Ternary and n-ary f-distributive Structures

Abstract We introduce and study ternary f-distributive structures, Ternary f-quandles and more generally their higher n-ary analogues. A classification of ternary f-quandles is provided in low dimensions. Moreover, we study extension theory and introduce a cohomology theory for ternary, and more generally n-ary, f-quandles. Furthermore, we give some computational examples.


Introduction
The rst instances of ternary operations appeared in the nineteenth century when Cayley considered cubic matrices. Ternary operations or more generally n-ary operations appeared naturally in various domains of theoretical and mathematical physics. The rst instances of ternary Lie algebras appeared in the Nambu's Mechanics when generalizing hamiltonian mechanics by considering more than one hamiltonian [1]. The algebraic formulation of Nambu's Mechanics was achieved by Takhtajan in [2]. Moreover, ternary algebraic structures appeared in String and Superstring theories when Basu and Harvey suggested to replace Lie algebra in the context of Nahm equations by a 3-Lie algebra. Furthermore, a ternary operation was used by Bagger-Lambert in the context of Bagger-Lambert-Gustavsson model of M2-branes and in the work of Okubo [3] on Yang-Baxter equation which gave impulse to signi cant development on n-ary algebras. In recent years, there has been a growth of interests in many generalizations of binary structures to higher n-ary contexts. In Lie algebra theory, for example, the bracket is replaced by a n-ary bracket and the Jacobi identity is replaced by its higher analogue, see [4]. Generalizations of quandles to the ternary case were done recently in [5]. One may also mention reference [6] where the author uses two ternary operators, providing a generalization of a Dehn presentation which assigns a relation to each crossing in terms of the regions of the diagram that surround the crossing. For example, by coloring the four regions respectively a, b, c and d (see gure 2 in [6]), the author obtains d as a ternary function T(a, b, c) = ab − c. This example of ternary operation was also considered in Example 2.8 in [5]. The author shows under certain conditions that ternary checkerboard colorings de ne link invariants. The paper deals with di erent algebraic structures considered in [5] and in this work.
In this paper we introduce and study a twisted version of ternary, respectively n-ary, generalizations of racks and quandles, where the structure is de ned by a ternary operation and a linear map twisting the distributive property. These type of algebraic structures, called sometimes Hom-algebras, appeared rst in quantum deformations of algebras of vector elds, motivated by physical aspects. A systematic study and mathematical aspects were provided for Lie type algebras by Hartwig-Larsson and Silvestrov in [7] whereas associative and other nonassociative algebras were discussed by the fourth author and Silvestrov in [8] and n-ary Hom-type algebras in [9]. The main feature of all these generalization is that the usual identities are twisted by a homomorphism. We introduce in this article the notions of ternary, respectively n-ary, f -shelf (resp. f -rack, f -quandle), give some key constructions and properties. Moreover, we provide a classi cation in low dimensions of f -quandles. We also study extensions and modules, as well as cohomology theory of these structures. For classical quandles theory, we refer to [10], see also [11][12][13][14][15][16][17][18][19]. For basics and some developments of Hom-type algebras, we refer to [8,[20][21][22][23][24][25][26].
This paper is organized as follows. In Section 2, we review the basics of f -quandles and ternary distributive structures and give the general n-ary setting. In Section 3, we discuss a key construction introduced by Yau, we show that given a n-ary f -shelf (resp. f -rack, f -quandle) and a shelf morphism then one constructs a new n-ary f -shelf (resp. f -rack, f -quandle) and we provide examples. In Section 4, we provide a classi cation of ternary f -quandles in low dimensions. Section 5 gives the extension theory of f -quandles and modules. Finally, in Section 6 we introduce the cohomology of n-ary f -distributive structures and give examples.

f -quandles and ternary (resp. n-ary) distributive structures
In this section we aim to introduce the notion of ternary and more generally n-ary f -quandles, generalizing the notion of f -quandle given in [27].

. A review of f -quandles and related structures
First, we review the basics of the binary f -quandles. We refer to [27] for the complete study. Classical theory of quandle could be found in De nition 2.
1. An f -shelf is a triple (X, *, f ) in which X is a set, * is a binary operation on X, and f : X → X is a map such that, for any x, y, z ∈ X, the identity -Given any set X and map f : X → X, then the operation x * y = f (x) for any x, y ∈ X gives a f -quandle. We call this a trivial f -quandle structure on X. -For any group G and any group endomorphism f of G, the operation x * y = y − xf (y) de nes a f -quandle structure on G. -Consider the Dihedral quandle Rn, where n ≥ , and let f be an automorphism of Rn. Then f is given by f (x) = ax + b, for some invertible element a ∈ Zn and some b ∈ Zn [16]. The binary operation called an Alexander f -quandle.
Remark 2.4. Axioms of De nition 2.1 give the following identity, We note that the two medial terms in this equation are swapped (resembling the mediality condition of a quandle). Note also that the mediality in the general context may not be satis ed for f -quandles. For example one can check that the f -quandle given in item (2) of Examples is not medial.

. Ternary and n-ary f -quandles
Now we introduce and discuss the analogous notion of a f -quandle in the ternary setting and more generally in the n-ary setting.
The previous condition is called right f -distributivity.
Remark 2.6. Using the diagonal map D : x, x), equation (4) can be written, as a map from Q × to Q, in the following form where id stands for the identity map. In the whole paper we denote by ρ : Q × → Q × the map de ned as ρ = p , • p , • p , where p i,j is the transposition i th and j th elements, i.e. A morphism of ternary quandles is a map ϕ : (Q, T) → (Q ′ , T ′ ) such that ϕ(T(x, y, z)) = T ′ (ϕ(x), ϕ(y), ϕ(z)).
A bijective ternary quandle endomorphism is called ternary f -quandle automorphism. Therefore, we have a category whose objects are ternary f -quandles and morphisms as de ned above. As in the case of the binary quandle there is a notion of medial ternary quandle This de nition of mediality can be written in term of the following commutative diagram We generalize the notion of ternary f -quandle to n-ary setting.
Example 2.16. Let (Q, *, f ) be an f -quandle and de ne an n-ary twisted operation on Q by where i = , · · · , n. It is straightforward to see that (Q, T, α) is an n-ary f -quandle where α = f n− .

Yau twist
The following proposition provides a way of constructing new n-ary f -shelf (resp. f -rack, f -quandle) along a shelf morphism. In particular, given an n-ary shelf (resp. rack, quandle) and a shelf morphism, one may obtain an n-ary f -shelf (resp. f -rack, f -quandle). Recall that this construction was introduced rst by Yau to deform a Lie algebra to a Hom-Lie algebra along a Lie algebra morphism. It was generalized to di erent situation, in particular to n-ary algebras in [9].

Example 3.2. Let (Q, *) be a quandle and de ne a ternary operation on Q by T(x, y, z)
is a ternary f -quandle. Example 3.6. Let (Q, *) be a quandle and de ne a n-ary operation on Q by

Classi cation of ternary f -quandles of low orders
We developed a simple program to compute all ternary f -quandles of orders 2 and 3. The results of which we used to obtain the complete list of isomorphism classes.
In the case of order 3, we found a total of 84 distinct isomorphism classes, including 30 ternary quandles (those such that f = id Q ).
Since for each xed a, b, the map x → τ(x, a, b) is a permutation, in the following table we describe all ternary f -quandles of order three in terms of the columns of the Cayley table. Each column is a permutation of the elements and is described in standard notation, that is by explicitly writing it in terms of products of disjoint cycles. Thus for a given z we give the permutations resulting from xing y = , , . For example, the ternary set τ (x, y, z) with the Cayley Table 1 will be represented with the permutations ( ), ( ), ( ); ( ), ( ), ( ); ( ), ( ), ( ). This will appear in Table 3 as shown in Table 2.    Table 4 lists the isomorphism classes, the rst table lists those that such that f = id Q , and the second table lists classes with members with a non-trivial twisting.

Extensions of f -quandles and modules
In this section we investigate extensions of ternary f -quandles. We de ne generalized ternary f -quandlecocycles and give examples. We give an explicit formula relating group -cocycles to ternary f -quandlecocycles, when the ternary f -quandle is constructed from a group.  T((x, a), (y, b), (z, c)) = (T(x, y, z), αx,y,z(a, b, c)), where T(x, y, z) denotes the ternary f -quandle product in X, if and only if α satis es the following conditions: 1. αx,x,x(a, a, a) = g(a) for all x ∈ X and a ∈ A; 2. αx,y,z (−, b, c) : A → A is a bijection for all x, y, z ∈ X and for all b, c ∈ A; (αx,u,v(a, d, e), αy,u,v(b, d, e), αz,u,v(c, d, e)) for all x, y, z, u, v ∈

X and a, b, c, d, e ∈ A. Such function α is called a dynamical ternary f -quandle cocycle or dynamical ternary f -rack cocycle (when it satis es above conditions).
The ternary f -quandle constructed above is denoted by X ×α A, and it is called extension of X by a dynamical cocycle α. The construction is general, as Andruskiewitch and Graña showed in [11].
Assume (X, T, F) is a ternary f -quandle and α be a dynamical f -cocycle. For x ∈ X, de ne Tx(a, b, c) :=  αx,x,x(a, b, c). Then it is easy to see that (A, Tx, F) is a ternary f -quandle for all x ∈ X. u,v(a, d, e), αx,u,v(b, d, e), αx,u,v(c, d, e)) for all a, b, c, d, e ∈ A. Now, we discuss Extensions with constant cocycles. Let (X, T, F) be a ternary f -rack and λ : T(z,u,v) λx,u,v we say λ is a constant ternary f -rack cocycle.

Remark 5.2. When x = y = z in condition (3) above, we get
If (X, T, F) is a ternary f -quandle and further satis es λx,x,x = id for all x ∈ X , then we say λ is a constant ternary f -quandle cocycle.
. Modules over ternary f -rack De nition 5.3. Let (X, T, F) be a ternary f -rack, A be an abelian group and f , g : X → X be homomorphisms. A structure of X-module on A consists of a family of automorphisms (η ijk ) i,j,k∈X and a family of endomorphisms (τ ijk ) i,j,k∈X of A satisfying the following conditions: In the n-ary case, we generalized the above de nition as follows.

Remark 5.5. If X is a ternary f -quandle, a ternary f -quandle structure of X-module on A is a structure of an X-module further satis es
and Furthermore, if f , g = id maps, then it satis es Remark 5.6. When x = y = z in (8), we get
Then, it can be veri ed directly that α is a dynamical cocycle and the following relations hold: De nition 5.8. When κ further satis es κx,x,x = in (23) for any x ∈ X, we call it a generalized ternary fquandle -cocycle.

(i) T(T(x, y, z), u, v) = T(x, T(y, z, u), v) = T(x, y, T(z, u, v)) (associativity), (ii) T(e, x, e) = T(x, e, e) = T(e, e, x) = x (existence of identity element), (iii) T(x, y, y) = T(y, x, y) = T(y, y, x) = e ( existence of inverse element).
Example 5.12. Here we provide an example of a ternary f -quandle module and explicit formula of the ternary f -quandle -cocycle obtained from a group -cocycle. Let G be a group and let → A → E → G → be a short exact sequence of groups where E = A θ G by a group -cocycle θ and A is an Abelian group. The multiplication rule in E is given by (a, x) · (b, y) · (c, z) = (a + x · b + y · c + θ(x, y, z), T(x, y, z)), where x · b means the action of A on G. Recall that the group 3-cocycle condition is θ(x, y, z) + xθ(y, z, u) + θ(x, yz, u) = θ(xy, z, u) + θ(x, y, zu). Now, let X = G be a ternary f -quandle with the operation T(x, y, z) = f (xy − z) and let g : A → A be a map on A so that we have a map F : E → E given by F(a, x) = (g(a), f (x)). Therefore the group E becomes a ternary f -quandle with the operation T ((a, x), (b, y), (c, z)) = F((a, x) · (b, y) − · (c, z)).

Cohomology theory of n-ary f -quandles
In this section we present a general cohomology for n-ary f -quandles, and include speci c examples, including the generalized ternary case, and speci c examples in both the ternary and binary case.
Let (X, T, f ) be a ternary f -rack where f : X → X is a ternary f -rack morphism. We will de ne the generalized cohomology theory of f -racks as follows: For a sequence of elements (x , x , x , x , . . . , x p+ ) ∈ X p+ de ne More generally, if we are considering an n-ary f -rack (X, T, f ), using the same notation T for the n-ary operation, we de ne the bracket as follows: Notice that for i = (p − )j + < n, we have This relation is obtained by applying the rst axiom of f -quandles p − i times, rst grouping the rst i − terms together, then iterating this process, again grouping and iterating each. We provide cohomology theory for the f -rack Proof. To prove that ∂ p+ ∂ p = , and thus ∂ is a coboundary map we will break the composition into pieces, using the linearity of η and τ i . First we will show that the composition of the i th term of the rst summand of ∂ p with the j th term of the rst summand of ∂ p+ cancels with the (j + ) th term of the rst summand of ∂ p with the i th term of the rst summand of ∂ p+ for i ≤ j. As the sign of these terms are opposite, we need only show that the compositions are equal up to their sign. For the sake of readability we will introduce the following, based on A and B above: This is precisely the (j + ) th term of the rst summand of ∂ p with the i th term of the rst summand of ∂ p+ .
Similar manipulations show that the composition of τ i from ∂ p with the i th term of the rst sum of ∂ p+ cancels with the composition of the (i + ) th term of the rst sum of ∂ p with τ i from ∂ p+ . For the sake of brevity we will omit showing these manipulations, but the table below presents all relations which are canceled by similar manipulations.
In the table, η i represents the i th summand of the rst sum, • i represents the i th summand of the second sum, with order of composition determining its origin in δp or δ p+ .
All these relations leave n + remaining terms, which cancel via the third axiom from the De nition.
We present the ternary case below, using the convention from the previous section, so τ and µ representing τ and τ respectively. Specializing further in Example 6.2, we obtain the following result. A direct computation gives H (X = Z , A = Z ) is -dimensional with basis { χ + χ , χ + χ }. As such the dim(Im(δ )) = , and additional calculation gives dim(ker(δ )) = , thus H is also -dimensional.
Lastly we consider a binary case, obtaining, as expected, a familiar result.

Example 6.4. Let η be the multiplication by T and τ be the multiplication by S in Example 2.11. The -cocycle condition is written for a function ϕ : X → A as
Tϕ(x) + Sϕ(y) − ϕ(x * y) = .
Note that this means that ϕ : X → A is a quandle homomorphism. For ψ : X × X → A, the -cocycle condition can be written as In [27], the groups H and H with coe cients in the abelian group Z of the f -quandle X = Z , T = , S = and f (x) = were computed. More precisely, H (Z , Z ) is -dimensional with a basis χ + χ and H is -dimension with a basis {χ ( , ) − χ ( , ) }.