X . Then: (i) The squaring map x x ∗ x is constant. (ii) If a ∗ c = a ∗ c , then L a L − c = L a L − c .Proof. (i) From (tW), ( x ∗ y ) ∗ ( x ∗ y ) = ( y ∗ y ) ∗ ( y ∗ y ) for every x, y ∈ X . Upon substituting x/ ∗ y for x , we see that x ∗ x is independent of x .(ii) Let us again write x ∗ y = xy . Note that ( xx )( x ( y \ z )) = ( yx )( y ( y \ z )) = ( yx ) z is aninstance of (tW) and therefore ( c c )( c ( a \ ( a x ))) = ( a c )( a x ) = ( a c )( a x ) = ( c c )( c x ),using (tW) again in the last step. Canceling the unique square and dividing on the left by c , weget a \ ( a x ) = c \ ( c x ). Substituting c \ x for x and multiplying on the left by a , we finally get a ( c \ x ) = a ( c \ x ). (cid:3) Proposition 3.4.
Every twisted Ward quasigroup is isotopic to a group. roof. Suppose that ( X, · ) is a twisted Ward quasigroup and the assumptions of the quadranglecriterion are satisfied. Applying Lemma 3.3(ii) to equalities a c = a c and b c = b c , weget d = a \ ( a d ) = a \ ( a d ) = a \ ( a ( c \ ( c d ))) = a \ ( a ( c \ ( c d ))) = c \ ( c d ) and thus b d = b ( c \ ( c d )) = b ( c \ ( c d )) = b d . We are done by Proposition 3.2. (cid:3) The left multiplication group of a quasigroup ( X, ∗ ) is the permutation group generated by allleft translations, i.e., LMlt( X ) = h L x : x ∈ X i . As in [2], we define the positive (resp. negative ) displacement group of ( X, ∗ ) as the subgroup ofLMlt( X ) generated by all positive (resp. negative) displacements, that is,Dis + ( X ) = h L x L − y : x, y ∈ X i , Dis − ( X ) = h L − x L y : x, y ∈ X i . The displacement group of ( X, ∗ ) is then the groupDis( X ) = h L x L − y , L − x L y : x, y ∈ X i . Note that Dis + ( X ) = h L e L − x : x ∈ X i for any fixed e ∈ X since L x L − y = ( L e L − x ) − ( L e L − y ), andDis − ( X ) = h L − x L e : x ∈ X i since L − x L y = ( L − x L e )( L − y L e ) − .A permutation group G acts regularly on a set X if for every x, y ∈ X there is a unique g ∈ G such that g ( x ) = y . Recall the following result of Dr´apal: Proposition 3.5. [4, Proposition 5.2]
A quasigroup X is isotopic to a group if and only if Dis + ( X ) acts regularly on X . In such a case, X is isotopic to Dis + ( X ) . It follows from Propositions 3.4 and 3.5 that every twisted Ward quasigroup X is isotopic to thegroup Dis + ( X ), which acts regularly on X . Here is an alternative proof of this fact which does notrefer to Propositions 3.2 and 3.5. Lemma 3.6.
Let X = ( X, ∗ ) be a twisted Ward quasigroup and let e denote the unique square in X . Then: (i) Dis( X ) = Dis + ( X ) = Dis − ( X ) . (ii) ( L − x L e )( L − y L e ) = L − x/ ∗ e ) ∗ ( e \ ∗ y ) L e and ( L − x L e ) − = L − e \ ∗ x ) ∗ e L e for every x, y ∈ X . (iii) Dis( X ) is equal to { L − x L e : x ∈ X } and is isomorphic to the group isotope ( X, ⋄ ) , where x ⋄ y = ( x/ ∗ e ) ∗ ( e \ ∗ y ) .Proof. (i) The identity (tW) says L x ∗ y L x = L y ∗ y L y = L e L y and hence is equivalent to L x L − y = L − x ∗ y L e , which shows that Dis + ( X ) ≤ Dis − ( X ). Replacing y with x \ ∗ y , we obtain the identity L x L − x \ ∗ y = L − y L e , (3.1)which implies Dis − ( X ) ≤ Dis + ( X ). Hence Dis( X ) = Dis + ( X ) = Dis − ( X ).(ii) Fix x, y ∈ X and let u = x/ ∗ e so that u \ ∗ x = e . By a repeated application of (3.1), we have( L − x L e )( L − y L e ) = ( L u L − u \ ∗ x )( L e L − e \ ∗ y ) = L u ( L − u \ ∗ x L e ) L − e \ ∗ y = L u L − e \ ∗ y = L u L − u \ ∗ ( u ∗ ( e \ ∗ y )) = L − u ∗ ( e \ ∗ y ) L e = L − x/ ∗ e ) ∗ ( e \ ∗ y ) L e . Using (3.1) again, we also have( L − x L e ) − = ( L e L − e \ ∗ x ) − = L e \ ∗ x L − e = L e \ ∗ x L − e \ ∗ x ) \ ∗ (( e \ ∗ x ) ∗ e ) = L − e \ ∗ x ) ∗ e L e . (iii) Part (ii) proves that Dis( X ) = Dis − ( X ) is equal to { L − x L e : x ∈ X } and is isomorphic to( X, ⋄ ), where x ⋄ y = ( x/ ∗ e ) ∗ ( e \ ∗ y ), which therefore has to be a group. Clearly, ( X, ⋄ ) is isotopicto ( X, ∗ ). (cid:3) We proceed to describe all twisted Ward quasigroups. heorem 3.7. Let ( X, ∗ ) be a quasigroup. Then ( X, ∗ ) is a twisted Ward quasigroup if and onlyif there is a group ( X, · ) , ψ ∈ Aut( X, · ) and c ∈ X such that x ∗ y = cψ ( x − y ) for every x, y ∈ X .Proof. We have verified the converse implication in Example 3.1. For the direct implication, supposethat ( X, ∗ ) is a twisted Ward quasigroup. By Proposition 3.4, ( X, ∗ ) is isotopic to a group ( X, · ),i.e., there are permutations ϕ , ψ of X such that x ∗ y = ϕ ( x ) ψ ( y ) for all x, y ∈ X . We may assumewithout loss of generality that ψ (1) = 1, otherwise set ¯ ϕ ( x ) = ϕ ( x ) ψ (1) and ¯ ψ ( y ) = ψ (1) − ψ ( y ) toobtain x ∗ y = ¯ ϕ ( x ) ¯ ψ ( y ) and ¯ ψ (1) = 1.Writing (tW) in terms of · , ϕ and ψ , and replacing z with ψ − ( z ), we have ϕ ( ϕ ( x ) ψ ( y )) · ψ ( ϕ ( x ) z ) = ϕ ( ϕ ( y ) ψ ( y )) · ψ ( ϕ ( y ) z )for all x, y, z ∈ X . Rearranging this, we have ϕ ( ϕ ( y ) ψ ( y )) − · ϕ ( ϕ ( x ) ψ ( y )) = ψ ( ϕ ( y ) z ) · ψ ( ϕ ( x ) z ) − . Note that the left hand side is independent of z . Comparing the right hand sides upon substituting z = 1 and z = ϕ ( x ) − , respectively, we obtain ψ ( ϕ ( y )) · ψ ( ϕ ( x )) − = ψ ( ϕ ( y ) ϕ ( x ) − ) · ψ ( ϕ ( x ) ϕ ( x ) − ) = ψ ( ϕ ( y ) ϕ ( x ) − )for all x, y ∈ X , where we have used ψ (1) = 1. Since ϕ is bijective, this is equivalent to ψ ( y ) ψ ( x ) − = ψ ( yx − )for all x, y ∈ X , and thus ψ is an automorphism of the group ( X, · ).Writing (tW) in terms of · , ϕ and ψ again and substituting y = z = 1, we have ϕ ( ϕ ( x )) · ψ ( ϕ ( x )) = ϕ ( ϕ (1)) · ψ ( ϕ (1))for every x ∈ X , with the right hand side being constant, say equal to c . Since ϕ is bijective, this isequivalent to ϕ ( x ) ψ ( x ) = c . Then ϕ ( x ) = cψ ( x ) − and x ∗ y = cψ ( x ) − ψ ( y ) = cψ ( x − y ) for every x , y ∈ X . (cid:3) Since isotopic groups are isomorphic [17, Corollary III.2.3], the following result solves the iso-morphism problem for twisted Ward quasigroups.
Proposition 3.8.
Let ( X, · ) be a group, ϕ, ψ ∈ Aut( X, · ) and c ∈ X . Then: (i) The mapping x cx is an isomorphism tWq( X, · , ϕ, → tWq( X, · , ϕ, c ) . (ii) The twisted Ward quasigroups tWq( X, · , ϕ, and tWq( X, · , ψ, are isomorphic if and onlyif ϕ , ψ are conjugate in Aut( X, · ) .Proof. Let us denote the multiplication in tWq( X, · , ϕ, c ) by ∗ ϕ,c .(i) For every x, y ∈ X , we have ( cx ) ∗ ϕ,c ( cy ) = cϕ (( cx ) − ( cy )) = cϕ ( x − y ) = c ( x ∗ ϕ, y ).(ii) If ψ = ρϕρ − for some ρ ∈ Aut( X, · ), then ρ ( x ∗ ϕ, y ) = ρϕ ( x − y ) = ψρ ( x − y ) = ψ ( ρ ( x ) − ρ ( y )) = ρ ( x ) ∗ ψ, ρ ( y ) . Conversely, if ρ is an isomorphism tWq( X, · , ϕ, → tWq( X, · , ψ,
1) then ρϕ ( x − y ) = ψ ( ρ ( x ) − ρ ( y )) (3.2)for every x, y ∈ X . Upon substituting x = 1 into (3.2) we obtain ρϕ = ψρ . Applying ψ − to bothsides of (3.2), we then get ρ ( x − y ) = ρ ( x ) − ρ ( y ), so ρ ∈ Aut( X, · ). (cid:3) For a group G , let cc ( G ) denote the number of conjugacy classes of G . Corollary 3.9.
The number of twisted Ward quasigroups of order n up to isomorphism is q ( n ) = X G cc (Aut( G )) , where the summation runs over all groups G of order n up to isomorphism. eturning to braidings, we deduce: Theorem 3.10.
Let r : X × X → X × X be given by r ( x, y ) = ( x ◦ y, x • y ) . Then r is an idempotentlatin braiding if and only if there is a group ( X, · ) , an automorphism ϕ ∈ Aut( X, · ) and c ∈ X suchthat r ( x, y ) = ( xϕ ( c ) − ϕ ( y ) , c ) . Moreover, up to isomorphism, we can take c = 1 and ϕ up to conjugation in Aut( X, · ) .Proof. By Theorem 2.4, r is an idempotent left nondegenerate braiding if and only if r ( x, y ) =( x \ ∗ y, ( x \ ∗ y ) ∗ ( x \ ∗ y )) for a twisted Ward left quasigroup ( X, ∗ ), and r is latin if and only if ( X, ∗ )is a twisted Ward quasigroup. Then, by Theorem 3.7, x ∗ y = cψ ( x − y ) for some group ( X, · ), ψ ∈ Aut( X, · ) and c ∈ X . Then certainly ( x \ ∗ y ) ∗ ( x \ ∗ y ) = c , and since x \ ∗ y = xψ − ( c − y ), we canwrite x ◦ y = x \ ∗ y = xϕ ( c ) − ϕ ( y ) by taking ϕ = ψ − . The last part follows by Proposition 3.8. (cid:3) The Cayley kernel, squaring and twisted Ward left quasigroups of prime order
For an equivalence relation R on a set X , let X R be a complete set of representatives of theequivalence classes of R and let [ x ] R denote the equivalence class of R containing the element x .On a left quasigroup ( X, · ) define two equivalence relations x ∼ y ⇔ L x = L y ,x ≡ y ⇔ xx = yy. The equivalence relation ∼ is usually called the Cayley kernel in this context. If ∼ is the fullequivalence X × X then ( X, · ) is called permutational , cf. Example 2.9. If ∼ is the equality relation { ( x, x ) : x ∈ X } then ( X, · ) is called faithful . Lemma 4.1.
Let ( X, · ) be a left quasigroup. Then the equivalence relations ∼ and ≡ intersecttrivially and the following inequalities hold for every x ∈ X : (i) | [ x ] ≡ | ≤ | X ∼ | , (ii) | [ x ] ∼ | ≤ | X ≡ | , (iii) | X | ≤ | X ∼ | · | X ≡ | .Proof. Let x ∼ y and x ≡ y . Then xy = yy = xx and left cancellation yields x = y . The inequalities(i) and (ii) are then immediate consequences. Finally, | X | = P x ∈ X ∼ | [ x ] ∼ | ≤ | X ∼ | · | X ≡ | by (ii). (cid:3) In general left quasigroups, neither of the two equivalences is a congruence. In finite twistedWard left quasigroups, the Cayley kernel is not always a congruence (cf. Example 4.2) but ≡ is acongruence (cf. Proposition 4.3). We do not know whether ≡ is a congruence in infinite Ward leftquasigroups, too. Example 4.2.
In the twisted Ward left quasigroup with multiplication table1 2 3 41 1 3 2 42 1 3 2 43 4 2 3 14 4 2 3 1the Cayley kernel is not a congruence: L = L and L · = L = L = L · . Proposition 4.3.
In a finite twisted Ward left quasigroup, the equivalence ≡ is a congruence.Proof. Since ( xz )( xz ) does not depend on x by (tW), we always have xz ≡ yz . If x ≡ y then( zx )( zx ) = ( xx )( xx ) = ( yy )( yy ) = ( zy )( zy ) by (tW), and hence zx ≡ zy . By finiteness, ≡ isinvariant under left division, too. (cid:3) e proceed towards a classification of twisted Ward left quasigroups of prime order. Lemma 4.4.
Let ( X, · ) be a twisted Ward left quasigroup and x, y ∈ X . If L x , L y agree at a pointthen L x = L y . In particular, every finite faithful twisted Ward left quasigroup is a quasigroup.Proof. Assume that xc = yc for some c ∈ X . Then for every z ∈ X we have ( xc )( xz ) = ( cc )( cz ) =( yc )( yz ) = ( xc )( yz ) and we obtain xz = yz by left cancellation. (cid:3) Proposition 4.5.
Let ( X, · ) be a twisted Ward left quasigroup. Then the following conditions holdfor every x ∈ X : (i) | [ x ] ≡ | = | X ∼ | , (ii) if X is finite then | [ x ] ∼ | = | X ≡ | , (iii) | X | = | X ≡ | · | X ∼ | .Proof. (i) Fix z ∈ X and observe that for every x ∈ X we have ( xz )( xz ) = ( zz )( zz ). Hence theelements R z ( X ) = { xz : x ∈ X } in the column indexed by z all have the same square. By Lemma4.4, the cardinality of R z ( X ) is equal to | X ∼ | . Hence | [ u ] ≡ | ≥ | X ∼ | for every u ∈ R z ( x ), and Lemma4.1(i) asserts equality. By varying z , we will encounter all elements of X in this fashion.(iii) By (i), all blocks of ≡ have the same size | X ∼ | , so | X | = | X ≡ | · | X ∼ | .(ii) By Lemma 4.1(ii) and by part (iii), we have | X | = P x ∈ X ∼ | [ x ] ∼ | ≤ | X ∼ | · | X ≡ | = | X | . Hencewe have an equality and | [ x ] ∼ | = | X ≡ | follows because | [ x ] ∼ | ≤ | X ≡ | for every x ∈ X . (cid:3) Example 4.6.
The sets X ∼ , X ≡ may have different cardinalities. For instance, in the twistedWard left quasigroup X with multiplication table1 2 3 4 5 61 2 1 4 3 5 62 3 4 1 2 6 53 2 1 4 3 5 64 3 4 1 2 6 55 2 1 4 3 5 66 3 4 1 2 6 5we have | [ x ] ∼ | = 3 and | [ x ] ≡ | = 2 for every x ∈ X . Note that X is neither permutational nor aquasigroup. Theorem 4.7.
Every twisted Ward left quasigroup of prime order is either permutational or aquasigroup.Proof.
Let X be a twisted Ward left quasigroup of prime order. By Proposition 4.5, all equivalenceclasses of ∼ have the same cardinality. Since X is of prime order, it follows that either ∼ hasa single equivalence class or all equivalence classes of ∼ are singletons. In the former case, X ispermutational. In the latter case, X is faithful and thus a quasigroup by Lemma 4.4. (cid:3) Corollary 4.8.
Let q ( n ) (resp. ℓ ( n ) ) denote the number of twisted Ward quasigroups (resp. twistedWard left quasigroups) of order n up to isomorphism. Let p ( n ) be the partition number, i.e., thenumber of ways in which n can be written as a sum of nonincreasing positive integers. Then ℓ ( n ) ≥ q ( n ) + p ( n ) if n > . Moreover, if n is prime then ℓ ( n ) = q ( n ) + p ( n ) = n − p ( n ) .Proof. Denote by (
X, f ) the permutational (Ward) left quasigroup with multiplication given by x ∗ y = f ( y ). It is easy to see that ( X, f ) is isomorphic to (
X, g ) if and only if f and g are conjugatein the symmetric group S X . Recall that there are p ( | X | ) conjugacy classes in S X . Moreover, if | X | > X, f ) is never a quasigroup. Hence ℓ ( n ) ≥ q ( n ) + p ( n ) if n > n is prime. By Theorem 4.7, ℓ ( n ) = q ( n ) + p ( n ). The only group of order n isthe cyclic group C n . Since Aut( C n ) is an abelian group of order n −
1, we have q ( n ) = n − (cid:3) he following table summarizes the numbers of twisted Ward left quasigroups ℓ ( n ) and twistedWard quasigroups q ( n ) of order n ≤
11 up to isomorphism, as well as the partition number p ( n ). n ℓ ( n ) 1 3 5 14 11 31 21 93 64 ? 66 q ( n ) 1 1 2 5 4 5 6 25 14 9 10 p ( n ) 1 2 3 5 7 11 15 22 30 42 56Neither of the sequences ( ℓ ( n )), ( q ( n )) appears in the Online Encyclopedia of Integer Sequences[16]. It is not difficult to calculate the numbers q ( n ) for small values of n by hand, using Corollary3.9. We can then calculate ℓ ( n ) for prime orders n from Corollary 4.8. The remaining values ℓ ( n )(and for independent verification also all other values except for ℓ (10) and ℓ (11)) were calculatedby the finite model builder Mace4 [14].We conclude the paper with a construction that yields all finite twisted left Ward quasigroupsin principle.
Proposition 4.9.
Let X and A be sets. For every x ∈ X , let f x be a bijection on X × A and letus write f x ( y, b ) = ( f [1] x ( y, b ) , f [2] x ( y, b )) . Define ( X × A, ∗ ) by ( x, a ) ∗ ( y, b ) = f x ( y, b ) . Then ( X × A, ∗ ) is a twisted Ward left quasigroup if and only if f f [1] x ( y,b ) f x (4.1) is independent of x . Moreover, every finite twisted Ward left quasigroup is isomorphic to one ofthis form.Proof. The groupoid ( X × A, ∗ ) is a left quasigroup. The identity (tW) requires that the expression(( x, a ) ∗ ( y, b )) ∗ (( x, a ) ∗ ( z, c )) is idependent of ( x, a ). Expanding the expression, we obtain f x ( y, b ) ∗ f x ( z, c ) = ( f [1] x ( y, b ) , f [2] x ( y, b )) ∗ f x ( z, c ) = f f [1] x ( y,b ) f x ( z, c ) . Hence ( X × A, ∗ ) satisfies (tW) if and only if (4.1) is independent of x .By Proposition 4.5, the Cayley kernel of a finite twisted Ward left quasigroup W has blocks ofthe same size, say each bijectively mapped onto a fixed set A . We can then represent the underlyingset of W as X × A for a suitable set X . The product ( x, a ) and ( y, b ) in W depends only on x , y and b . Moreover, for a fixed ( x, a ), the left translation by ( x, a ) in W is a bijection of X × A depending on x only, and this is how we obtain the mappings f x . (cid:3) References [1] N. Andruskiewitsch and M. Gra˜na,
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