A Survey of Racks and Quandles: Some recent developments
aa r X i v : . [ m a t h . R A ] O c t A SURVEY OF RACKS AND QUANDLES: SOME RECENTDEVELOPMENTS.
MOHAMED ELHAMDADI
Abstract.
This short survey contains some recent developments of thealgebraic theory of racks and quandles. We report on some elements ofrepresentation theory of quandles and ring theoretic approach to quan-dles. Introduction
The intent of this article is to summarize some recent progress on thetheory of quandles and racks from an algebraic point of view. Quandlesare in general non-associative structures whose axioms correspond to thealgebraic axiomatization of the three Reidemeister moves in knot theory [13].Quandles and racks appeared in the literature with many different names. In1942 Mituhisa Takasaki [27] introduced the notion of kei as an abstraction ofthe notion of symmetric transformation. The earliest known work on racksis contained in the 1959 correspondence between John Conway and GavinWraith who studied racks in the context of the conjugation operation in agroup. Around 1982, Joyce [20] and Matveev [25] introduced independentlythe notion of a quandle. Joyce and Matveev associated to each orientedknot a certain quandle in such a way that one of the main problems inknot theory, the problem of equivalence of knots, is transformed into theproblem of isomorphism of quandles. Since then, quandles have been ofmuch interest to topologists as well as algebraists. A Cohomology theoryfor racks was introduced in [18] and was extended to a quandle cohomologyin [8]. This quandle cohomology was used to define state-sum invariant forknots and paved the road to prove some properties of knots and knottedsurfaces, such as invertibility [7, 8] and the minimal triple point numbersof knotted surfaces [26]. Quandles were also investigated from an algebraicpoint of view and relations to other algebraic structures such as Lie algebras[4, 5], Frobenius algebras and Yang-Baxter equation [6], Hopf algebras [2, 5],quasigroups and Moufang loops [15], representation theory [11] and ringtheory [3]. For more details on racks, quandles and some other relatedstructures (birack, biquandles) we refer the reader to [13].The outline of this article is as follows. In Section 2, we review thebasics of quandles and give examples. Section 3 deals with some notionsof representation theory of quandles. Precisely, we review the concept offinitely stable quandles, show the existence of non-trivial stabilizing familiesfor the core quandle
Core ( G ) and discuss strong irreducible representations.In section 4 we relate quandles to ring theory by focusing on the quandle ringof quandles. We present the key elements to show that non trivial quandles are not power associative and we survey doubly transitive quandles anddiscuss the problem of isomorphism of quandle rings.2. Background on Racks and Quandles
Since this paper is meant to be a short overview, we will not discuss herethe use of quandles in low dimensional topology and knot theory. We referthe interested reader, for example, to the book [13].Let ( X, ⊲ ) be a set with a binary operation. If the set X is closed under theoperation ⊲ , that is for all x, y in X , x ⊲ y ∈ X , then ( X, ⊲ ) is called a magma .For x ∈ X , the right multiplication by x is the map R x : X → X given by R x ( u ) = u ⊲ x, ∀ u ∈ X . Now we give the following definition of a rack Definition 2.1. [13] A rack is a magma ( X, ⊲ ) such that for all x in X , theright multiplication R x is an automorphism of ( X, ⊲ ) .If in addition every element is idempotent, then ( X, ⊲ ) is called a quandle .The following are some examples of racks and quandles Example . (1) A rack X is trivial if ∀ x ∈ X , R x is the identity map.(2) For any abelian group G , the operation x ⊲ y = − x defines aquandle structure on G called Takasaki quandle. In particular, if G = Z n (integers modulo n ), it is called dihedral quandle and denotedby R n .(3) Let X be a module over the ring Λ = Z n [ t ± , s ] / ( s − ( − t ) s ) . Then X is a rack with operation x ⊲ y = tx + sy . If t + s = , then this rackis not a quandle. But if s = − t , then this rack becomes a quandlecalled Alexander quandle (also called affine quandle).(4) The conjugation x ⊲ y = yxy − in a group G makes it into a quandle,denoted Conj ( G ) .(5) The operation x ⊲ y = yx − y in a group G makes it into a quandle,denoted Core ( G ) .(6) Let G be a group and let ψ be an automorphism of G , then theoperation x ⊲ y = ψ ( xy − ) y defines a quandle structure on G . Fur-thermore, if H is a subgroup of G such that ψ ( h ) = h, for all h ∈ H ,then the operation Hx ⊲ Hy = Hψ ( xy − ) y gives a quandle structureon G/H .For each x ∈ X , the left multiplication by x is the map denoted by L x : X → X and given by L x ( y ) := x ⊲ y . A function f : ( X, ⊲ ) → ( Y, ⋄ ) is aquandle homomorphism if for all x, y ∈ X, f ( x ⊲ y ) = f ( x ) ⋄ f ( y ) . Given aquandle ( X, ⊲ ) , we will denote by Aut(X) the automorphism group of X . Thesubgroup of Aut(X), generated by the automorphisms R x , is called the inner automorphism group of X and denoted by Inn ( X ) . The subgroup of Aut(X),generated by R x R − , for all x, y ∈ X , is called the transvection groupof X denoted by Transv ( X ) . It is well known [20] that the transvection group is a normal subgroup of the inner group and the latter group is anormal subgroup of the automorphism group of X . The quotient groupInn ( X ) / Transv ( X ) is a cyclic group. To every quandle, there is a groupassociated to it which has a universal property and called enveloping groupof the quandle. The following is the definition. SURVEY OF RACKS AND QUANDLES: SOME RECENT DEVELOPMENTS. 3
Definition . Let ( X, ⊲ ) be a quandle. The enveloping group of X is definedby G X = F ( X ) / < x ∗ y = yxy − , x, y ∈ X > , where F ( X ) denotes the freegroup generated by X .Note that there is a natural map ι : X − → G X . In the following, we showa functoriality property of this group. Proposition . [17] Let ( X, ⊲ ) be a quandle and G be any group. Given anyquandle homomorphism ϕ : X − → Conj ( G ) . Then, there exists a uniquegroup homomorphism e ϕ : G X → G which makes the following diagram com-mutative X ι / / ϕ (cid:15) (cid:15) G X e ϕ (cid:15) (cid:15) Conj ( G ) id / / G Thus one obtains the natural identification expressing that the functor en-veloping from the category of quandles to the category of groups is a left ad-joint to the functor conjugation:
Hom qdle ( X, Conj ( G )) ∼ = Hom grp ( G X , G ) .The following are some properties and definitions of some quandles . • A quandle X is involutive , or a kei , if, ∀ x ∈ X, R x is an involution. • A quandle is connected if Inn ( X ) acts transitively on X . • A quandle is faithful if x → R x is an injective mapping from X toInn ( X ) . • A Latin quandle is a quandle such that for each x ∈ X , the leftmultiplication L x by x is a bijection. That is, the multiplicationtable of the quandle is a Latin square. • A quandle X is medial if ( x ⊲ y ) ⊲ ( z ⊲ w ) = ( x ⊲ z ) ⊲ ( y ⊲ w ) for all x, y, z, w ∈ X . It is well known that a quandle is medial if and only ifits tranvection group is abelian. For this reason, sometimes medialquandles are called abelian . For example, every Alexander quandleis medial. • A quandle X is called simple if the only surjective quandle homo-morphisms on X have trivial image or are bijective.3. Elements of Representation Theory of Quandles
Recently, elements of representation theory of racks and quandles wereintroduced in [11]. A certain class of quandles called finitely stable quandles was introduced as a generalization of the notion of a stabilizer defined in [12].A stabilizer in a quandle X is an element x ∈ X such that R x is the identitymap. From the definition of the quandle operation of Conj ( G ) , one seesimmediately that an element of G is a stabilizer if and only if it belongsto the center Z ( G ) of G . On the other hand, an element x ∈ Core ( G ) isa stabilizer if and only if ∀ y ∈ G, ( xy − ) = , and thus the core quandle Core ( G ) of a group G has no stabilizers if G has no non-trivial –torsions.This observation suggested that the property of having stabilizers is toostrong to capture the identity and center of a group in the category ofquandles. This property was then weakened to give the notion of stabilizingfamilies . MOHAMED ELHAMDADI
Finitely Stable Quandles.
First we recall the following definition.
Definition 3.1. [11] Let X be a rack or quandle. A stabilizing family oforder n (or n – stabilzer ) for X is a finite subset { x , . . . , x n } of X such thatthe product R x n R x n − . . . R x is the identity element in the group Inn ( X ) .For any positive integer n , we define the n –center of X , denoted by S n ( X ) ,to be the collection of all stabilizing families of order n for X . The collection S ( X ) := S n ∈ N S n ( X ) of all stabilizing families for X is called the center ofthe quandle X . Now we state the definition of finitely stable quandle. Definition 3.2. [11] A quandle X is said to be finitely stable if S ( X ) isnon-empty. It will be called n – stable if it has a stabilizing family of order n . Remark . Note that if every element of X is an n –stabilzer, then werecover the definition of an n – quandle of Joyce [20], Definition 1.5, page 39. Example . From Definition 3.2, one sees that any finite quandle X isfinitely stable. Example . Consider the integers Z with the quandle structure x ⊲ y = − x, x, y ∈ Z . Given any family { x , . . . , x n } of integers we get the formula y ⊲ ( x i ) ni = = n − X i = (− ) i x n − i + (− ) n y, ∀ y ∈ Z . One then obtain the stabilizing family { x , x , · · · , x n , x n } of order andthus Z admits an infinitely many stabilizing families of even orders.Now we show how to obtain non-trivial stabilizing families of the corequandle Core ( G ) . Proposition 3.6. [11]
Let G be a non-trivial group. Then for all evennatural number ≤ | G | , there exists a non-trivial stabilizing family of order for the core quandle Core ( G ) . In particular, if G is infinite, Core ( G ) admits infinitely many stabilizing families. For the stabilizing families of odd order of the core quandle, we have thefollowing characterization.
Theorem 3.7. [11]
Let G be a group. The core quandle of G is ( + ) –stable if and only if all elements of G are –torsions; in other words, G isisomorphic to L i ∈ I Z , for a certain finite or infinite set I . Sometimes, a quandle may have no stabilizing family at all as shown inthe following example.
Example . Let V be a vector space equipped with the quandle structure x ⊲ y = x + y2 . Then S ( V ) = ∅ .Let G be a group and n a positive integer. The n – core of G was definedby Joyce [20] as the subset of the Cartesian product G n consisting of all SURVEY OF RACKS AND QUANDLES: SOME RECENT DEVELOPMENTS. 5 tuples ( x , . . . , x n ) such that x · · · x n = . Moreover, the n -core of G has anatural quandle structure defined by the formula ( x , . . . , x n ) ⊲ ( y , . . . , y n ) := ( y − x n y , y − x y , . . . , y − − x n − y n ) (1)A notion of the n - pivot of a group as a generalization of the n -core isgiven by the following definition. Definition 3.9.
Let G be a group and let Z ( G ) be its center. For n ∈ N ,we define the n -pivot of G to be the subset of G n given by P n ( G ) = { ( x , . . . , x n ) ∈ G n | x · · · x n ∈ Z ( G ) } . It is straightforward to see that the formula (1) defines a quandle structureon P n ( G ) . Moreover, we have the following proposition. Proposition 3.10. ([11])
Let G be a non-trivial group. Then for all n ∈ N ,there is a bijection S n ( Conj ( G )) ∼ = P n ( G ) . In particular, S n ( Conj ( G )) is naturally equipped with a quandle structure.Furthermore, if G has a trivial center, then S ( Cong ( G )) = Core ( G ) .Example . If G has trivial center, then for all n ≥ , S n ( Conj ( G )) coincides with the n –core of G . In particular S ( Cong ( G )) = Core ( G ) .3.2. Quandle Actions and Representations.
Recall that in [12] a rackaction of a rack X on a space M consists of a map M × X ∋ ( m, x ) − → m · x ∈ M such that(I) ∀ x ∈ X , the map m − → m · x is a bijection of M ; and(II) ∀ m ∈ M, x, y ∈ X, we have ( m · x ) · y = ( m · y ) · ( x ⊲ y ) . (2)If { x i } si = is a family of elements in X , we will write m · ( x i ) i for ( · · · ( m · x ) · · · · ) · x s . We will require two additional axioms that generalize, in an appropriateway, the concept of group action. Precisely we give the following.
Definition 3.12. ([11]) An action of the rack ( X, ⊲ ) on the space M consistsof a map M × X ∋ ( m, x ) − → m · x ∈ M satisfying equation (2) and suchthat for all { u , . . . , u s } ∈ S ( X ) , m · ( u i ) i = m · ( u σ ( i ) ) i , ∀ m ∈ M, (3)for all cycle σ in the subgroup of S n generated by the cycle ( ) . Example . Let G be a group acting (on the right) on a space M . Wethen get a rack action of Conj ( G ) on M by setting m · g := mg − . It is easyto check that for all { g i } i ∈ P n ( G ) , we have m · ( g i ) i = m · ( g σ ( i ) ) i , for all m ∈ M and σ in the subgroup generated by the cycle ( ) . Notice however that a rack action of
Conj ( G ) does not necessarily definea group action of G . MOHAMED ELHAMDADI
Definition 3.14.
Let X be a rack acting on a non-empty set M and let m ∈ M . For { x i } si = ∈ X , we define:(i) the orbit of { x i } i as M · ( x i ) i = { m · ( x i ) si = | m ∈ M } , and(ii) the stabilizer of m as X [ m ] = { x ∈ X | m · x = m } .Then, it follows immediately from equation (2), that all stabilizers aresubquandles of X . Definition 3.15.
A quandle action of X on a nonempty set M is faithful iffor any m ∈ M , the map ϕ m : X → M sending x to m · x is injective, i.e. m · x = m · y for x = y . Lemma 3.16. ([11])
Let X be a nonempty set. A quandle structure ⊲ on X is trivial if and only if ( X, ⊲ ) acts faithfully on a nonempty set M and theaction satisfies ( m · x ) · y = ( m · y ) · x for all x, y ∈ X, m ∈ M . Now we give a definition of various properties of the action (see [11]).
Definition 3.17.
Let X be a quandle acting on a non-empty set M .(i) An approximate unit for the quandle action is a finite subset { t i } ri = ⊂ X such that m · ( t i ) i = m for all m ∈ M .(ii) An element t ∈ X is an r – unit for the quandle action if the family { t } ri = is an approximate unit for the rack action.(iii) The quandle action of X on M is called r –periodic if each t ∈ X isan r –unit for the quandle action.(iv) A quandle action is said to be strong if every stabilizing family ofthe quandle is an approximate unit. Example . Let ( X, ⊲ ) be a quandle. Then we get a rack action of X onits underlying set by defining m · x := m ⊲ x, m, x ∈ X . It is immediate thata finite subset { t i } i ⊂ X is an approximate unit for this quandle action ifand only if it is a stabilizing family for the quandle structure, hence it is astrong action. Furthermore, this action is n –periodic if and only if X is an n –quandle in the sense of Joyce [20], Definition 1.5, p 39.Now we define the notion of representation of a quandle (see [11]). Definition 3.19. A representation of a quandle X consists of a vector space V and a quandle homomorphism ρ : X − → Conj ( GL ( V )) . In other words, wehave ∀ x, y ∈ X, ρ x ⊲ y = ρ y ρ x ρ − , where the isomorphim ρ ( x ) of the vectorspace V is denoted by ρ x . Example . Any representation ( V, ρ ) of a group G induces, in a naturalway, a representation of the quandle Conj ( G ) .Let us give an example of quandle representation analogous to the regularrepresentation in group theory. Example . Let X be a finite quandle and let C [ X ] be the vector spaceof complex valued functions on X , seen as the space of formal sums f = P x ∈ X a x x , where a x ∈ C , x ∈ X . We now construct the regular representa-tion of a quandle Xρ : X − → Conj ( GL ( C [ X ])) , given by ρ x ( f )( y ) := f ( R − ( y )) . SURVEY OF RACKS AND QUANDLES: SOME RECENT DEVELOPMENTS. 7
Definition 3.22. ([11]) Let V and W be representations of the quandle X .A linear map φ : V − → W is called an intertwining map of representationsif for each x ∈ X the following diagram commutes V ρ Vx / / φ (cid:15) (cid:15) V φ (cid:15) (cid:15) W ρ Wx / / W. When φ is an isomorphism then it is called an equivalence of quandlerepresentations . If V is a representation of the quandle X , a subspace W ⊂ V is called a subrepresentation if for any finite family { x i } i ⊆ X , W · ( x i ) i ⊆ W . Definition 3.23.
A quandle representation V of X is said to be irreducible if it has no proper subrepresentations.3.3. Strong Representations.Definition 3.24. [11] A strong representation of X is a representation V such that the quandle action is strong.We denote by Rep s ( X ) the set of equivalence classes of strong irreduciblefinite dimensional representations of X .One can check that as in the group case, Rep s ( X ) is an abelian groupunder tensor product of strong irreducible representations. Remark . In view of Proposition 3.10, we notice that if G is an abeliangroup, the only strong representation of Conj ( G ) is the trivial one. Example . The regular representation ( C [ X ] , ρ ) of a rack X , defined inExample 3.21, is clearly strong. Example . Let ( Z = {
1, 2, 3 } , ⊲ ) , with x ⊲ y = − x , be the dihedralquandle. Define ρ : Z − → Conj ( GL ( C )) as the rack representation inducedby the reflections on C = Span { e , e , e } ρ = ( ) , ρ = ( ) , and ρ = ( ) . Then ρ is a strong representation of Z as a quandle, although it is clearlynot a group representation. Note, however, that this is a reducible represen-tation since we have the following complete decomposition into irreduciblesubspaces C [ X ] = C < e + e + e > ⊕ C < e − e , e − e > . Example . If X is an involutive quandle then, every x ∈ X is a –stabilizerfor X . Then every pair ( V, τ ) , where V is a vector space and τ : V − → V is alinear involution, gives rise to a strong representation ˜ τ : X − → Conj ( GL ( V )) by setting ˜ τ x ( v ) = τ ( v ) for all x ∈ X, v ∈ V . Proposition 3.29. [11]
Suppose the quandle X is finite, involutive, andconnected. Then, the regular representation of X corresponds to a conjugacyclass of the symmetric group S n where n = X . More generally, if X has k connected components, then the regular representation corresponds to k conjugacy classes in S n . Theorem 3.30.
Let X be a finite quandle. Then every irreducible strongrepresentation of X is either trivial or finite dimensional. MOHAMED ELHAMDADI
We shall observe that in the case of involutive quandles the above re-sult becomes more precise with regard to the dimension of the irreduciblerepresentations. We then have the following theorem (see [11])
Theorem 3.31.
Every irreducible strong representation of an involutiveconnected finite quandle X is one-dimensional. Quandles and Ring Theory
In [3], a theory for quandle rings was proposed for quandles by analogyto the theory of group rings for groups. In [10], quandles were investigatedfrom the point of view of ring theory. Precisely, some basic properties ofquandle rings were investigated and some open questions which were raisedin [3] were solved. In particular, examples of quandles were provided forwhich the quandle rings k [ X ] and k [ Y ] are isomorphic, but the quandles X and Y are not isomorphic.4.1. Power Associativity of Quandle Rings.
We associate to everyquandle ( X, ⊲ ) and an associative ring k with unity, a nonassociative ring k [ X ] . First, we start by recalling the following Definition and Proposition. Definition and Proposition 4.1.
Let ( X, ⊲ ) be a quandle and k be anassociative ring with unity. Let k [ X ] be the set of elements that are uniquelyexpressible in the form P x ∈ X a x x , where x ∈ X and a x = for almost all x . Then the set k [ X ] becomes a ring with the natural addition and themultiplication given by the following, where x, y ∈ X and a x , a y ∈ k , ( X x ∈ X a x x ) · ( X y ∈ X b y y ) = X x,y ∈ X a x b y ( x ⊲ y ) . In [3], power associativity of dihedral quandles was investigated and thequestion of determining the conditions under which the quandle ring R [ X ] is power associative was raised. In this section we give a complete solutionto this question. Precisely, we prove that quandle rings are never powerassociative when the quandle is non-trivial and the ring has characteristiczero.Let first let’s recall the following definition from [1] Definition 4.2.
A ring k in which every element generates an associativesubring is called a power-associative ring. Example . Any alternative algebra is power associative. Recall that analgebra A is called alternative if x · ( x · y ) = ( x · x ) · y and x · ( y · y ) =( x · y ) · y, ∀ x, y ∈ A , (for more details see [16]).In non-associative algebras, the product x · x is also denoted by x , as inthe associative case.It is well known [1] that a ring k of characteristic zero is power-associativeif and only if, for all x ∈ k , x · x = x · x , (4)and x · x = ( x · x ) · x. (5) SURVEY OF RACKS AND QUANDLES: SOME RECENT DEVELOPMENTS. 9
Theorem 4.4. [10]
Let k be a ring of characteristic zero and let ( X, ⊲ ) bea non-trivial quandle. Then the quandle ring k [ X ] is not power associative. The proof of this theorem is based on the use of the two identities (4)and (5). The proof consists of two steps:(1) show that if there exist x, y ∈ X such that x = y and x ⊲ y = y ⊲ x ,then k [ X ] is not power associative.(2) Assume ∀ x, y ∈ X such that x = y , we have x ⊲ y = y ⊲ x . If k [ X ] ispower associative, then x ⊲ y = x .A complete proof can be found in [10].4.2. Higher Transitivity in Groups and Quandles.
First we recall thenotion of n -transitivity of groups [9] and use it in the context of quandles.Let G be a group acting (the right) on a set X , thus G acts naturally on anycartesian product X n of X by ( x , · · · , x n ) g = ( x g, · · · , x n g ) . Note that thesubset X n := { ( x , · · · , x n ) ∈ X n | ∀ i = j, x i = x j } , of X n , is G -invariant. Wethen state the following definition of n -transitive action. Definition 4.5. [9] Let G be a group, X be a non-empty set and n be apositive integer. The group G acts n -transitively on X if it acts transitivelyon the subset X n of X n .It is clear from this definition that an n -transitive action implies an ( n − ) -transitive action. It is also well known that Sym ( X ) acts n -transitively on X when n ≤ X .Now we state the following definition of n -transitive quandles. Definition 4.6. [23] A finite quandle X , with cardinality | X | > 2 , is called n -transitive if the action of the inner group Inn ( X ) is n -transitive action on X . The following is the definition of primitive action. Definition 4.7. [9] Let G be a group acting on (the right) on a finite non-empty set X . We say that G preserve an equivalence relation ∼ on X if thefollowing condition is satisfied: x ∼ y if and only if xg ∼ yg. The action of G on X is said to be primitive if the group G acts transitivelyon X and if G preserves no non-trivial partition of X .It is well known [9] that any -transitive group is primitive, but the con-verse is not true in general. Corollary 1.5A on page 14 of [9] states that agroup G is primitive if and only if each point stabilizer is a maximal subgroupof G . Thus, the study of finite primitive permutation groups is equivalentto the study of the maximal subgroups of finite groups.McCarron showed in Proposition 5 of [23] that if n ≥ and X is a finite n -transitive quandle with at least four elements, then n = . Furthermore, heshowed that the dihedral quandle of three elements is the only -transitivequandle, and there are no -transitive quandles with at least four elements.Now we turn to the notion of quandles of cyclic type and we will give theirrelastionship to -transitive quandles (called also two point-homogeneous in[28]). Definition 4.8.
A finite quandle X is of cyclic type if for each x ∈ X, thepermutation R x acts on X \ { x } as a cycle of length | X | − , where | X | > 2 denotes the cardinality of X. Example . The following list of quandles of cyclic type is taken from table1 in [21]. • Among the quandles of order three, the dihedral quandle is the onlyone of cyclic type. • Among the quandles of order four, the quandle X = {
1, 2, 3, 4 } with R = ( ) , R = ( ) , R = ( ) and R = ( ) is the only oneof cyclic type. • On the set of five elements {
1, 2, 3, 4, 5 } , there are exactly two quan-dles X and Y of cyclic type: X with R = ( ) , R = ( ) , R = ( ) R = ( ) and R = ( ) , and Y with R = ( ) , R = ( ) , R = ( ) R = ( ) and R = ( ) .The classification of quandles of cyclic type was investigated in [21]. Theirmain theorem states that the isomorphism classes of cyclic quandles of order n are in one-to-one correspondence with the permutations of the symmetricgroup S n satisfying certain conditions. Furthermore, they obtained theclassification of the quandles of cyclic type with cardinality up to order .They also stated the following conjecture: "A quandle with at lease elements is of cyclic type if and only if its cardinality is a power of a prime".This conjecture was later proved in [29] (see Theorem and also Corollary ). These types of quandles appeared earlier in [22], where they were calledquandles with constant profile ( {
1, n − } , . . . , {
1, n − } ) .In [28], the author defined a quandle X to be two-point homogeneous ifthe inner automorphism group Inn ( X ) acts doubly-transitive on X . Fur-thermore, the author classified two-point homogeneous quandles of primecardinality by proving that they are linear Alexander quandles. This wasgeneralized in [29, 30] to the case of prime powers giving the following result:A finite quandle ( X, ⊲ ) is -transitive if and only if it is of cyclic type, if andonly if it is isomorphic to an Alexander quandle ( F q , ⊲ ) over the finite field F q with operation x ⊲ y = ax + ( − a ) y where a is a primitive element (i.e.a generator of the multiplicative group F × q ) .The notion of -transitive quandles was also used recently in [10] to studyquandles from a ring theoretical approach.4.3. Isomorphisms of Quandle Rings.
In [10], a notion of partition-typeof quandles was introduced and it was shown that if the quandle rings k [ X ] and k [ Y ] are isomorphic and the quandles X and Y are orbit -transitive,then X and Y are of the same partition type. Now let us recall the definitionof partition type. Definition 4.10. ([10]) Let X be a finite quandle of cardinality n . The partition type of X is λ = ( λ , . . . , λ n ) with λ j being the number of orbits ofcardinality j in X . SURVEY OF RACKS AND QUANDLES: SOME RECENT DEVELOPMENTS. 11
Example . Let X = {
1, 2, 3, 3, 5, 6, 7 } be a quandle with the orbit de-composition X = { } ∐ {
2, 3 } ∐ {
4, 5, 6 } ∐ { } . Then the partition type of thequandle X is λ = (
2, 1, 1, 0, . . . ) .In order to state the next result about the same partition type for quan-dles with isomorphic quandle rings, we need the following theorem whichgives the list of subgroups G S n for which the representation V st is ir-reducible, where V st ⊂ V = k [ X ] be the subspace orthogonal to the vector v triv = P x ∈ X i x . The groups ( i ) − ( v ) are respectively the affine and projectivegeneral semilinear groups, projective semilinear unitary groups, Suzuki andRee groups. We refer the reader to [9, Chapter 7] for definitions of thesegroups. Theorem 4.12. [10](a) If char ( k ) = then V st is an irreducible representation of the sub-group G < S n if and only if G is -transitive and A n G . (b) In case char ( k ) = p > 3 then V st is an irreducible representationof the subgroup G < S n if and only if G is -transitive and A n G except (i) G AΓL ( m, q ) , and p divides q ; (ii) G PΓL ( m, q ) , m ≥ and p divides q ; (iii) G PΓU (
3, q ) , and p divides q + ; (iv) G Sz ( q ) , and p divides q + + m , where m = ; (v) G Re ( q ) , and p divides ( q + )( q + + m ) , where m = . The following theorem was proved in [10].
Theorem 4.13.
Assume char ( k ) =
2, 3 . If the quandle rings k [ X ] and k [ Y ] are isomorphic and the quandles X and Y are orbit -transitive ( G X i ’s are notamong the groups from ( i ) − ( v ) in Theorem 4.12 in case char ( k ) = p > 3 ),then X and Y are of the same partition type. The following two examples were provided in [10] as answers to
Question of [3] on the existence of two nonisomorphic quandles X and Y withisomorphic quandle rings.
Example . Let k be a field with char ( k ) = and let X = Y = {
1, 2, 3, 4 } as sets. Define a quandle structure on the set X by R and R are the identitymaps and R and R are both equal to the transposition ( ) . Define aquandle operation on the set Y by R being equal to the transposition ( ) and all others R i are the identity map. Note that the orbit decomposition {
1, 2 } ∐ { } ∐ { } is the same for both quandles X and Y . An isomorphism f between X and Y must satisfy f ( ) ∈ {
1, 2 } . The sets { x, 1 ⊲ x = } and { x, 2 ⊲ x = } have cardinality as subsets of X but cardinality as subsetsof Y , thus the quandles X and Y are not isomorphic. The ring isomorphismis given by ϕ : k [ X ] ∼ → k [ Y ] , where ϕ = . Example . Let k be a field with characteristic zero and assume X = Y = {
1, 2, 3, 4, 5, 6, 7 } as sets. Define a quandle structure on the set X by R = ( ) , R = ( )( ) and all the other R i , i =
5, 6, are the identitymap. Define a quandle operation on the set Y by R = ( ) , R = ( ) andall the other R i , i =
5, 6, are the identity map. A similar argument to theone given in the previous example shows that X and Y are not isomorphicas quandles. One family of ring isomorphisms is given by ϕ : k [ X ] ∼ → k [ Y ] ,where ϕ = − . There is an easy way to obtain other examples from the one we justprovided (see [10] for more details).
Acknowledgement:
The author would like to thank Neranga Fernando,El-kaïoum M. Moutuou and Boris Tsvelikhovskiy for many fruitful sugges-tions which improved the paper.
References [1] A. A. Albert,
Power-associative rings , Trans. Amer. Math. Soc. (1948), 552–593.[2] Nicolás Andruskiewitsch and Matías Graña, From racks to pointed Hopf algebras , Adv.Math. (2003), no. 2, 177–243.[3] V.G. Bardakov, I.B.S. Passi, and M. Singh,
Quandle rings , J. Algebra Appl. (2019),1950157 (23 pages).[4] J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi, and Masahico Saito,
Coho-mology of categorical self-distributivity , J. Homotopy Relat. Struct. (2008), no. 1,13–63.[5] , Cohomology of the adjoint of Hopf algebras , J. Gen. Lie Theory Appl. (2008),no. 1, 19–34.[6] J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi, Enver Karadayi, and MasahicoSaito, Cohomology of Frobenius algebras and the Yang-Baxter equation , Commun. Con-temp. Math. (2008), no. suppl. 1, 791–814.[7] J. Scott Carter, Mohamed Elhamdadi, Matias Graña, and Masahico Saito, Cocycleknot invariants from quandle modules and generalized quandle homology , Osaka J.Math. (2005), no. 3, 499–541.[8] J. Scott Carter, Daniel Jelsovsky, Seiichi Kamada, Laurel Langford, and MasahicoSaito, Quandle cohomology and state-sum invariants of knotted curves and surfaces ,Trans. Amer. Math. Soc. (2003), no. 10, 3947–3989.[9] John D. Dixon and Brian Mortimer,
Permutation groups , Graduate Texts in Mathe-matics, vol. 163, Springer-Verlag, New York, 1996.[10] Mohamed Elhamdadi, Neranga Fernando, and Boris Tsvelikhovskiy,
Ring theoretic as-pects of quandles , J. Algebra (2019), 166–187, doi: 10.1016/j.jalgebra.2019.02.011.MR3915329[11] Mohamed Elhamdadi and El-kaïoum M. Moutuou,
Finitely stable racks and rackrepresentations , Comm. Algebra (2018), no. 11, 4787–4802.[12] Mohamed Elhamdadi and El-Kaïoum M. Moutuou, Foundations of topological racksand quandles , J. Knot Theory Ramifications (2016), no. 3, 1640002, 17.[13] Mohamed Elhamdadi and Sam Nelson, Quandles—an introduction to the algebra ofknots , Student Mathematical Library, vol. 74, American Mathematical Society, Provi-dence, RI, 2015.[14] Mohamed Elhamdadi, Jennifer Macquarrie, and Ricardo Restrepo,
Automorphismgroups of quandles , J. Algebra Appl. (2012), no. 1, 1250008, 9. SURVEY OF RACKS AND QUANDLES: SOME RECENT DEVELOPMENTS. 13 [15] Mohamed Elhamdadi,
Distributivity in quandles and quasigroups , Algebra, geometryand mathematical physics, Springer Proc. Math. Stat., vol. 85, Springer, Heidelberg,2014, pp. 325–340.[16] Mohamed Elhamdadi and Abdenacer Makhlouf,
Cohomology and formal deforma-tions of alternative algebras , J. Gen. Lie Theory Appl. (2011), Art. ID G110105,10.[17] Roger Fenn and Colin Rourke, Racks and links in codimension two , J. Knot TheoryRamifications (1992), no. 4, 343–406.[18] Roger Fenn, Colin Rourke, and Brian Sanderson, The rack space , Trans. Amer. Math.Soc. (2007), no. 2, 701–740.[19] A. Ishii, M. Iwakiri, Y. Jang, and K. Oshiro, A G -family of quandles and handlebody-knots , Illinois J. Math. (2013), no. 3, 817–838.[20] David Joyce, A classifying invariant of knots, the knot quandle , J. Pure Appl. Algebra (1982), no. 1, 37–65.[21] Seiichi Kamada, Hiroshi Tamaru, and Koshiro Wada, On classification of quandles ofcyclic type , Tokyo J. Math. (2016), no. 1, 157–171.[22] Pedro Lopes and Dennis Roseman, On finite racks and quandles , Comm. Algebra (2006), no. 1, 371–406.[23] James McCarron, Small homogeneous quandles , ISSAC 2012—Proceedings of the37th International Symposium on Symbolic and Algebraic Computation, ACM, NewYork, 2012, pp. 257–264.[24] ,
Connected quandles with order equal to twice an odd prime, arXiv:1210.2150.[25] S. V. Matveev,
Distributive groupoids in knot theory , Mat. Sb. (N.S.) (1982), no. 1, 78–88, 160 (Russian).[26] Shin Satoh and Akiko Shima,
The 2-twist-spun trefoil has the triple point numberfour , Trans. Amer. Math. Soc. (2004), no. 3, 1007–1024.[27] Mituhisa Takasaki,
Abstraction of symmetric transformations , Tôhoku Math. J. (1943), 145–207 (Japanese).[28] Hiroshi Tamaru, Two-point homogeneous quandles with prime cardinality , J. Math.Soc. Japan (2013), no. 4, 1117–1134.[29] Leandro Vendramin, Doubly transitive groups and cyclic quandles , J. Math. Soc.Japan (2017), no. 3, 1051–1057.[30] Koshiro Wada, Two-point homogeneous quandles with cardinality of prime power ,Hiroshima Math. J. (2015), no. 2, 165–174. Department of Mathematics, University of South Florida, Tampa, FL 33620,U.S.A.
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