Source document: https://docs-lawep.s3.us-east-2.amazonaws.com/1697750728959.pdf (Georgia)
Journal of Homotopy and Related Structures, vol.

3(1), 2008, pp.13–63 COHOMOLOGY OF CATEGORICAL SELF-DISTRIBUTIVITY J. SCOTT CARTER, ALISSA S. CRANS, MOHAMED ELHAMDADI and MASAHICO SAITO (communicated by James Stasheﬀ) Abstract We deﬁne self-distributive structures in the categories of coalgebras and cocommutative coalgebras.

We obtain exam- ples from vector spaces whose bases are the elements of ﬁnite quandles, the direct sum of a Lie algebra with its ground ﬁeld, and Hopf algebras.
The self-distributive operations of these structures provide solutions of the Yang–Baxter equation, and, conversely, solutions of the Yang–Baxter equation can be used to construct self-distributive operations in certain categories.

Moreover, we present a cohomology theory that encom- passes both Lie algebra and quandle cohomologies, is analogous to Hochschild cohomology, and can be used to study deforma- tions of these self-distributive structures.
All of the work here is informed via diagrammatic computations.

1.

Introduction In the past several decades, operations satisfying self-distributivity: (a b) c = (a c) (b c) have secured an important role in knot theory.

Such operations not only provide solutions of the Yang–Baxter equation and satisfy a law that is an algebraic distilla- tion of the type (III) Reidemeister move, but they also capture one of the essential properties of group conjugation.
Sets possessing such a binary operation are called shelves.

Adding an axiom corresponding to the type (II) Reidemeister move amounts to the property that the set acts on itself (on the right) bijectively and thus gives the structure of a rack.

Further introducing a condition corresponding to the type (I) Reidemeister move has the eﬀect of making each element idempotent and gives the structure of a quandle.

Keis, or involutory quandles, satisfy an extra involutory condition.
Such structures were discussed as early as the 1940s [25].

Supported in part by NSF Grants DMS #0301089 and #0301095.

Received March 01, 2007, revised December 03, 2007; published on March 15, 2008.

2000 Mathematics Subject Classiﬁcation: 16W30, 57T05, 57T10, 57M27.

Key words and phrases: Categorical internalization, self-distributivity, quandle, Lie Algebra, Hopf Algebra, Hochschild Cohomology, coalgebra, Yang Baxter equation, trigonometric coalgebras, rack.
c 2008, J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi and Masahico Saito.

Permission to copy for private use granted.

Journal of Homotopy and Related Structures, vol.

3(1), 2008 14 The primordial example of a self-distributive operation comes from group conju- gation: x y = y−1xy.

This operation satisﬁes the additional quandle axioms which are stated in the sequel.
Quandle cohomology has been studied extensively in connection with applications to knots and knotted surfaces [10, 11].

Analogues of self-distributivity in a variety of categorical settings have been discussed as adjoint maps in Lie algebras [12] and quantum group theories (see for example [20, 19]).

In particular, the adjoint map of Hopf algebras x ⊗ y → S(y(1))xy(2) is a direct analogue of group conjugation.
Thus, analogues of self-distributive oper- ations are found in a variety of algebraic structures where cohomology theories are also deﬁned.

In this paper, we study how quandles and racks and their cohomology theories are related to these other algebraic systems and their cohomology theories.

Speciﬁcally, we treat self-distributive maps in a uniﬁed manner via a categorical technique called internalization [13].
Then we develop a cohomology theory and provide explicit relations to rack and Lie algebra cohomology theories.

Furthermore, this cohomology theory can be seen as a theory of obstructions to deformations of self-distributive structures.

The organization of this paper is as follows: Section 2 consists of a review of the fundamentals of quandle theory, internalization in a category, and the deﬁnition of a coalgebra.
Section 3 contains a collection of examples that possess a self-distributive binary operation.

In particular, a motivating example built from a Lie algebra is presented.

In Section 4 we relate the ideas of self-distributivity to solutions of the Yang-Baxter equation, and demonstrate connections of these ideas to Hopf alge- bras.

Section 5 contains a review of Hochschild cohomology from the diagrammatic point of view and in relation to deformations of algebras.
These ideas are imitated in Section 6 where the most original and substantial ideas are presented.

Herein a cohomology theory for shelves in the category of coalgebras is deﬁned in low dimensions.

The theory is informed by the diagrammatic representation of the self- distributive operation, the comultiplication, their axioms, and their relationships.

Section 7 contains the main results of the paper.


Theorems 7.4 through 7.9 state that the cohomology theory is non-trivial, and that non-trivial quandle cocycles and Lie algebra cocycles give non-trivial shelf cocycles and non-trivial deformations in dimension 2 and 3.

Acknowledgements In addition to the National Science Foundation who supported this research ﬁnancially, we also wish to thank our colleagues whom we engaged in a number of crucial discussions.
The topology seminar at South Alabama listened to a series of talks as the work was being developed.

Joerg Feldvoss gave two of us a wonderful lecture on deformation theory of algebras and helped provide a key example.

John Baez was extremely helpful with some fundamentals of categorical constructions.

Journal of Homotopy and Related Structures, vol.

3(1), 2008 15 We also thank N. Apostolakis, L.H.

Kauﬀman, D. Radford, and J.D.


Stasheﬀ for valuable conversations, as well as the referee for comments that helped improve the exposition.

2.

Internalized Shelves 2.1. Review of Quandles A quandle, X, is a set with a binary operation (a, b) → a b such that (I) For any a ∈ X, a a = a.

(II) For any a, b ∈ X, there is a unique c ∈ X such that a = c b.

(III) For any a, b, c ∈ X, we have (a b) c = (a c) (b c).
A rack is a set with a binary operation that satisﬁes (II) and (III).

Racks and quandles have been studied extensively in, for example, [6, 14, 16, 23].

The following are typical examples of quandles: A group G with conjugation as the quandle operation: a b = b−1ab, denoted by X = Conj(G), is a quandle.

Any subset of G that is closed under such conjugation is also a quandle.

More generally if G is a group, H is a subgroup, and s is an automorphism that ﬁxes the elements of H (i.e.
s(h) = h ∀h ∈ H), then G/H is a quandle with deﬁned by Ha Hb = Hs(ab−1)b.

Any Λ(= Z[t, t−1])-module M is a quandle with a b = ta + (1 − t)b, for a, b ∈ M , and is called an Alexander quandle.

Let n be a positive integer, and for elements i, j ∈ {0, 1, .

.

.

, n − 1}, deﬁne i j ≡ 2j − i (mod n).


Then deﬁnes a quandle structure called the dihedral quandle, Rn, that coincides with the set of reﬂections in the dihedral group with composition given by conjugation.

The third quandle axiom (a b) c = (a c) (b c), which corresponds to the type (III) Reidemeister move, can be reformulated to make sense in a more general setting.
In fact, for our work here we do not need the full-ﬂedged structure of a quandle; we simply need a structure having a binary operation satisfying the self-distributive law.

We call a set together with a binary operation satisfying the self-distributive axiom (III) a shelf.
We reformulate the self-distributive operation of a shelf as follows: Let X be a shelf with the shelf operation denoted by a map q : X × X → X. Deﬁne ∆ : X → X × X by ∆(x) = (x, x) for any x ∈ X, and τ : X × X → X × X by a transposition τ (x, y) = (y, x) for x, y ∈ X.

Then axiom (III) above can be written as: q(q × 1) = q(q × q)(1 × τ × 1)(1 × 1 × ∆) : X 3 → X.

It is natural and useful to formulate this axiom for morphisms in certain cate- gories.
This approach was explored in [12] (see also [2]) and involves a technique known as internalization.

2.2.

Internalization All familiar mathematical concepts were deﬁned in the category of sets, but most of these can live in other categories as well.

This idea, known as internalization, is actually very familiar.

For example, the notion of a group can be enhanced by looking at groups in categories other than Set, the category of sets and functions between them.
We have the notions of topological groups, which are groups in Journal of Homotopy and Related Structures, vol.

3(1), 2008 16 the category of topological spaces, Lie groups, groups in the category of smooth manifolds, and so on.
Internalizing a concept consists of ﬁrst expressing it completely in terms of commutative diagrams and then interpreting those diagrams in some suﬃciently nice ambient category, K. In this paper, we consider the notion of a shelf in the categories of coalgebras and cocommutative coalgebras.

Thus, we deﬁne the notion of an internalized shelf, or shelf in K. This concept is also known as a shelf object in K or internal shelf.
Given two objects X and Y in an arbitrary category, we deﬁne their product to be any object X × Y equipped with morphisms π1 : X × Y → X and π2 : X × Y → Y called projections, such that the following universal property is satisﬁed: for any object Z and morphisms f : Z → X and g : Z → Y, there is a unique morphism h : Z → X × Y such that f = π1h and g = π2h.

Note that this product does not necessarily exist, nor is it unique.
However, it is unique up to canonical isomorphism, which is why we refer to the product when it exists.

We say a category has binary products when every pair of objects has a product.

Trinary products (X × Y ) × Z and X × (Y × Z) are deﬁned similarly, are canonically isomorphic, and denoted by X ×Y ×Z if the isomorphism is the identity.

Inductively, n-ary products are deﬁned.

We say a category has ﬁnite products if it has n-ary products for all n 0.
Note that whenever X is an object in some category for which the product X × X exists, there is a unique morphism called the diagonal D : X → X ×X such that π1D = 1X and π2D = 1X .

In the category of sets, this map is given by D(x) = (x, x) for all x ∈ X.

In a category with ﬁnite products, we also have a transposition morphism given by τ : X × X → X × X by τ = (π2 × π1)DX×X .

Deﬁnition 2.1.

Let X be an object in a category K with ﬁnite products.
A map q : X × X → X is a self-distributive map if the following diagram commutes: X × X × X 1×1×∆ q×1 X × X × X × X X × X 1×τ ×1 X × X × X × X 1×1×q X × X × X q X q X × X q×1 where ∆ : X → X × X is the diagonal morphism in K and τ : X × X → X × X is the transposition.
We also say that a map q satisﬁes the self-distributive law.

Deﬁnition 2.2.

Let K be a category with ﬁnite products.

A shelf in K is a pair (X, q) such that X is an object in K and q : X × X → X is a morphism in K that satisﬁes the self-distributive law of Deﬁnition 2.1.

Example 2.3.
Example 2.3.

A quandle (X, q) is a shelf in the category of sets, with the cartesian products and the diagonal map D : X → X × X deﬁned by D(x) = (x, x) for Journal of Homotopy and Related Structures, vol.

3(1), 2008 17 Figure 1: Internal Shelf Axioms all x ∈ X.
Thus the language of shelves and self-distributive maps in categories uniﬁes all examples discussed in this paper, in particular those constructed from Lie algebras.

Remark 2.4.
Throughout this paper, all of the categories considered have ﬁnite products: • Set, the category whose objects are sets and whose morphisms are functions • Vect, the category whose objects are vector spaces over a ﬁeld k and whose morphisms are linear functions • Coalg, the category whose objects are coalgebras with counit over a ﬁeld k and whose morphisms are coalgebra homomorphisms and compatible with counit • CoComCoalg, the category whose objects are cocommutative coalgebras with counit over a ﬁeld k and whose morphisms are cocommutative coalgebra ho- momorphisms and compatible with counit It is convenient for calculations to express the maps and axioms of a shelf in K diagrammatically as we do in the left and right of Fig.
1, respectively.

Note that the self-distributive map q is not the multiplication map and therefore requires a diﬀerent diagrammatic representation, which can be found on the far left in Fig.

1.

The composition of the maps is read from right to left (gf )(x) = g(f (x)) in text and from bottom to top in the diagrams.

In this way, when reading from left to right one can draw from top to bottom and when reading a diagram from top to bottom, one can display the maps from left to right.
The argument of a function (or input object from a category) is found at the bottom of the diagram.

2.3.

Coalgebras A coalgebra is a vector space C over a ﬁeld k together with a comultiplication ∆ : C → C ⊗ C that is linear and coassociative: (∆ ⊗ 1)∆ = (1 ⊗ ∆)∆.

A coalgebra is cocommutative if the comultiplication satisﬁes τ ∆ = ∆, where τ : C ⊗ C → C ⊗ C is the transposition τ (x ⊗ y) = y ⊗ x.
A coalgebra with counit is a coalgebra with a linear map called the counit : C → k such that ( ⊗ 1)∆ = 1 = (1 ⊗ )∆ via XXXXXXXXXXXXXDqtXXXXSelf−distributivelawX Journal of Homotopy and Related Structures, vol.
3(1), 2008 18 k ⊗ C ∼= C. Diagrammatically, this condition says that the following commutes: k × C C ⊗ C 1× C × k ×1 ∆ C Note that if (C, ∆, ) is a coalgebra with counit, then so is the tensor product C ⊗ C. Lemma 2.5.
If C is a coalgebra with counit, the comultiplication ∆C : C → C ⊗ C is the diagonal map in the category of coalgebras with counits.

Proof.
Since C ⊗ C is the product in the category of coalgebras with counits, there is a diagonal, that is a unique morphism φ : C → C ⊗ C which makes the following diagram commute: 1 C 1 φ C C ⊗ C π1 π2 C where the π1 and π2 are projection maps deﬁned by π1 := A ⊗ B π2 := A ⊗ B 1⊗B A⊗1 A ⊗ k k ⊗ B ∼ ∼ A B where A and B are the counit maps for coalgebras A and B.
Since the comulti- plication ∆C satisﬁes the same property as φ and φ is unique, they must coincide.

 A linear map f between coalgebras is said to be compatible with comultiplication, or preserves comultiplication, if it satisﬁes the condition ∆f = (f ⊗ f )∆.
Diagram- matically, the following commutes: C f D ∆C C ⊗ C f ⊗f ∆D D ⊗ D A linear map f between coalgebras is said to be compatible with counit, or pre- serves counit, if it satisﬁes the condition f = , which, diagrammatically says the Journal of Homotopy and Related Structures, vol.
3(1), 2008 19 following diagram commutes: C C f k D D In particular, if (C, ∆, ) is a coalgebra with counit, a linear map q : C ⊗ C → C between coalgebras is compatible with comultiplication if and only if it satisﬁes ∆q = (q ⊗ q)(1 ⊗ τ ⊗ 1)(∆ ⊗ ∆), and it is compatible with counit if and only if it satisﬁes q = ⊗ .
A morphism f in the category of coalgebras with counit is a linear map that preserves comultiplication and counit.

As suggested by the categories listed in Re- mark 2.4, we will focus our main attention on coalgebras with counits.

Thus, we use the word ‘coalgebra’ to refer to a coalgebra with counit and the phrase ‘coalgebra morphism’ to refer to a linear map that preserves comultiplication and counit.
On the other hand, we wish to consider examples in which the self-distributive map is not compatible with the counit (see the sequel).

For categorical hygiene, we are distinguishing a function that satisﬁes self-distributivity and is compatible with comultiplication from a morphism in the category Coalg.

3.

Self-Distributive Maps for Coalgebras In this section we give concrete and broad examples of self-distributive maps for cocommutative coalgebras.
Speciﬁcally, we discuss examples constructed from quandles/racks used as bases, Lie algebras, and Hopf algebras.

3.1.

Self-Distributive Maps for Coalgebras Constructed From Racks In this section we note that quandles and racks can be used to construct self- distributive maps in CoComCoalg simply by using their elements as basis.

Let X be a rack.

Let V = kX be the vector space over a ﬁeld k with the elements of X as basis.
Then V is a cocommutative coalgebra with counit, with comultiplication ∆ induced by the diagonal map ∆(x) = x ⊗ x, and the counit induced by (x) = 1 for x ∈ X.

This is a standard construction of a coalgebra with counit from a set.

Set W = k ⊕ kX.

We denote an element of W = k ⊕ kX by a + x∈X axx or axx.
Extend more brieﬂy by a + ∆ and on V = kX to W by linearly extending ∆(1) = 1 ⊗ 1 and (1) = 1 for 1 ∈ k. More explicitly, x axx, and when context is understood by a + ∆(a + and (a + is an object in CoComCoalg.

axx) = a + axx) = a(1 ⊗ 1) + ax(x ⊗ x), ax.
With these deﬁnitions, one can check that (W, ∆, ) Deﬁne q : W ⊗ W → W by linearly extending q(x ⊗ y) = x y, q(1 ⊗ x) = 1, Journal of Homotopy and Related Structures, vol.

3(1), 2008 20 q(x ⊗ 1) = 0, and q(1 ⊗ 1) = 0.

More explicitly, q( (a + axx) ⊗ (b + byy) ) = y aby + x,y axby(x y).

Proposition 3.1.

The extended map q given above is a self-distributive linear map compatible with comultiplication.
Proof.

The proof is by calculations.

For example, the LHS of the self-distributivity is computed as q(q ⊗ 1)( (a + czz) ) axx) ⊗ (b + byy) ⊗ (c + aby + axby(x y)) ⊗ (c + czz) ) y = q( ( = y,z abycz + x,y axbycz((x y) z), x,y,z which is compared with the RHS.

Similarly, the compatibility is proved by verifying ∆q = (q ⊗ q)(1 ⊗ τ ⊗ 1)(∆ ⊗ ∆).
The pair (W, q) falls short of being a shelf in CoComCoalg due to the following: Proposition 3.2.

The extended map q deﬁned above is not compatible with the counit, but satisﬁes q = q( ⊗ 1).

Proof.

The counit has as its image k ⊂ W .

Thus the image of ⊗ 1 is in W ⊗ W .


We compute the following three quantities: q( (a + axx) ⊗ (b + byy) ) = ( aby + axby(x y)) = a by + x,y axby, ⊗ ( (a + q( ⊗ 1)( (a + axx) ⊗ (b + axx) ⊗ (b + The ﬁrst and third coincide.

 3.2.
3.2.

Lie Algebras byy) ) = (a + ax)(b + by) x,y ax + byy) ) = ab + a by + b axby, and byy) ) = q( (a + ax) ⊗ (b + = (a + ax) by.
A Lie algebra g is a vector space over a ﬁeld k of characteristic other than 2, with an antisymmetric bilinear form [·, ·] : g × g → g that satisﬁes the Jacobi identity [[x, y], z] + [[y, z], x] + [[z, x], y] = 0 for any x, y, z ∈ g. Given a Lie algebra g over k we can construct a coalgebra N = k ⊕ g. We will denote elements of N as either (a, x) or a + x, depending on clarity, where a ∈ k and x ∈ g. In fact, N is a cocommutative coalgebra with comultiplication and counit given by ∆(x) = x ⊗ 1 + 1 ⊗ x for x ∈ g and ∆(1) = 1 ⊗ 1, (1) = 1, (x) = 0 for x ∈ g. Journal of Homotopy and Related Structures, vol.
3(1), 2008 21 In general we compute, for a ∈ k and x ∈ g, ∆((a, x)) = ∆(a + x) = ∆(a) + ∆(x) = a(1 ⊗ 1) + x ⊗ 1 + 1 ⊗ x = (a ⊗ 1 + x ⊗ 1) + 1 ⊗ x = (a + x) ⊗ 1 + 1 ⊗ x = (a, x) ⊗ (1, 0) + (1, 0) ⊗ (0, x).

The following map is found in quantum group theory (see for example, [19], and studied in [12] in relation to Lie 2-algebras).
Deﬁne q : N ⊗ N → N by linearly extending q(1 ⊗ (b + y)) = (b + y), q((a + x) ⊗ 1) = a + x and q(x, y) = [x, y] for a, b ∈ k and x, y ∈ g, i.e., q((a, x) ⊗ (b, y)) = q((a + x) ⊗ (b + y)) = ab + bx + [x, y] = (ab, bx + [x, y]).
Since the solution to the classical YBE follows from the Jacobi identity, and the YBE is related to self-distributivity (see next section) via the third Reidemeister move, it makes sense to expect that there is a relation between the Lie bracket and the self-distributivity axiom.

Lemma 3.3.

The above deﬁned q satisﬁes the self-distributive law in Deﬁnition 2.1.

Proof.
Proof.

We compute q(q ⊗ 1)((a, x) ⊗ (b, y) ⊗ (c, z)) = q((ab + bx + [x, y]) ⊗ (c, z)) = abc + bcx + c[x, y] + b[x, z] + [[x, y], z], q(q ⊗ q)(1 ⊗ τ ⊗ 1)(1 ⊗ 1 ⊗ ∆)((a + x) ⊗ (b + y) ⊗ (c + z)) = q(q ⊗ q)(1 ⊗ τ ⊗ 1)((a + x) ⊗ (b + y) ⊗ {(c + z) ⊗ 1 + 1 ⊗ z}) = q(q ⊗ q)((a + x) ⊗ (c + z) ⊗ (b + y) ⊗ 1 + (a + x) ⊗ 1 ⊗ (b + y) ⊗ z = q((ac + cx + [x, z]) ⊗ (b + y)) + q((a + x) ⊗ [y, z]) = (abc + bcx + c[x, y] + b[x, z] + [[x, z], y]) + [x, [y, z]], and the Jacobi identity in g veriﬁes the condition.
Lemma 3.4.

The map q constructed above is a coalgebra morphism.

Proof.

We compute: ∆q((a + x) ⊗ (b + y)) = (ab + bx + [x, y]) ⊗ 1 + 1 ⊗ (bx + [x, y]).
On the other hand, we have (q ⊗ q)(1 ⊗ τ ⊗ 1)(∆ ⊗ ∆)((a + x) ⊗ (b + y)) = (q ⊗ q)(1 ⊗ τ ⊗ 1)((a + x) ⊗ 1 + 1 ⊗ x) ⊗ ((b + y) ⊗ 1 + 1 ⊗ y) = q((a + x) ⊗ (b + y)) ⊗ q(1 ⊗ 1) + q(1 ⊗ 1) ⊗ q(x ⊗ y) + (b + y) ⊗ x +(a + x) ⊗ (y) = ((a + bx + [x, y]) ⊗ 1) + 1 ⊗ [x, y] + b ⊗ x + (a + x) ⊗ 0 = ((a + bx + [x, y]) ⊗ 1) + 1 ⊗ (bx + [x, y]) For the counit, we compute: q( (a + x) ⊗ (b + y) ) = (ab + bx + [x, y]) = ab = ( ⊗ )( (a + x) ⊗ (b + y) ).
Journal of Homotopy and Related Structures, vol.

3(1), 2008 22 Combining these two lemmas, we have: Proposition 3.5.

The coalgebra N together with map q given above deﬁnes a shelf (N, q) in CoComCoalg.

Groups have quandle structures given by conjugation, and their subcategory of Lie groups are related to Lie algebras through tangent spaces and exponential maps.
In the above proposition we constructed shelves in CoComCoalg from Lie algebras, so we see this proposition as a step in completing the following square of relations.

Lie groups Lie algebras Quandles ?

??

3.3.
??

3.3.

Hopf Algebras A bialgebra is an algebra A over a ﬁeld k together with a linear map called the unit η : k → A, satisfying η(a) = a1 where 1 ∈ A is the multiplicative identity and with an associative multiplication µ : A ⊗ A → A that is also a coalgebra such that the comultiplication ∆ is an algebra homomorphism.

A Hopf algebra is a bialgebra C together with a map called the antipode S : C → C such that µ(S ⊗ 1)∆ = η = µ(1 ⊗ S)∆, where is the counit.
The reader can construct commutative diagrams similar to those found in Section 2.3 for the notions of bialgebra and Hopf algebra.

Our diagrammatic conventions for these maps are depicted in Fig.

2.

Recall that the diagrams are read from bottom to top.

These diagrams have been used (see for example [18, 27]) for proving facts about Hopf algebras and related invariants.

We review the diagrammatic representation of Hopf algebra axioms.
For conve- nience, assume that the underlying vector space of A is ﬁnite dimensional with or- dered basis (e1, e2, .

.

.

, en).

Then the multiplication µ and comultiplication ∆ are de- termined by the values, Λ ij(e), and ∆(e) = Y ij ei ⊗ ej.

Note that summation conventions are being applied, and so, for example, Λ ij(e).

Similarly, the unit can be written as i Aiei.


The counit can be written as (ei) = Vi ∈ k, so that for a general η(1) = i Ai)(ei) = i aiVi.
Finally, the antipode i αiei, we have ( vector, is a linear map so S(ei) = sj i ∈ k. ∈ k, of the structure constants: µ(ei ⊗ej) = Λ n i ej for constants sj ij(e) = αiei) = ( ij, Y ij =1 Λ Thus the axioms of a (ﬁnite dimensional) Hopf algebra can be formulated in terms of the structure constants.

The table below summarizes these formulations.


Again summation convention applies, and all super, and subscripted letters are constants in the ground ﬁeld.

 Journal of Homotopy and Related Structures, vol.
3(1), 2008 23 associativity coassociativity unit counit Compatibility Antipode j = Λq Λq pΛp irΛr ij Y ij q = Y j p Y p r Y ir q ijAj = AjΛ ji = δ i = Y ji Vi = δj r Λr j = Y pq i Y vw uwY tu ij pY rp j = ViAj rqsq j = Λi Λ V iY ij tvΛq pY pq Λp rqsr Λi In the table above, δ i denotes a Kronecker delta function.

It is a small step now to translate these.


Speciﬁcally, the multiplication tensor Λ is diagrammatically represented by the leftmost trivalent vertex read from bottom to top.

The letter choices Λ, Y , A and V are meant to suggest the graphical depictions of these operators.

A composition of maps corresponds to a contraction of the corresponding indices of tensors which, in turn, corresponds to connecting end points of diagrams together vertically.
Figures 2 and 3 represent such diagrammatic conventions of maps that appear in the deﬁnition of a Hopf algebra and their axioms.

The gap in the ‘Antipode condition’ diagram in Fig.

3 should be interpreted as follows: The counit results in a constant which is then reimbedded in the algebra via the unit map.

Figure 2: Operations in Hopf algebras Figure 3: Axioms of Hopf algebras Let H be a Hopf algebra.
Deﬁne q : H ⊗ H → H by q = µ(1 ⊗ µ)(S ⊗ 1 ⊗ 1)(τ ⊗ 1)(1 ⊗ ∆) where µ, ∆, and S denote the multiplication, comultiplication, xyxDxx(1)(2)ehmxS( )xxytxyComultiplicationMultiplicationUnitCounitAntipodeTranspositionxySAssociativityUnit CounitCounit is an algebra homCompatibilityCoassociativityAntipode conditionUnit is a coalgebra hom SS Journal of Homotopy and Related Structures, vol.

3(1), 2008 24 and antipode, respectively.


If we adopt the common notation ∆(x) = x(1) ⊗ x(2) and µ(x ⊗ y) = xy, then q is written as q(x ⊗ y) = S(y(1))xy(2).

This appears as an adjoint map in [26, 20], and its diagram is depicted in Fig.

4.

Notice the analogy with the group conjugation as a quandle: in a group ring, ∆(y) = y ⊗ y and S(y) = y−1, so that q(x ⊗ y) = y−1xy, and therefore, is of a great interest from point of view of quandles.
Figure 4: Self-distributive map in Hopf algebras Figure 5: Proof of self-distributivity in Hopf algebras Proposition 3.6.

The above deﬁned linear map q : H ⊗ H → H satisﬁes the self-distributive law in Deﬁnition 2.1.

Proof.

In Fig.

5, it is indicated that this follows from two properties of the adjoint map: q(q ⊗ 1) = q(1 ⊗ µ) (which is used in the ﬁrst and the third equalities in the ﬁgure), and µ = µ(1 ⊗ q)(τ ⊗ 1)(1 ⊗ ∆) (which is used in the second equality).
It is known that these properties are satisﬁed, and proofs are found in [26, 15].

Here we include diagrammatic proofs for the reader’s convenience in Fig.

6 and Fig.

7, respectively.

The dotted loops in Fig.

6, and all those that follow, indicate where changes are made to the diagram.

 Remark 3.7.

The deﬁnition of q above contains an antipode, which is a coalgebra anti-homomorphism and not necessarily a coalgebra morphism.

Thus, (H, q) is not a shelf in Coalg in general.

3.4.
3.4.

Other Examples In this section we observe that there are plenty of examples of self-distributive lin- ear maps for 2-dimensional cocommutative coalgebras and shelves in CoComCoalg.

Let V be the two dimensional vector space over k with basis {x, y}.

Deﬁne a coalgebra structure on V using the diagonal map ∆(z) = z ⊗ z for z ∈ {x, y} and extending it linearly.

S Journal of Homotopy and Related Structures, vol.

3(1), 2008 25 Figure 6: q(q ⊗ 1) = q(1 ⊗ µ) Figure 7: µ = µ(1 ⊗ q)(τ ⊗ 1)(1 ⊗ ∆) Lemma 3.8.
A linear map q : V ⊗ V → V is self-distributive and compatible with comultiplication if and only if q is one of the functions indicated via any column in the table below.

The values are determined on the basis elements x ⊗ x, through y ⊗ y as indicated.
q(x ⊗ x) = 0 q(x ⊗ y) = 0 q(y ⊗ x) = 0 q(y ⊗ y) = Among these, (V, q) is a shelf in CoComCoalg if and only if q(a, b) = 0 for any a, b ∈ {x, y}.

Proof.
Proof.

Let q(x ⊗ x) = γ1x + γ2y for some constants γ1, γ2 ∈ k. The compatibility condition ∆q(x ⊗ x) = (q ⊗ q)(1 ⊗ τ ⊗ 1)(∆ ⊗ ∆)(x ⊗ x) implies that γ1γ2 = 0 and γ2 1 = γ1, γ2 2 = γ2, i.e., q(x ⊗ x) = 0, x, or y.

The same Assoc.Anti−hom.AssociativityAssoc.NaturalityCompatibilityCancelSSSSSSSSSSSS Journal of Homotopy and Related Structures, vol.

3(1), 2008 26 holds for x ⊗ y, y ⊗ x and y ⊗ y, so that the value of q for a pair of basis elements is either a basis element (x or y), or 0.
A case by case analysis (facilitated by Mathematica and/or Maple) provides self-distributivity.

When (x) = (y) = 1, the only cases for which q = ⊗ are those for which q(a, b) = 0 for all four choices of a, b.
Another famous example of a cocommutative coalgebra is the trigonometric coal- gebra, T , generated by a and b with comultiplication given by: ∆(a) = a ⊗ a − b ⊗ b ∆(b) = a ⊗ b + b ⊗ a with counit (a) = 1, (b) = 0, in analogy with formulas for cos(x+y) and sin(x+y) and cos(0) = 1, sin(0) = 0.

Lemma 3.9.
Lemma 3.9.

Let T denote the trigonometric coalgebra over C. Let q : T ⊗ T → T be a linear map deﬁned by: q(a ⊗ a) = α1a + β1b,q(a ⊗ b) = α2a + β2b, q(b ⊗ a) = α3a + β3b,q(b ⊗ b) = α4a + β4b.

Then such a linear map q is self-distributive and compatible with comultiplication if and only if the coeﬃcients are found in Table 1, where i = −1.

√ Among these, (V, q) is a shelf in CoComCoalg if and only if (α1, α2, α3, α4) = (1, 0, 0, 0).

Proof.
Proof.

This result is a matter of verifying the conditions for self-distributivity and compatibility over all possible choices of inputs.

We generated solutions by both Maple and Mathematica.

For the compatibility condition we established a system of 12 quadratic equations in eight unknowns.

Originally there were 16 such equations, but 4 of these are duplicates.

In the Mathematica program we used the command “Solve” to generate a set of necessary conditions.
The self-distributive condition gave a system of cubic equations in the unknowns.

We checked these subject to the necessary conditions, and found the 21 solutions above.

Expressing as a (1×2) matrix and q as the 2×4 matrix α1 α2 α3 α4 β1 β2 β3 β4 We compute q = (α1, α2, α3, α4) and ⊗ = (1, 0, 0, 0).

The result follows.

 .

 4.
.

 4.

Yang–Baxter Equation and Self-Distributive Maps for Co- algebras In this section, we discuss relationships between solutions to the Yang-Baxter equations and self-distributive maps.

4.1.

A Brief Review of YBE The Yang–Baxter equation makes sense in any monoidal category.

Originally mathematical physicists concentrated on solutions in the category of vector spaces with the tensor product, obtaining solutions from quantum groups.
Journal of Homotopy and Related Structures, vol.
3(1), 2008 27 2 α1 α2 0 1 0 0 2 − i 1 β4 β3 β2 β1 α4 α3 0 0 −1 0 0 0 0 0 0 0 0 0 2 − i 1 0 0 0 0 1 i 1 0 0 0 0 2 2 2 1 0 0 0 0 0 1 1 1 1 0 − 1 0 0 2 2 2 2 1 0 0 0 0 1 0 4 − 1 4 − 1 4 − i 4 − 1 4 − i 4 − i 1 4 − 1 − i 4 − i 4 4 − i 4 − − 1 4 − 1 4 i 1 4 i 4 1 4 4 − i 4 − 1 4 − i 4 − i 4 0 0 1 0 0 − − 1 1 4 2 − 1 i 4 i 4 − i − i 2 − i 2 2 − 1 i 4 − i 4 4 − 1 4 − 1 1 4 i 4 1 4 − i 4 2 1 2 2 0 − 1 2 1 1 2 1 2 0 − 1 2 1 1 2 2 0 0 0 0 i 2 0 2 − i 2 − i 2 0 i 2 0 0 i 2 i 2 −i i Table 1: List of self-distributive maps in the trigonometric coalgebra Let V be a vector space and R : V ⊗ V → V ⊗ V an invertible linear map.
We say R is a Yang–Baxter operator if it satisﬁes the Yang–Baxter equation, (YBE), which says that: (R ⊗ 1)(1 ⊗ R)(R ⊗ 1) = (1 ⊗ R)(R ⊗ 1)(1 ⊗ R).
In other words, the YBE says that the following diagram commutes: V ⊗ V ⊗ V 1⊗R R⊗1 V ⊗ V ⊗ V V ⊗ V ⊗ V R⊗1 1⊗R V ⊗ V ⊗ V 1⊗R V ⊗ V ⊗ V R⊗1 V ⊗ V ⊗ V A solution to the YBE is also called a braiding.
In general, a braiding operation provides a diagrammatic description of the pro- Journal of Homotopy and Related Structures, vol.

3(1), 2008 28 cess of switching the order of two things.
This idea is formalized in the concept of a braided monoidal category, where the braiding is an isomorphism If we represent R : V ⊗ V → V ⊗ V by the diagram: RX,Y : X ⊗ Y → Y ⊗ X. then the Yang–Baxter equation is represented by: This diagram represents the third Reidemeister move in classical knot theory [7], and it gives the most important relations in Artin’s presentation of the braid group [4].

As a result, any invertible solution of the Yang–Baxter equation gives an invari- ant of braids.

4.2.
4.2.

Shelves in Coalg and Solutions of the YBE We now demonstrate the relationship between self-distributive maps in Coalg and solutions to the Yang–Baxter equation.

Figure 8: Solutions to YBE and shelves in Coalg Deﬁnition 4.1.

Let X be a coalgebra and q : X ⊗ X → X a linear map.

Then the linear map Rq : X ⊗ X → X ⊗ X deﬁned by Rq = (1X ⊗ q)(τ ⊗ 1X )(1X ⊗ ∆) is said to be induced from q. Conversely, let R : X ⊗ X → X ⊗ X be a linear map.
Then the linear map qR : X ⊗ X → X deﬁned by qR = ( ⊗ 1X )R is said to be induced from R. V VV VRRqRqqR Journal of Homotopy and Related Structures, vol.

3(1), 2008 29 Figure 9: Hypotheses of Theorem 4.2 Figure 10: Proof of Theorem 4.2 Diagrammatically, constructions of one of these maps from the other are depicted in Fig.

8.

Our goal is to relate solutions of the YBE and self-distributive maps in certain categories via these induced maps.

Theorem 4.2.
Theorem 4.2.

Let R : X ⊗ X → X ⊗ X be a solution to the YBE on a coalgebra X with counit.

Suppose R satisﬁes ( ⊗ )R = ( ⊗ ) and RqR = R. Then (X, qR) is a shelf in Coalg.

Proof.

The conditions in the assumption are presented in Fig.

9.

A proof is presented in Fig.

10.

 Theorem 4.3.

Let X be an object in CoComCoalg.

Suppose a self-distributive linear map q : X ⊗ X → X is compatible with comultiplication.

Then Rq is a solution to the YBE.

Proof.


The cocommutativity of ∆ is depicted in Fig.

11.

A proof, then, is depicted in Fig.

12.

Note here the condition that q is compatible with comultiplication is: Figure 11: Cocommutativity RRRRRRRRRRRRRRCo-commut.

Journal of Homotopy and Related Structures, vol.

3(1), 2008 30 Figure 12: Proof of Theorem 4.3 ∆(q(a⊗b)) = q(a(1) ⊗b(1))⊗q(a(2) ⊗b(2)) or, equivalently, ∆q = q(1⊗τ ⊗1)(∆⊗∆).

This is applied in Fig.


12 on the bottom row with the equal sign indicated to follow from compatibility.

 Propositions 3.1 and 3.5 and Theorem 4.3 imply the following: Corollary 4.4.

Let q be a map deﬁned from a quandle/rack as in Proposition 3.1 or from a Lie algebra as in Proposition 3.5.

Then the induced map Rq is a solution to the YBE.

In the Lie algebra case, the map is given as follows: Rq((a, x) ⊗ (b, y)) = (b, y) ⊗ (a, x) + (1, 0) ⊗ (0, [x, y]).

This appears, for example, in [12, 19].


Remark 4.5.

Next we focus on the case of the adjoint map in Hopf algebras.

Remark 3.7 states that the self-distributive map q(x ⊗ y) = S(y(1))xy(2) is not compatible with comultiplication, and therefore, Theorem 4.3 cannot be applied.

However, the induced map Rq does, indeed, satisfy the YBE.
This is of course for Compati−bilitySelf−distributivityCocommutativityNaturalityNaturalityCocommutativityCoassociativityNaturalityNaturalityAssoc.NaturalityNaturalityCoassociativity Journal of Homotopy and Related Structures, vol.

3(1), 2008 31 Figure 13: Conclusion of Prop.

4.6/ Hypothesis of Prop.


4.7 Figure 14: Proof of Proposition 4.6, second equation diﬀerent reasons, and proved in [26], which was interpreted in [15] as a restriction of a regular representation of the universal R-matrix of a quantum double.
Since it is of a great interest why the same construction gives rise to solutions to the YBE for diﬀerent reasons, we include their proofs in diagrams for the reader’s convenience, and we specify two conditions from [26] in our point of view, to construct Rq from q, and make a restatement of a theorem in [26] as follows: Proposition 4.6.

In a Hopf algebra, let q = µ(1 ⊗ µ)(S ⊗ 1 ⊗ 1)(τ ⊗ 1)(1 ⊗ ∆).

Then q(q ⊗ 1) = q(1 ⊗ µ) and (q ⊗ µ)(1 ⊗ τ ⊗ 1)(∆ ⊗ ∆) = (1 ⊗ µ)(τ ⊗ 1)(1 ⊗ ∆)(1 ⊗ q)(τ ⊗ 1)(1 ⊗ ∆).

Proof.
Proof.

The proofs are indicated in Figs.

6 and 14, respectively.

 Recall from Section 3.3 that in a Hopf algebra, the map q = µ(1 ⊗ µ)(S ⊗ 1 ⊗ 1)(τ ⊗ 1)(1 ⊗ ∆) satisﬁes self-distributivity.

Proposition 4.7.

Suppose X is a Hopf algebra and q is any linear map that satisﬁes q(q ⊗ 1) = q(1 ⊗ µ) and (q ⊗ µ)(1 ⊗ τ ⊗ 1)(∆ ⊗ ∆) = (1 ⊗ µ)(τ ⊗ 1)(1 ⊗ ∆)(1 ⊗ q)(τ ⊗ 1)(1 ⊗ ∆).

Then Rq is a solution to the YBE.

Proof.

The required conditions are depicted in Fig.

13.

And the proof is given in Fig.

15.


 In particular, the above proposition applies when q(x ⊗ y) = S(y(1))xy(2).

Def.CounitUnitDef.antip.compat.RedrawCounitUnitco-assoc.assoc.assoc.co-assoc.co-assoc.antipodecompatiblityassoc.NaturalityCo-assoc.SSSSSSSSSSSSSSSSS Journal of Homotopy and Related Structures, vol.

3(1), 2008 32 Figure 15: Proof of Proposition 4.7 5.
Graph Diagrams for Bialgebra Hochschild Cohomology The analogue of group cohomology for associative algebras is Hochschild coho- mology.

Then a natural question is, “What is an analogue of quandle cohomology for shelves in Coalg?” Since we have developed diagrammatic methods to study self-distributivity in Coalg, we apply these methods to seek such a cohomology the- ory, in combination with the interpretations of cocycles in bialgebra cohomology in terms of deformation theory of bialgebras.
The ﬁrst step toward this goal is to establish diagrammatic methods for Hochschild cohomology in terms of graph dia- grams.

Such approaches are found for homotopy Lie algebras and operads [21].

On the other hand, a diagrammatic method using polyhedra for bialgebra cohomology was given in [22].

In this section we follow the exposition in [22] of cocycles that appear in bialgebra deformation theory, and establish tree diagrams that can be used to prove cocycle conditions.
First we recall the Hochschild cohomology for bialgebras from [22].

Let A = (V, µ, ∆) be a bialgebra over a ﬁeld k, where µ, ∆ are multiplication and comul- tiplication, respectively, and dH : Hom(V ⊗p, V ⊗q) → Hom(V ⊗(p+1), V ⊗q) is the Hochschild diﬀerential dH (f ) = µ(1 ⊗ f ) + p−1 (−1)i+1f (1i ⊗ µ ⊗ 1n−i−1) + (−1)p+1µ(f ⊗ 1) i=0 where the left and right module structures are given by multiplication.
Du- ally dC : Hom(V ⊗p, V ⊗q) → Hom(V ⊗p, V ⊗(q+1)) denotes the coHochschild NaturalityCoassociativityNaturalityHypothesisNaturalityCoassociativitymmq(q 1)=q(1 )q(q 1)=q(1 ) Journal of Homotopy and Related Structures, vol.

3(1), 2008 33 diﬀerential.

These deﬁne the total complex (C ∗ b (A; A) = i=1Hom(V ⊗(n−i+1), V ⊗i).

For example, for a 1-cochain f ∈ Hom(V, V ), dH (f )(x⊗ ⊕n y) = xf (y) − f (xy) + f (x)y and dC(f )(x) = x(1) ⊗ f (x(2)) − f (x)(1) ⊗ f (x)(2) + f (x(1)) ⊗ x(2).
b (A; A), D), where C n For the rest of this section, we establish graph diagrams for Hochschild cohomol- ogy and review their aspects in deformation theory of bialgebras.

5.1.

Graph Diagrams for Hochschild Diﬀerentials A 1-cochain f ∈ Hom(V, V ) is represented by a circle on a vertical segment as shown in Fig.

16, where the images of f under the ﬁrst diﬀerentials dH (f ) and dC(f ), as computed above, are also depicted.
In general, a (m + n − 1)-cochain in Hom(V ⊗m, V ⊗n) is represented by a diagram in Fig.

17.

Figure 16: Hochschild 1-diﬀerentials Figure 17: Hochschild (m + n)-cochains Let τ be the transposition.

In general, τi indicates the transposition of the ith and (i + 1)st factors; the notation is used when type-setting gets complicated.
For (φ1, φ2), where φ1 ∈ Hom(V ⊗2, V ) and φ2 ∈ Hom(V, V ⊗2), the diﬀerentials are dH (φ1) = µ(1 ⊗ φ1) − φ1(µ ⊗ 1) + φ1(1 ⊗ µ) − µ(φ1 ⊗ 1), dC(φ1) = (µ ⊗ φ1)τ2(∆ ⊗ ∆) − ∆(φ1) + (φ1 ⊗ µ)τ2(∆ ⊗ ∆), dH (φ2) = (µ ⊗ µ)τ2(∆ ⊗ φ2) − φ2µ + (µ ⊗ µ)τ2(φ2 ⊗ ∆), dC(φ2) = (1 ⊗ φ2)∆ − (∆ ⊗ 1)(φ2) + (1 ⊗ ∆)(φ2) − (φ2 ⊗ 1)∆.

(1) (2) (3) (4) The 2-cocycle conditions are dH (φ1) = 0, dC(φ1) = dH (φ2), and dC(φ2) = 0.

The diﬀerential D of the total complex is D = dH − dC, D(φ1, φ2) = dH (φ1) + [dH (φ2) − dC(φ1)] − dC(φ2).
We demonstrate a proof that (φ1, φ2) = (dH (f ), dC(f )) satisﬁes dC(φ1) = dH (φ2) using graph diagrams.

First, we use encircled vertices as depicted in Fig.

17 to rep- resent an element of Hom(V ⊗m, V ⊗n).

Then dC(φ1) and dH (φ2) are represented on the top line of Fig.

18.

Substituting (φ1, φ2) = (dH (f ), dC(f )), that are represented ( ) =H+dC+,( ) =dnm Journal of Homotopy and Related Structures, vol.

3(1), 2008 34 Figure 18: Hochschild 2-diﬀerentials diagrammatically as in Fig.
16, we perform diagrammatic computations as in the rest of Fig.

18, and the equality follows because multiplication and comultiplication are compatible.

In particular each diagram in the left of the ﬁgure for which the vertex is external to the operations corresponds to a similar diagram on the right, but the correspondence is given after considering the compatible structures.
Figure 19: Hochschild 3-cocycle conditions For 3-cochains ψi ∈ Hom(V ⊗3, V ), ψ2 ∈ Hom(V ⊗2, V ⊗2) and ψ3 ∈ Hom(V, V ⊗3), the 3-cocycle condition is explicitly written as dH (ψ1) = 0, dC(ψ1) = dH (ψ2), dC(ψ2) = dH (ψ3), and dC(ψ3) = 0, see [22] dH (ψ1) = µ(1 ⊗ ψ1) − ψ1(µ ⊗ 12) + ψ1(1 ⊗ µ ⊗ 1) − ψ1(12 ⊗ µ) + µ(ψ1 ⊗ 1), dC(ψ1) = (µ(1 ⊗ µ) ⊗ ψ1)τ (∆ ⊗ ∆ ⊗ ∆) − ∆(ψ1) + (ψ1 ⊗ µ(µ ⊗ 1))τ (∆ ⊗ ∆ ⊗ ∆), dH (ψ2) = (µ ⊗ µ)τ2(∆ ⊗ ψ2) − ψ2(µ ⊗ 1) + ψ2(1 ⊗ µ) − (µ ⊗ µ)τ2(ψ2 ⊗ ∆), dC(ψ2) = (µ ⊗ ψ2)τ2(∆ ⊗ ∆) − (∆ ⊗ 1)(ψ2) + (1 ⊗ ∆)(ψ2) − (ψ2 ⊗ µ)τ2(∆ ⊗ ∆), dH (ψ3) = (µ ⊗ µ ⊗ µ)τ ((1 ⊗ ∆)∆ ⊗ ψ3) − ψ3(µ) + (µ ⊗ µ ⊗ µ)τ (ψ3 ⊗ (∆ ⊗ 1)∆), dC(ψ3) = (1 ⊗ ψ3)∆ − (∆ ⊗ 12)(ψ3) + (1 ⊗ ∆ ⊗ 1)(ψ3) − (12 ⊗ ∆)(ψ3) + (ψ3 ⊗ 1)∆, =d()==HdC()=0 Journal of Homotopy and Related Structures, vol.
3(1), 2008 35 where τ = τ4τ3τ2 and τ = τ5τ2τ3.

The ﬁrst two 3-cocycle conditions, dH (ψ1) = 0 and dC(ψ1) = dH (ψ2), are de- picted in Fig.

19.

Note that the ﬁrst is the pentagon identity for associativity.

In particular, ψ1 can be regarded as an obstruction to associativity.

The morphism ψ1 is assigned the diﬀerence between the two diagrams that represent the two ex- pressions (ab)c and a(bc).
Thus ψ1 and its diagram are assigned to the change of diagrams corresponding to associativity, and can be seen to form an actual pen- tagon, as depicted in Fig.

20.

In Fig.

20 the usual pentagon relation is depicted and adorned.

Consider, for example, the graph at the top of the pentagon (which corresponds to the associ- ated string ((ab)c)d) and its descendent on the far left (which corresponds to the associated string (a(bc))d).
The top left arrow that represents the parenthetical regrouping is decorated by a graph (encircled by a dotted arc) with a Neptune’s trident on the lower left.

The solid circle at the trident’s junction represents a 3- cocycle.

The trident, then, corresponds to the unassociated string abc.

The dotted encircling of graphs with tridents is given to indicate that these correspond to the arrows of the pentagon relation.
The 3-cocycle condition is written so that the sum of the three cochains on the left of the ﬁgure is equal to the sum of the two on the right.

In general, when graph transformations are given the arrows are denoted by graphs encircled by dotted arcs and with distinguished (singular) vertices indicated by solid colors or solid circles at the vertex.
Figure 20: Hochschild 3-cocycles as movies, part I Similarly, the second condition dC(ψ1) = dH (ψ2) can be represented as a se- quence of applications of the associativity and compatibility conditions as depicted in Fig.

21.

Furthermore, the relations dC(ψ2) = dH (ψ3) and dC(ψ3) = 0 can be obtained by turning the equations in Fig.

19 upside-down.

Similarly, the “movie- moves” in Figs.

20 and 21 can be turned upside-down.

Thus, dC(ψ3) = 0 when the Journal of Homotopy and Related Structures, vol.
3(1), 2008 36 pentagon identity for coassociativity holds, and dC(ψ2) = dH (ψ3) when compati- bility and coassociativity are compared.

Figure 21: Hochschild 3-cocycles as movies, part Il 5.2. Review of Cocycles in Deformation Theory Next we follow [22] for deformation of bialgebras.
A deformation of A = (V, µ, ∆) is a k[[t]]-bialgebra At = (Vt, µt, ∆t), where Vt = V ⊗ k[[t]] and At/(tAt) ∼= A. Deformations of µ and ∆ are given by µt = µ + tµ1 + · · · + tnµn + · · · : Vt ⊗ Vt → Vt and ∆t = ∆ + t∆1 + · · · + tn∆n + · · · : Vt → Vt ⊗ Vt where µi : V ⊗ V → V , ∆i : V → V ⊗ V , i = 1, 2, · · ·, are sequences of maps.
Suppose ¯µ = µ + · · · + tnµn and ¯∆ = ∆+· · ·+tn∆n satisfy the bialgebra conditions (associativity, compatibility, and coassociativity) mod tn+1, and suppose that there exist µn+1 : V ⊗ V → V and ∆n+1 : V → V ⊗ V such that ¯µ + tn+1µn+1 and ¯∆ + tn+1∆n+1 satisfy the bialgebra conditions mod tn+2.

Deﬁne ψ1 ∈ Hom(V ⊗3, V ), ψ2 ∈ Hom(V ⊗2, V ⊗2), and ψ3 ∈ Hom(V, V ⊗3) by ¯µ(¯µ ⊗ 1) − ¯µ(1 ⊗ ¯µ) = tn+1ψ1 mod tn+2, ¯∆¯µ − (¯µ ⊗ ¯µ)τ2( ¯∆ ⊗ ¯∆) = tn+1ψ2 mod tn+2, ( ¯∆ ⊗ 1) ¯∆ − (1 ⊗ ¯∆) ¯∆ = tn+1ψ3 mod tn+2.
(5) (6) (7) For the associativity of ¯µ + tn+1µn+1 mod tn+2 we obtain: (¯µ+tn+1µn+1)((¯µ+tn+1µn+1)⊗1)−(¯µ+tn+1µn+1)(1⊗(¯µ+tn+1µn+1)) = 0 mod tn+2 Journal of Homotopy and Related Structures, vol.

3(1), 2008 37 which is equivalent by degree calculations to: dH (µn+1) = µ(1 ⊗ µn+1) − µn+1(µ ⊗ 1) + µn+1(1 ⊗ µn+1) − µ(µn+1 ⊗ 1) = ψ1.

Similarly, we obtain: (ψ1, ψ2, ψ3) = D(µn+1, ∆n+1).

The cochains (ψ1, ψ2, ψ3), de- ﬁned by deformations (5,6,7) then, satisfy the 3-cocycle condition D(ψ1, ψ2, ψ3) = 0.
This concludes the review of deformation for the 2-cocycle conditions cited from [22].

6.

Towards a Cohomology Theory for Shelves in Coalg Let (X, q) be a coalgebra with a self-distributive linear map.

In this section we present low-dimensional cocycle conditions for q.

We justify our cocycle condi- tions through the use of analogy with Hochschild bialgebra cohomology using dia- grammatics and the deformation theories reviewed in the preceding section.
Both analogies are used interchangeably throughout this section, both in deﬁnitions and computations.

6.1.

Chain Groups Following the diagrammatics of the preceding section, we deﬁne shelf chain groups, for positive integers n and i = 1, .

.

.

, n by: C n,i sh (X; X) = Hom(X ⊗(n+1−i), X ⊗i), i=1C n,i sh(X; X) = ⊕n C n where the subscript ‘sh’ denotes shelf.
Speciﬁcally, the chain groups in low dimen- sions of our concern are: sh (X; X), C 1 C 2 C 3 sh(X; X) = Hom(X, X), sh(X; X) = Hom(X ⊗2, X) ⊕ Hom(X, X ⊗2), sh(X; X) = Hom(X ⊗3, X) ⊕ Hom(X ⊗2, X ⊗2) ⊕ Hom(X, X ⊗3).

To help keep track of the chain groups and their indices, we include the diagram in Fig.

22 and the following explanation.

The reader should be warned that this is not the standard indexing of double complexes.
Instead, the chain groups C n,i are located at position (n + 2 − i, i) in the positive quadrant of the integer lattice.

The chain groups C j are the direct sum of the groups along lines of slope (−1).

Unlike spectral sequences, the diﬀerential dn,i is deﬁned on multiple factors (instead of a single factor C n,i) of C n and has its image in the factor C n+1,i.

Hence dn,i raises the ﬁrst subscript of the cochain groups by 1, and the second subscript indicates the image factor.
In the remaining sections we will deﬁne diﬀerentials that are homomorphisms between the chain groups: dn,i : C n sh(X; X) → C n+1,i sh (X; X)(= Hom(X ⊗(n+2−i), X ⊗i)) and will be deﬁned individually for n = 1, 2, 3 and i = 1, .

.

.

, n + 1, and D1 = d1,1 − d1,2 : C 1 sh(X; X) → C 2 sh(X; X), Journal of Homotopy and Related Structures, vol.
3(1), 2008 38 Figure 22: The lattice of chain groups and diﬀerentials D2 = d2,1 + d2,2 + d2,3 : C 2 D3 = d3,1 + d3,2 + d3,3 + d3,4 : C 3 sh(X; X) → C 3 sh(X; X), sh(X; X) → C 3 sh(X; X).

6.2.

First Diﬀerentials We take d1,2 : Hom(X; X)(= C 1,1 sh (X; X)) → Hom(X, X ⊗2)(= C 2,2 sh (X; X)) to be the coHochschild diﬀerential for the comultiplication d1,2(f ) = (1 ⊗ f )∆ − ∆f + (f ⊗ 1)∆.
Again by analogy with the diﬀerential for multiplication, we take: d1,1 : Hom(X, X)(= C 1,1 sh (X; X)) → Hom(X ⊗2, X)(= C 2,1 to be d1,1(f ) = q(1 ⊗ f ) − f q + q(f ⊗ 1).

Then deﬁne D1 : C 1 by D1 = d1,1 − d1,2.

sh (X; X)) sh(X; X) → C 2 sh(X; X) 6.3.

Second Diﬀerentials We derive second diﬀerentials by analogy with deformation theory, and then show that our deﬁnitions carry through in diagrammatics.
Recall that the self-distributivity, compatibility, and coassociativity are written as: q(q ⊗ 1) = q(q ⊗ q)τ2(1 ⊗ 1 ⊗ ∆), ∆q = (q ⊗ q)τ2(∆ ⊗ ∆), (∆ ⊗ 1)∆ = (1 ⊗ ∆)∆.

where τ2 is the transposition acting on the second and third tensor factors.
As before let Xt = X ⊗ k[[t]] and suppose we have partial deformations ¯q = q + · · · + tnqn and ¯∆ = ∆ + · · · + tn∆n satisfying the above three conditions mod tn+1, and suppose there are qn+1 and ∆n+1 such that ¯q + qn+1 and ¯∆ + ∆n+1 satisfy the three conditions mod tn+2.

Setting ¯q(¯q ⊗ 1) − ¯q(¯q ⊗ ¯q)τ2(1 ⊗ 1 ⊗ ¯∆) = tn+1ξ1 mod tn+2, iC1C2C3C412,11,13,14,12,23,33,24,24,3123234n+2−i Journal of Homotopy and Related Structures, vol.
3(1), 2008 39 ¯∆¯q − (¯q ⊗ ¯q)τ2( ¯∆ ⊗ ¯∆) = tn+1ξ2 mod tn+2, ( ¯∆ ⊗ 1) ¯∆ − (1 ⊗ ¯∆) ¯∆ = tn+1ξ3 mod tn+2, we obtain: [q(qn+1 ⊗ 1) + qn+1(q ⊗ 1)] − [qn+1(q ⊗ q)τ2(1 ⊗ 1 ⊗ ∆)+ q(qn+1 ⊗ q)τ2(1 ⊗ 1 ⊗ ∆) + q(q ⊗ qn+1)τ2(1 ⊗ 1 ⊗ ∆) + q(q ⊗ q)τ2(1 ⊗ 1 ⊗ ∆n+1)] = ξ1, [∆qn+1 + ∆n+1q] − [(qn+1 ⊗ q)τ2(∆ ⊗ ∆) + (q ⊗ qn+1)τ2(∆ ⊗ ∆)+ (q ⊗ q)τ2(∆n+1 ⊗ ∆) + (q ⊗ q)τ2(∆ ⊗ ∆n+1)] = ξ2, [(∆n+1 ⊗ 1)∆ + (∆ ⊗ 1)∆n+1] − [(1 ⊗ ∆n+1)∆ + (1 ⊗ ∆)∆n+1] = ξ3.
A natural requirement is D2(qn+1, ∆n+1) = (ξ1, ξ2, ξ3), so we will deﬁne D2 : sh(X; X) → C 3 sh (X; X)(= sh(X; X) by D2 = d2,1 + d2,2 + d2,3.

Let η1 ∈ C 2,1 sh (X; X)(= Hom(X, X ⊗2)) be cochains.
Then, C 2 Hom(X ⊗2, X)) and η2 ∈ C 2,2 d2,1(η1, η2) = [q(η1 ⊗ 1) + η1(q ⊗ 1)] − [η1(q ⊗ q)τ2(1 ⊗ 1 ⊗ ∆) + q(η1 ⊗ q)τ2(1 ⊗ 1 ⊗ ∆) +q(q ⊗ η1)τ2(1 ⊗ 1 ⊗ ∆) + q(q ⊗ q)τ2(1 ⊗ 1 ⊗ η2)] d2,2(η1, η2) = [∆η1 + η2q] − [(η1 ⊗ q)τ2(∆ ⊗ ∆) + (q ⊗ η1)τ2(∆ ⊗ ∆) +(q ⊗ q)τ2(η2 ⊗ ∆) + (q ⊗ q)τ2(∆ ⊗ η2)] d2,3(η1, η2) = [(η2 ⊗ 1)∆ + (∆ ⊗ 1)η2] − [(1 ⊗ η2)∆ + (1 ⊗ ∆)η2] .

In fact, d2,3 = dC, the same as the coHochschild 2-diﬀerential for the comultiplica- tion.
Figure 23: Diagrams for 2-cochains Figure 24: The ﬁrst 2-diﬀerential d2,1 The diagrammatic conventions for q, a 2-cochain η1 ∈ Hom(X ⊗2, X), and ∆, a 2-cochain η2 ∈ Hom(X, X ⊗2) are depicted from left to right, respectively, in Fig.

23.

The ﬁrst and second diﬀerentials d2,1(η1, η2), d2,2(η1, η2) are depicted in Fig.

24 and Fig.

25, respectively.

Here we note that these diagrams agree with those for h1h2qD Journal of Homotopy and Related Structures, vol.
3(1), 2008 40 Figure 25: The second 2-diﬀerential d2,2 Figure 26: d2,1(D1(f )) = 0 Hochschild bialgebra cohomology in the sense that they are obtained by the follow- ing process: (1) Consider the diagrams of the equality in question (in this case the self-distributivity condition and the compatibility), (2) Mark exactly one vertex of such a diagram, (3) Take a formal sum of such diagrams over all possible markings.

In Fig.
In Fig.

24, the ﬁrst two terms correspond to the LHS of q(q ⊗1) = q(q ⊗q)τ2(12 ⊗∆), and one of the two white triangular vertices is marked by a black vertex, representing the 2-cochain η1, while the remaining white vertex represents q.

The negative four terms correspond to the RHS, and the last term has a circle, representing η2 while unmarked ones in the rest represent ∆.

The same procedure for the compatibility gives rise to Fig.

25.

Lemma 6.1.

For any f ∈ C 1 sh(X; X), we have D2D1(f ) = 0.

Proof.
Proof.

A proof is depicted in Fig.

26 and Fig.

27.

By assumption, η1 = d1,1(f ) and η2 = d1,2(f ).

Therefore, as in the case of Hochschild homology, marked vertices representing η1 and η2 are replaced by formal sum of three diagrams representing Journal of Homotopy and Related Structures, vol.

3(1), 2008 41 Figure 27: d2,2(D1(f )) = 0 d1,1(f ) and d1,2(f ), see Fig.

16.

The situation in which the ﬁrst two terms are replaced by three terms each is depicted in the top two lines of Fig.

26.
26.

A white circle on an edge represents f .

The bottom three lines show replacements for the remaining four negative terms.

Then the terms represented by identical graphs cancel directly.

If a white circle representing f appears near the boundary, then we use the self-distributive axiom to relate this to another term.

For example, the ﬁrst term on the top left cancels with the third term on the bottom row since f is on the second tensor factor at the bottom of each.
To facilitate the reader’s understanding of the computation we present the fol- lowing sequences: 1, −2, 3, 4, −5, 2 and −6, 5, −7, −8, 7, −3, −9, 6, −1, 9, −4, 8.

Label the diagrams below the arrows in Fig.

26 in order with these numbers.

The mi- nus sign indicates the sign of the given term on the given side of the equation, and the number indicates which diagrams cancel which.

A similar labelling can be accomplished in Fig.

27.

 We also note the following restricted version: Lemma 6.2.
Let f ∈ C 1 Hom(X; X ⊗2), then D2(d1,1(f ), 0) = 0. sh(X; X) = Hom(X; X).

If d1,2(f ) = 0 ∈ C 2,2 sh (X; X) = Proof.

The conclusion is restated by the following condition: d2,i(d1,1(f ), 0) = 0 for i = 1, 2, since d1,1(f ) is not in the domain of the diﬀerential d2,3.

Then one computes d2,i(η1, 0) for η1 = d1,1(f ) either directly, or diagrammatically using Figs.

26, and 27, without trivalent vertices that are encircled.

 Journal of Homotopy and Related Structures, vol.

3(1), 2008 42 6.4.
3(1), 2008 42 6.4.

Third Diﬀerentials Throughout this section, we consider only self-distributive linear maps for co- commutative coalgebras with counits.

The map q needs not be compatible with the counit (cf.

Proposition 3.2), but there must be such a counit present because sometimes the diagonal has to be deﬁned in a categorical context.
In any case, 3-diﬀerentials d3,i : C 3 sh(X; X)(= ⊕3 j=1Hom(X ⊗(4−j), X ⊗j)) → C 4,i sh (X; X)(= Hom(X ⊗(n+2−i), X ⊗i)) are deﬁned in Appendix AppendixA, for i = 1, 2, 3.

For i = 4, it is deﬁned by the same map as the diﬀerential for ∆ for co-Hochschild cohomology (the pentagon identity for the comultiplication).
These diﬀerentials are deﬁned by direct analogues with Hochschild diﬀerentials in diagrammatics, and we will justify our deﬁnition in two more ways: (1) 2-cochains vanish under these maps, (2) 3-cocycles of quandle and Lie algebra cohomology are realized in these formulas as discussed in the next section.

The deﬁning formulas and diagrammatic methods to derive them are de- ferred to Appendix AppendixA, and we proceed with statements of lemmas we need to continue with this cohomology theory.

Lemma 6.3.
Lemma 6.3.

Let (η1, 0) ∈ Hom(X ⊗2, X) ⊂ C 2 d3,iD2(η1, 0) = 0 for i = 1, 2, 3. sh(X; X) (so that η2 = 0).

Then Proof and diagrams are included in Appendix AppendixB.

6.5.

Cohomology Groups Now we use these diﬀerentials to deﬁne cohomology groups for self-distributive linear maps for objects in Coalg.

Let (X, ∆) be an object in Coalg, and q : X ⊗X → X be a self-distributive linear map.

Then Lemmas 6.1 implies Corollary 6.4.

0 → C 1 sh(X; X) D1→ C 2 sh(X; X) D2→ C 3 sh(X; X) is a chain complex.
This enables us to deﬁne the following cohomology related groups: Deﬁnition 6.5.

The 1-cocycle and cohomology groups are deﬁned by: sh (X; X) | d1,i(f ) = 0} sh (X; X) = {f ∈ C 1,i sh (X; X) = Z 1,i H 1,i for i = 1, 2, and For dimension 2, we deﬁne Z 2 H 2 sh(X; X) = Z 2 sh(X; X)/B2 sh(X; X).

H 1 sh(X; X) = Z 1,1 sh (X; X) ⊕ Z 1,2 sh(X; X) = Ker(D2), B2 sh (X; X).
sh(X; X) = Im(D1), and Since the 2-cocycle conditions were formulated directly from a deformation theory formulation, we have the following: Proposition 6.6.

Let Xt = X ⊗ k[[t]] and suppose we have partial deformations ¯q = q + · · · + tnqn and ¯∆ = ∆ + · · · + tn∆n satisfying the above three conditions Journal of Homotopy and Related Structures, vol.

3(1), 2008 43 mod tn+1, so that they deﬁne a self-distributive map in Coalg mod tn+1.
Then there exist qn+1 : X ⊗ X → X and ∆n+1 : X → X ⊗ X such that ¯q + tn+1qn+1 and ¯∆ + tn+1∆n+1 satisfy the three conditions mod tn+2, so that they deﬁne a self- distributive linear map mod tn+2, if and only if (qn+1, ∆n+1) satisfy the 2-cocycle condition: D2(qn+1, ∆n+1) = 0.

To extend the chain complex to dimension 3, we need to make some restrictions.
The reason the following restriction are made (on the chain groups C n,1 and the diﬀerential maps) is that in the deﬁnition of the 3-diﬀerential, we assumed that the comultiplication is cocommutative and is not deformed (η2 = 0).

We do not know if a general 3-diﬀerential is deﬁned in the case of a deformed comultiplication.

Thus, for 3-cocycles, we assume that (X, ∆, q) consists of an object (X, ∆) in sh (X; X)) n = CoComCoalg, with a self-distributive linear map q.
Let dn,i be the restriction of dn,i to C n,1 n+1 sh (X; X) = Hom(X ⊗n, X), and D 1 = dn,i|(C n,1 1 = d1,1, D for n = 2, 3 and i = 1, 2, 3.

Then consider the sequence i=1 dn,i 1 C : 0 → Z 1,2 sh (X; X) D 1→ C 2,1 sh (X; X) D 2→ C 3,1 sh (X; X) D 3→ C 4,1 sh (X; X).

The prime in the following notation D i is just a convention to indicate that they are restricted maps.

Proof.

The condition D 1 to Z 1,2 tion of D Theorem 6.7.
Let (X, ∆) be an object in CoComCoalg and q : X ⊗ X → X be a self-distributive linear map.

Then C is a chain complex.

2D 1 = 0 follows from Lemma 6.2, as the domain restric- sh (X; X) is the same as the assumption of the lemma.

3D Now we prove D 2 = 0.

First note that the domain restriction of D 2 means 2(η10) = D2(η1, 0), when we set η2 = 0.

Note also that the image of d2,1 does that D not land in the domain of d3,4.
Hence it is suﬃcient to prove that d3,iD2(η1, 0) = 0 for i = 1, 2, 3.

This is Lemma 6.3.

 This enables us to deﬁne: Deﬁnition 6.8.

The 1-cocycle and cohomology group are deﬁned as: H 1,1 sh (X; X) = {f ∈ Z 1,2 and the 2- and 3-coboundary, cocycle, and cohomology groups are deﬁned as: sh (X; X) | d1,1(f ) = 0}, sh (X; X) = Z 1,1 Bj,1(X; X) = Image(D Z j,1(X; X) = Ker(D H j,1(X; X) = Z j,1(X; X)/Bj,1(X; X) j−1), j), for j = 2, 3.
The cocycles in these theories are called shelf cocycles.

The name is a bit of a notational compromise.

They should be called “cocycles for self-distributive linear maps for objects in the category of cocommutative coalgebras with counit,” which would inevitably get shortened to cocococo-cycles.

There are two points here.

First, the analogy “quandle is to rack as rack is to shelf” does not extend to the terminology for shelf-cohomology.
More importantly, we do not require q to be compatible with counit in deﬁning cohomology theories, yet we call them shelf cocycles for short.

Journal of Homotopy and Related Structures, vol.

3(1), 2008 44 7.

Relations to Other Cohomology Theories In this section we examine relations of these cocycles to those in other cohomology theories, speciﬁcally the original quandle cohomology theories [10] and Lie algebra cohomology.

7.1.
7.1.

Quandle Cohomology In this section we present procedures that produce shelf 2- and 3-cocycles from quandle 2- and 3-cocycles, respectively, and show that non-triviality is inherited by these processes.

First we brieﬂy review the deﬁnition of quandle 2- and 3-cocycles.
A quandle 2-cocycle is a linear function φ deﬁned on the free abelian group generated by pairs of elements (x, y) taken from a quandle X such that φ(x, y) − φ(x, z) + φ(x y, z) − φ(x z, y z) = 0, ∀x, y, z ∈ X and φ(x, x) = 0 for all x ∈ X.

The function φ takes values in some ﬁxed abelian group A.
Similarly a 3-cocycle is a function θ with the properties that θ(x, y, z)+θ(xz, yz, w)+θ(x, z, w) = θ(xy, z, w)+θ(x, y, w)+θ(xw, yw, z w), and θ(x, x, y) = θ(x, y, y) = 0 for all x, y, z, w ∈ X. Quandle cohomology groups H n these conditions, see [10, 11] for details.

Q(X; A) were deﬁned based on These cocycles were used to develop invariants of classical knots and knotted surfaces.

We summarize the construction as follows.


Given a quandle homomor- phism from the fundamental quandle of a codimension 2 embedding to the ﬁnite quandle X, and given a cocycle (φ or θ), we evaluate the cocycle at the incom- ing quandle elements near each 0-dimensional multiple point (crossing and triple point, respectively), in the projection of the knot or knotted surface.
These values are added together in the abelian group A, and the collection of the results are formally collected together as a multiset over all homomorphisms.

The cocycle in- variants are fairly powerful in determining properties of knots and knotted surfaces.

Generalizations have been discovered [1, 8, 9].

Recall that W = k ⊕ kX (V = kX) is the direct sum of the ﬁeld k and the vector space whose basis is comprised of the elements in X, and the self-distributive map q deﬁned on V was extended to W .
Theorem 7.1.

For a quandle 2-cocycle φ with the coeﬃcient group A = k, deﬁne ˆφ : W ⊗ W → W by linearly extending ˆφ(x ⊗ y) = φ(x, y), ˆφ(1 ⊗ x) = 1, and ˆφ(x ⊗ 1) = ˆφ(1 ⊗ 1) = 0 for x, y ∈ X.

Then ˆφ satisﬁes d2,1( ˆφ, 0) = 0.

Proof.

We write expressions such as (a + (a + Ax).

Then x axx) in the more compact form (a + x axx) ⊗ (b + y ayy) ⊗ (c + z azz) Journal of Homotopy and Related Structures, vol.
3(1), 2008 45 = abc(1 ⊗ 1 ⊗ 1) + Abc(x ⊗ 1 ⊗ 1) + aBc(1 ⊗ y ⊗ 1) + ABc(x ⊗ y ⊗ 1) + abC(1 ⊗ 1 ⊗ z) + AbC(x ⊗ 1 ⊗ z) + aBC(1 ⊗ y ⊗ z) + ABC(x ⊗ y ⊗ z) In order to compute d2,1( ˆφ, 0) on expressions such as the one above, we must compute it on the eight tensor products (1 ⊗ 1 ⊗ 1) through (x ⊗ y ⊗ z).

These calculations are summarized in the table below (juxtaposition or commas are used in place of ⊗ for typesetting purposes).
0 0 0 0 0 0 1 φ(x, y) 0 ⊗ 1 0 ⊗ 1 1 ⊗ 1 φ(x, y) ⊗ 1 0 ⊗ z 0 ⊗ z 1 ⊗ z φ(x, y) ⊗ z 0 0 0 0 0 0 1 φ(x y, z) 0 ⊗ 1 0 ⊗ 1 1 ⊗ 1 (x y) ⊗ 1 0 ⊗ z 0 ⊗ z 1 ⊗ z (x y) ⊗ z φ(x, z) φ(x, z), 1 1, y z φ(x, z), y z 0 1 φ(x z, y z) x z, 1 1, y z x z, y z 0 0 0 x z, 1 1, φ(y, z) x z, φ(y, z) 1111 x111 11y1 x1y1 1z1z xz1z 1zyz xzyz 1⊗1⊗ 1 x⊗1⊗ 1 1⊗y ⊗ 1 x⊗y ⊗1 1⊗ 1⊗z x⊗ 1 ⊗ z 1⊗y ⊗z x⊗ y ⊗z Thus the calculation becomes: d2,1( ˆφ, 0) (a + x axx) ⊗ (b + y ayy) ⊗ (c + z azz) = d2,1( ˆφ, 0) (abc(1 ⊗ 1 ⊗ 1) + Abc(x ⊗ 1 ⊗ 1) + aBc(1 ⊗ y ⊗ 1) + ABc(x ⊗ y ⊗ 1) + (abC(1 ⊗ 1 ⊗ z) + AbC(x ⊗ 1 ⊗ z) + aBC(1 ⊗ y ⊗ z) + ABC(x ⊗ y ⊗ z)) = [ 2aBC + ABC φ(x, y) + ABC φ(x y, z) ] − [ 2aBC + ABCφ(x z, y z) + ABCφ(x, z) ] Remark 7.2.
On the other hand, without the factor k in W , the original 2-cocycles do not give rise to shelf cocycles.

Consider V to have as its basis the trivial quandle X and let q : V ⊗ V → V be induced from so that q(x ⊗ y) = x for all x, y ∈ X.

If η2 = 0 and η1 is any linear function, then d2,1(η1, 0)(x ⊗ y ⊗ z) = −x = 0 ∈ V .

But in quandle cohomology any function is a cocycle.

Theorem 7.3.
Theorem 7.3.

For the cocycles in Theorem 7.1, the following holds: If φ is not a coboundary, then ˆφ is not a coboundary.

In particular, if H 2 Q(X; k) = 0, then H 2,1 sh (W ; W ) = 0.

Proof.

A function φ is a coboundary if and only if there is a 1-cochain such that δg = φ, which is written as φ(x, y) = g(x) − g(x y) for any x, y ∈ X (see [10]).

Suppose ˆφ is a coboundary, then there is a 1-cochain f such that D1(f ) = ˆφ.
A 1-cochain f , in this case, is a map f : W → W (= k ⊕ kX), which is written as Journal of Homotopy and Related Structures, vol.

3(1), 2008 f (a+ k, and f1(a + axx) = f0(a+ axx)+f1(a+ axx), where a ∈ k, x ∈ X, f0(a+ axx) ∈ kX.
The condition D1(f ) = ˆφ, then, is written as: 46 axx) ∈ ˆφ( (a + axx) ⊗ (b + byy) ) = a = D1(f )( (a + axx) ⊗ (b + ayy) ) by + x,y axby φ(x, y) = {q(1 ⊗ f ) − f q + q(f ⊗ 1)}( (a + axx) ⊗ (b + ayy) ) In particular, for (a + axx, b + byy) = (x, y), we obtain: φ(x, y) = (x f1(y)) − (f0(x y) + f1(x y)) + (f0(x) + f1(x) y), and by comparing the k and kX factors, this reduces to φ(x, y) = f0(x) − f0(x y) and f1(x y) = x f1(y) + f1(x) y.
In particular, the ﬁrst equation implies that φ is a coboundary and causes a contradiction.

 Next we consider 3-cocycles.

Theorem 7.4.

For a quandle 3-cocycle θ with the coeﬃcient group A = k, deﬁne ˆθ : W ⊗ W ⊗ W → W by linearly extending ˆθ(x ⊗ y ⊗ z) = θ(x, y, z), ˆθ(1 ⊗ y ⊗ z) = 1, and ˆθ(x⊗y⊗1) = ˆθ(x⊗1⊗z) = ˆθ(x⊗1⊗1) = ˆθ(1⊗y⊗1) = ˆθ(1⊗1⊗z) = ˆθ(1⊗1⊗1) = 0 for x, y, z ∈ X.

Then ˆθ is a shelf 3-cocycle: d3,1(ˆθ, 0, 0) = 0.

The proof is found in Appendix AppendixD.

Theorem 7.5.
Theorem 7.5.

For the cocycles in Theorem 7.4, the following holds: If θ is not a coboundary, then ˆθ is not a coboundary.

In particular, if H 3 Q(X; k) = 0, then H 3,1 sh (W ; W ) = 0.

Proof.

The proof is similar to that of Theorem 7.3.

The cochain θ is a coboundary if and only if there is a 2-cochain φ such that δφ = θ, which is written as θ(x, y, z) = φ(x, y) + φ(x y, z) − φ(x, z) − φ(x z, y z) for any x, y, z ∈ X (see [10]).

Suppose ˆθ is a coboundary.


Then there is a 2-cochain f such that D2(f ) = ˆθ.

A 2-cochain f , in this case, is a map f : W ⊗ W → W (= k ⊕ kX), that is written as: f ( (a + axx) ⊗ (b + byy) ) = f0( (a + axx) ⊗ (b + byy) )+ f1( (a + axx) ⊗ (b + byy) ), where a ∈ k, x, y ∈ X, f0(a+ axx) ∈ kX.

We take speciﬁc axx) ∈ k, and f1(a+ values and compute ˆθ = D2(f ) evaluated at x⊗y⊗z.
We have ˆθ(x⊗y⊗z) = θ(x, y, z), and D2(f )(x ⊗ y ⊗ z) = [ f0(x ⊗ y) + f1(x ⊗ y) z + f0((x y) ⊗ z) + f1((x y) ⊗ z) ] − [ f0((x z) ⊗ (y z)) + f1((x z) ⊗ (y z)) + f0(x ⊗ z) +f1(x ⊗ z) (y z) + (x z) f1(y ⊗ z) ], Journal of Homotopy and Related Structures, vol.
3(1), 2008 47 and comparing the elements on k, we obtain θ(x, y, z) = f0(x ⊗ y) + f0((x y) ⊗ z) − f0((x z) ⊗ (y z)) − f0(x ⊗ z), so that by deﬁning φ(x, y) = f0(x ⊗ y) for any x, y ∈ X, we obtain a contradiction θ = δφ.

 7.2.

Lie Algebra Cohomology Let q : N ⊗ N → N be the map deﬁned in Lemma 3.2, where N = k ⊕ g for a Lie algebra g over a ground ﬁeld k. Let ψ : g × g → g be a Lie algebra 2-cocycle, with adjoint action.
Then ψ is bilinear and satisﬁes ψ(y, x) = −ψ(x, y), [ψ(x, y), z] + [ψ(y, z), x] + [ψ(z, x), y] + ψ([x, y], z) + ψ([y, z], x) + ψ([z, x], y) = 0.

It deﬁnes a linear map ψ : g ⊗ g → g. The following result says that a Lie algebra 2-cocycle gives rise to a shelf 2-cocycle, when the comultiplication is ﬁxed and undeformed (η2 = 0).

Theorem 7.6.

Let ψ : g × g → g be a Lie algebra 2-cocycle with adjoint action.
Deﬁne ˆψ : N ⊗ N → N by ˆψ((a + x) ⊗ (b + y)) = ψ(x ⊗ y) for a, b, c ∈ k, x, y, z ∈ g. Then ˆψ is a shelf 2-cocycle: d2,1( ˆψ, 0) = d2,2( ˆψ, 0) = 0.

Proof.
Proof.

One computes: d2,1( ˆψ, 0)( (a + x) ⊗ (b + y) ⊗ (c + z) ) = { q( ψ(x, y) ⊗ (c + z) ) + ˆψ( (ab + bx + [x, y]) ⊗ (c + z) )} − { { ˆψ(q ⊗ q) + q( ˆψ ⊗ q) + q(q ⊗ ˆψ) }τ2( (a + x) ⊗ (b + y) ⊗( (c + z) ⊗ 1 + 1 ⊗ z ) ) } = { (cψ(x, y) + [ψ(x, y), z] ) + ( bψ(x, z) + ψ([x, y], z) ) } − { ( cψ(x, y) + ψ([x, z], y) + ψ(x, [y, z]) ) +( bψ(x, z) + [ψ(x, z), y] ) + ( [x, ψ(y, z)] ) } = 0.

The other equality d2,2( ˆψ, 0) = 0 is checked similarly.
Next we consider Lie algebra 2-cocycles ψ : g × g → k with the trivial rep- resentation on the ground ﬁeld k. In this case the 2-cocycle condition is being skew-symmetric and satisfying the Jacobi identity: ψ([x, y], z) + ψ([y, z], x) + ψ([z, x], y) = 0. ψ(y, x) = −ψ(x, y), Let g = kγ + g where γ ∈ g and [γ, z] = 0 for all z ∈ g. Then g is a Lie algebra with Lie bracket given by [aγ + x, bγ + y] = [x, y] ∈ g ⊂ g.
For a given 2-cocycle ψ : g × g → k, deﬁne ψ : g × g → g by ψ(aγ + x, bγ + y) = ψ(x, y)γ ∈ g.

Then we claim that ψ satisﬁes the 2-cocycle condition with adjoint action.

We compute: [ψ(aγ + x, bγ + y), cγ + z] = [ψ(x, y)γ, cγ + z] = [ψ(x, y)γ, z] = 0.

Journal of Homotopy and Related Structures, vol.

3(1), 2008 48 Therefore the ﬁrst three terms involving the adjoint action, in fact, vanish by con- struction.
The last three terms reduce to the 2-cocycle condition of ψ, since ψ([aγ + x, bγ + y], cγ + z) = ψ([x, y], cγ + z) = ψ([x, y], z).

Hence this reduces to the previous case.

We summarize this situation as: Theorem 7.7.

A Lie algebra 2-cocycle valued in the ground ﬁeld with trivial rep- resentation gives rise to a shelf 2-cocycle.

Next we investigate relations for 3-cocycles.


A Lie algebra 3-cocycle with adjoint action is a totally skew-symmetric trilinear map ζ : g × g × g → g for a Lie algebra g that satisﬁes [ζ(x, y, z), w] − [ζ(x, y, w), z] + [ζ(x, z, w), y] − [ζ(y, z, w), x] − ζ([x, y], z, w) + ζ([x, z], y, w) − ζ([x, w], y, z) + ζ([y, z], x, w) − ζ([y, w], x, z) + ζ([z, w], x, y) = 0.

This deﬁnes a linear map ζ : g ⊗ g ⊗ g → g. Recall that we deﬁned N = k ⊕ g. Theorem 7.8.
Let ζ : g × g × g → g be a Lie algebra 3-cocycle with adjoint action.

Deﬁne ˆζ : N ⊗ N ⊗ N → N by ˆζ((a + x) ⊗ (b + y) ⊗ (c + z)) = ζ(x ⊗ y ⊗ z).

Then ˆζ satisﬁes d3,1(ˆζ, 0, 0) = 0.

Proof.

There are four positive (L1, L2, L3, L4) and three negative (R1, R2, R3) terms in d3,1(ˆζ, 0, 0) (the last negative term vanishes because ξ2 = 0 in (ξ1, ξ2, ξ3) = (ˆζ, 0, 0)).

We evaluate each term for a general element (a + x) ⊗ (b + y) ⊗ (c + z) ⊗ (d + w) as before.
The ﬁrst term L1 is q( ζ(x, y, z) ⊗ (d + w) ) = d ζ(x, y, z) + [ζ(x, y, z), w].
The second term L2 is ˆζ(q ⊗ q ⊗ 1)( (a + x) ⊗ {(c + z) ⊗ (b + y) ⊗ 1 + 1 ⊗ (b + y) ⊗ z} ⊗ w ) = ˆζ( (ac + cx + [x, z]) ⊗ (b + y) ⊗ w + (a + x) ⊗ [y, z] ⊗ w ) = c ζ(x, y, w) + ζ([x, z], y, w) + ζ(x, [y, z], w) By similar calculations the remaining terms give L3 : b ζ(x, z, w) + [ζ(x, z, w), y] L4 : [x, ζ(y, z, w)] R1 : b ζ(x, z, w) + ζ([x, y], z, w) R2 : c ζ(x, y, w) + [ζ(x, y, w), z] R3 : d ζ(x, y, z) + ζ([x, w], y, z) + ζ(x, [y, w], z) + ζ(x, y, [z, w]) and the result (L1 + L2 + L3 + L4) − (R1 + R2 + R3) = 0 follows.
Journal of Homotopy and Related Structures, vol.

3(1), 2008 49 Theorem 7.9.

For the cocycles in Theorem 7.7, the following holds: If ψ is not a coboundary, then ˆψ is not a coboundary.

In particular, if the second cohomology group of the Lie algebra cohomology with adjoint action is non-trivial (H 2 Lie(g; g) = 0), then H 2 sh(N ; N ) = 0.

Proof.

The proof is similar to that of Theorem 7.3.


If ψ is a coboundary, then there is a 1-cochain g such that δg = ψ, which is written as ψ(x, y) = [x, g(y)] + [g(x), y] − g([x, y]) for any x, y ∈ g. Suppose ˆφ is a coboundary, then there is a 1-cochain f such that D1(f ) = ˆψ.
A 1-cochain f , in this case, is a linear map f : N → N (= k ⊕ g), that is written as f (a+x) = f0(a+x)+f1(a+x), where a ∈ k, x ∈ g, f0(a+x) ∈ k, and f1(a+x) ∈ g. The condition D1(f ) = ˆφ, then, is written as ˆψ( (a + x) ⊗ (b + y) ) = ψ(x, y) = D1(f )( (a + x) ⊗ (b + y) ) = {q(1 ⊗ f ) − f q + q(f ⊗ 1)}( (a + x) ⊗ (b + y) ) In particular, for (a + x, b + y) = (x, y), we obtain ψ(x, y) = q(x ⊗ f1(y)) − (f0(q(x ⊗ y)) + f1(q(x ⊗ y))) + (f0(x) + q(f1(x) ⊗ y)) = [x, f1(y)] − f0([x, y]) − f1([x, y]) + f0(x) + [f1(x), y].
Comparing the elements in k and g in the image, we obtain 0 = −f0([x, y]) + f0(x), ψ(x, y) = [x, f1(y)] − f1([x, y]) + [f1(x), y], and the second implies that ψ is a coboundary.

 Let Wp be the Witt algebra, a Lie algebra over the ﬁeld Fp with p elements for a prime p > 3.

Speciﬁcally, Wp has basis ea, a ∈ Fp and has bracket deﬁned by [ea, eb] = (b − a)ea+b.
Then it is known [5] (we thank J. Feldvoss for informing us) that the Lie algebra cohomology with trivial action H 2 Lie(Wp; Fp) is one-dimensional and generated by the Virasoro cocycle c(ea, e−a) = a(a2 − 1) (otherwise zero).

Let W p be the object in CoComCoalg with a self- distributive linear map q constructed in Section 3.1.

Then we have: p = kγ ⊕ Wp, N (W p) = k ⊕ W Corollary 7.10.

H 2 sh(N (W p); N (W p)) = 0.

8.
8.

A Compendium of Questions What are more precise relationships among the Lie bracket, self-distributivity, solutions to the Yang-Baxter equations, Hopf algebras, and quantum groups?

Can the cocycles constructed herein be used to construct invariants of knots and knotted surfaces?

Can the coboundary maps be expressed skein theoretically?

Is there a spectral sequence that is associated to a ﬁltration of the chain groups?

If so, what are the diﬀerentials?

What does it compute?


Are there non-trivial cocycles among any of the trigonometric shelves?

The proofs of the main theorems come from grinding through computation.

Are there more conceptual proofs?

As the referee asked, are Journal of Homotopy and Related Structures, vol.

3(1), 2008 50 there maps among homology groups, such as H 2 sh?

How can the theory be extended to higher dimensions, such as to higher dimensional Lie algebras, or Lie 2-algebras?
How, if at all, do the Zamolodchikov tetrahedron equation and the Jacobiator identity of a Lie 2-algebra, relate to shelf cohomology?

Can it be shown to be a cohomology theory in the case when ξ2 and ξ3 are non-zero?

Is there a spin-foam interpretation of the 3-cocycle conditions?

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Journal of Homotopy and Related Structures, vol.

3(1), 2008 52 A. Deﬁnition and diagrammatics of the third diﬀerentials First we explain the diagrammatics.
Recall that 3-cocycle conditions in Hochschild cohomology correspond to two diﬀerent sequences of relations applied to graphs that change one graph to another.

At the top of Fig.

28, a graph representing q(q ⊗ 1)(q ⊗ 12) is depicted.

There are two ways to apply sequences of self-distributivity to this map to get the map represented by the bottom graphs.

Let ξj ∈ Hom(X ⊗(4−j), X ⊗j) ⊂ C 3 sh(X; X), j = 1, 2, 3.
A 3-cochain ξ1 represented by a black triangular vertex with three bottom edges and a single top edge corre- sponds to applying the self-distributivity relation to change a graph to another, and corresponds to where the self-distributivity relation was applied.

The two diﬀerent sequences are shown at the left and right of the ﬁgure.

These sequences give rise to the LHS and RHS of d3,1(ξ1, ξ2, ξ3).

Similar graphs are obtained as shown in Figs.

29 and 30.
29 and 30.

The third diﬀerential is deﬁned as D3 = d3,1 + d3,2 + d3,3 + d3,4.
The diﬀerentials thus obtained are: d3,1(ξ1, ξ2, ξ3) = [q(ξ1 ⊗ 1) + ξ1(q ⊗ q ⊗ 1)(1 ⊗ τ ⊗ 12)(12 ⊗ ∆ ⊗ 1) + q(ξ1 ⊗ q)(12 ⊗ τ ⊗ 1)(13 ⊗ ∆)(12 ⊗ q ⊗ 1) (1 ⊗ τ ⊗ 12)(12 ⊗ ∆ ⊗ 1) + q(q ⊗ ξ1)(q ⊗ q ⊗ 13)(1 ⊗ τ ⊗ 14) (12 ⊗ ∆ ⊗ 13)(12 ⊗ τ ⊗ 12)(1 ⊗ τ ⊗ τ ⊗ 1)(12 ⊗ ∆ ⊗ ∆) + q(q ⊗ q)(q ⊗ q ⊗ q ⊗ q)(1 ⊗ τ ⊗ 12 ⊗ τ ⊗ 1)(12 ⊗ ∆ ⊗ 13) (12 ⊗ τ ⊗ 12)(1 ⊗ τ ⊗ τ ⊗ 12)(12 ⊗ ∆ ⊗ ξ3)] − [ξ1(q ⊗ 12) + q(ξ1 ⊗ q)(12 ⊗ τ ⊗ 1)(13 ⊗ ∆) + ξ1(q ⊗ q ⊗ q)(1 ⊗ τ ⊗ 13)(12 ⊗ ∆ ⊗ 12)(12 ⊗ τ ⊗ 1)(13 ⊗ ∆) + q(q ⊗ q)(1 ⊗ τ ⊗ 1)(q ⊗ q ⊗ 12)(12 ⊗ ∆ ⊗ 12)(13 ⊗ ξ2) (12 ⊗ τ ⊗ 1)(13 ⊗ ∆) + q(q ⊗ q)(q ⊗ q ⊗ q ⊗ q)(1 ⊗ τ ⊗ τ ⊗ 13)(14 ⊗ τ ⊗ τ ⊗ 12) (1 ⊗ τ ⊗ ∆ ⊗ τ ⊗ 1)(12 ⊗ ∆ ⊗ ξ3)] d3,2(ξ1, ξ2, ξ3) = [∆ξ1 + ξ2(q ⊗ q)(1 ⊗ τ ⊗ 1)(12 ⊗ ∆) + (q ⊗ q)(1 ⊗ τ ⊗ 1)(ξ2 ⊗ ∆)(12 ⊗ q)(1 ⊗ τ ⊗ 1)(12 ⊗ ∆) + (q ⊗ q)(1 ⊗ τ ⊗ 1)(∆ ⊗ 12))(q ⊗ ξ2)(1 ⊗ τ ⊗ 1)(12 ⊗ ∆) + (q ⊗ q)(q ⊗ τ ⊗ q)(12 ⊗ q ⊗ q ⊗ 12)(1 ⊗ τ ⊗ τ ⊗ τ ⊗ 1) (∆ ⊗ 12 ⊗ τ ⊗ ∆)(1 ⊗ τ ⊗ 13)(1 ⊗ ∆ ⊗ ξ3)] − [ξ2(q ⊗ 1) + (q ⊗ q)(1 ⊗ τ ⊗ 1)(ξ2 ⊗ ∆) + (ξ1 ⊗ q)(13 ⊗ q ⊗ 1)(12 ⊗ τ ⊗ 12)(1 ⊗ τ ⊗ τ ⊗ 1)(∆ ⊗ ∆ ⊗ ∆) + (q ⊗ ξ1)(q ⊗ 14)(12 ⊗ τ ⊗ 12)(1 ⊗ τ ⊗ τ ⊗ 1)(∆ ⊗ ∆ ⊗ ∆) + (q ⊗ q)(q ⊗ q ⊗ q ⊗ q)(1 ⊗ τ ⊗ τ ⊗ τ ⊗ 1)(12 ⊗ τ ⊗ τ ⊗ ∆) (1 ⊗ τ ⊗ τ ⊗ 12)(∆ ⊗ ∆ ⊗ ξ3)] d3,3(ξ1, ξ2, ξ3) = [(∆ ⊗ 1)ξ2 + (ξ2 ⊗ q)(1 ⊗ τ ⊗ 1)(∆ ⊗ ∆) Journal of Homotopy and Related Structures, vol.
3(1), 2008 53 + (q ⊗ q ⊗ q)(1 ⊗ τ ⊗ 13)(12 ⊗ ∆ ⊗ 12)(12 ⊗ τ ⊗ 1)(ξ3 ⊗ ∆)] − [ξ3q + (1 ⊗ ∆)ξ2 + (q ⊗ ξ2)(1 ⊗ τ ⊗ 1)(∆ ⊗ ∆) + (q ⊗ q ⊗ q)(1 ⊗ τ ⊗ τ ⊗ 1)(12 ⊗ τ ⊗ 12)(1 ⊗ ∆ ⊗ 13)(∆ ⊗ ξ3)] Figure 28: First 3-diﬀerential, d3,1 1211112((xw )(z w ))((yz )w )22((xw )(zw ) )((yw )(zw ) )11211222((xw )(z w ))((yw )(z w ))1111221222((xw )(z w ))((yw )(z w ))1121112222((xy)z)w((xz )(yz ))w1211((xz )w )((yz )w )22((xy)w )(zw )12((xw )(yw ))(zw )211 Journal of Homotopy and Related Structures, vol.
3(1), 2008 54 Figure 29: Second 3-diﬀerential, d3,2 Figure 30: Third 3-diﬀerential, d3,3 Journal of Homotopy and Related Structures, vol.

3(1), 2008 55 B.

Proving D 3D 2 = 0 Proof of Lemma 6.3.

This is proved by calculations that seem complicated with- out diagrammatics.

We sketch our computational method.

For ξ1 ∈ Hom(X ⊗3, X) ⊂ C 3 sh(X; X), the ﬁrst two terms of d3,1(ξ1) are q(ξ1 ⊗ 1) and ξ1(q ⊗ q ⊗ 1)(1 ⊗ τ ⊗ 12)(12 ⊗ ∆ ⊗ 1), that are diagrammatically represented by left of Fig.
31, (1) and (2), respectively.

The black triangular four-valent vertex represents ξ1.

On the other hand, the ﬁrst term q(ξ1 ⊗ 1) corresponds to the change of the diagrams represented in (A) and (B).

Such a change of diagrams corresponds to d(2) 1 (η1) as depicted in Fig.

24.

Therefore the ﬁrst terms of d3,1d2,1(η1, 0) are q(ξ1 ⊗ 1) = q(d2,1(η1, 0) ⊗ 1) consisting of ﬁve terms represented by the diagrams on the right top two rows in Fig.

31.
31.

The third row consists of the positive terms of the second term (2), d2,1(η1, 0)(q ⊗ q ⊗ 1)(1 ⊗ τ ⊗ 12)(12 ⊗ ∆ ⊗ 1).

Thus to prove this lemma, we write out all terms and check that they cancel.

For example, the terms on the right of Fig.

31 labelled with (a) and (b) cancel.

Figure 31: A strategy for a proof The essential steps in the proofs are found in Figs.

32 and 33 below.

The ﬁrst rows of Fig.

32 coincide with those of Fig.

31.


The remaining left columns indicate the diﬀerent diagrams that are obtained by replacing the four-valent black vertices by the two sides of the self-distributive law.

The right-hand entries are the expansions of the terms in the next diﬀerential.

The terms are numbered and those in Fig.

32 and Fig.

33 cancel.

It is somewhat diﬃcult to see the cancellation of the terms labelled 7, 10, 11, 12, 13, and 14.
The terms labelled 15 coincide by applications of coassociativity and co- commutativity.

The identity between these terms becomes obvious after one works through the preceding terms.

The proofs that the diagrams represent the same linear maps are provided below in Appendix AppendixC.

 The next illustrations represent the proof of Lemma 6.3.

(b)(1)(2)(A)(B)(C)(a) Journal of Homotopy and Related Structures, vol.
3(1), 2008 56 Figure 32: d3,1D2(η1, 0), LHS Figure 33: d3,1D2(η1, 0), RHS 1513452368798411101259131415161617182171911141916187101312 Journal of Homotopy and Related Structures, vol.

3(1), 2008 57 Figure 34: d3,2D2(η1, 0), LHS Figure 35: d3,2D2(η1, 0), RHS 111234536745891011212131141514121513681079 Journal of Homotopy and Related Structures, vol.

3(1), 2008 58 Figure 36: d3,3D2(η1, 0) = 0 C. Proving identities between terms in Fig.
32 and 33 The next illustrations give the outlines of the proofs that the terms labelled 7, 10, 11, 12, 13, and 14 represent the same functions in Figs.

32 and 33.

Figure 37: The term 7 645223113564Self−distributiveDistantoperatorscommuteSelf−distributiveDistantoperatorscommuteDistantoperatorscommuteLHSRHSCoassociativeCocommutativeCoassociativeCompatible Journal of Homotopy and Related Structures, vol.
3(1), 2008 59 Figure 38: The term 10 Figure 39: The term 11 Figure 40: The term 12 DistantoperatorscommuteSelf-distributiveDistantoperatorscommuteDistantoperatorscommuteDistantoperatorscommuteLHSRHSCoassociativeCocommutativeCompatibleCoassociativeDistantoperatorscommuteDistantoperatorscommuteSelf−distributiveDistantoperatorscommuteLHSRHSCompatibleCoassociativeCocommutativeDistantoperatorscommuteSelf−distributiveDistantoperatorscommuteLHSRHSCoassociativeCoassociative Journal of Homotopy and Related Structures, vol.
3(1), 2008 60 Figure 41: The term 13 Figure 42: The term 14 D. Proof of Theorem 7.4 Proof of Theorem 7.4.
In a manner similar to the proof of Theorem 7.1, we begin by expanding: (a + axx) ⊗ (b + byy) ⊗ (c + czz) ⊗ (d + dww) = (a + Ax) ⊗ (b + By) ⊗ (c + Cz) ⊗ (d + Dw) = abcd(1 ⊗ 1 ⊗ 1 ⊗ 1) + Abcd(x ⊗ 1 ⊗ 1 ⊗ 1) + aBcd(1 ⊗ y ⊗ 1 ⊗ 1) + ABcd(x ⊗ y ⊗ 1 ⊗ 1) + abCd(1 ⊗ 1 ⊗ z ⊗ 1) + AbCd(x ⊗ 1 ⊗ z ⊗ 1) + aBCd(1 ⊗ y ⊗ z ⊗ 1) + ABCd(x ⊗ y ⊗ z ⊗ 1) + abcD(1 ⊗ 1 ⊗ 1 ⊗ w) + AbcD(x ⊗ 1 ⊗ 1 ⊗ w) + aBcD(1 ⊗ y ⊗ 1 ⊗ w) + ABcD(x ⊗ y ⊗ 1 ⊗ w) + abCD(1 ⊗ 1 ⊗ z ⊗ w) + AbCD(x ⊗ 1 ⊗ z ⊗ w) + aBCD(1 ⊗ y ⊗ z ⊗ w) + ABCD(x ⊗ y ⊗ z ⊗ w) DistantoperatorscommuteLHSRHSCoassociativeCompatibleCoassociativeCoassociativeDistantoperatorscommuteDistantoperatorscommuteSelf−distributiveLHSRHSCoassociativeCocommutativeCompatible Journal of Homotopy and Related Structures, vol.
3(1), 2008 61 In a table similar to the one above, the values of the various operators q(ˆθ ⊗ 1) and so forth can be evaluated on each of the sixteen tensors (1 ⊗ 1 ⊗ 1 ⊗ 1) through (x ⊗ y ⊗ z ⊗ w).

Most of these evaluations give 0 (a result we leave to the reader).

The exceptions are the values on (1 ⊗ y ⊗ z ⊗ w) and (x ⊗ y ⊗ z ⊗ w).

We remind the reader that ξ2 = 0 and ξ3 = 0, so those terms do not appear below.
We compute: q(ˆθ ⊗ 1)(1 ⊗ y ⊗ z ⊗ w) = ˆθ(q ⊗ q)(1 ⊗ τ ⊗ 12)(12 ⊗ ∆ ⊗ 1)(1 ⊗ y ⊗ z ⊗ w) = q(ˆθ ⊗ q)(12 ⊗ τ ⊗ 1)(12 ⊗ q ⊗ ∆)(1 ⊗ τ ⊗ 12)(12 ⊗ ∆ ⊗ 1)(1 ⊗ y ⊗ z ⊗ w) = (ˆθ)(q ⊗ 12)(1 ⊗ y ⊗ z ⊗ w) = q(ˆθ ⊗ q)(12 ⊗ τ ⊗ 1)(13 ⊗ ∆)(1 ⊗ y ⊗ z ⊗ w) = ˆθ(q ⊗ q ⊗ 1)(1 ⊗ τ ⊗ 1 ⊗ q)(12 ⊗ ∆ ⊗ 12)(12 ⊗ τ ⊗ 1)(13 ⊗ ∆)(1 ⊗ y ⊗ z ⊗ w) and = 1, q(q ⊗ ˆθ)(q ⊗ q ⊗ 13)(1 ⊗ τ ⊗ 14)(12 ⊗ ∆ ⊗ 13)(12 ⊗ τ ⊗ 12)(1 ⊗ τ ⊗ τ ⊗ 1) (12 ⊗ ∆ ⊗ ∆)(1 ⊗ y ⊗ z ⊗ w) = q(q ⊗ q)(1 ⊗ τ ⊗ 1)(q ⊗ q ⊗ ξ2)(1 ⊗ τ ⊗ 13)(12 ⊗ ∆ ⊗ 12)(12 ⊗ τ ⊗ 1) (13 ⊗ ∆)(1 ⊗ y ⊗ z ⊗ w) = 0.
The last equality follows trivially since ξ2 = 0.

A scheme for making these compu- tations is illustrated in Fig 43.
Meanwhile, q(ˆθ ⊗ 1)(x ⊗ y ⊗ z ⊗ w) = θ(x, y, z) ˆθ(q ⊗ q)(1 ⊗ τ ⊗ 12)(12 ⊗ ∆ ⊗ 1)(x ⊗ y ⊗ z ⊗ w) = θ(x z, y z, w) q(ˆθ ⊗ q)(12 ⊗ τ ⊗ 1)(12 ⊗ q ⊗ ∆)(1 ⊗ τ ⊗ 12)(12 ⊗ ∆ ⊗ 1)(x ⊗ y ⊗ z ⊗ w) = θ(x, z, w) (ˆθ)(q ⊗ 12)(x ⊗ y ⊗ z ⊗ w) = θ(x y, z, w) q(ˆθ ⊗ q)(12 ⊗ τ ⊗ 1)(13 ⊗ ∆)(x ⊗ y ⊗ z ⊗ w) = θ(x, y, w) ˆθ(q ⊗ q ⊗ 1)(1 ⊗ τ ⊗ 1 ⊗ q)(12 ⊗ ∆ ⊗ 12)(12 ⊗ τ ⊗ 1)(13 ⊗ ∆)(x ⊗ y ⊗ z ⊗ w) = and θ(x w, y w, z w), q(q ⊗ ˆθ)(q ⊗ q ⊗ 13)(1 ⊗ τ ⊗ 14)(12 ⊗ ∆ ⊗ 13)(12 ⊗ τ ⊗ 12)(1 ⊗ τ ⊗ τ ⊗ 1) (12 ⊗ ∆ ⊗ ∆)(x ⊗ y ⊗ z ⊗ w) Journal of Homotopy and Related Structures, vol.
3(1), 2008 62 = q(q ⊗ q)(1 ⊗ τ ⊗ 1)(q ⊗ q ⊗ ξ2)(1 ⊗ τ ⊗ 13)(12 ⊗ ∆ ⊗ 12)(12 ⊗ τ ⊗ 1) (13 ⊗ ∆)(x ⊗ y ⊗ z ⊗ w) = 0.

The result follows.

 Figure 43: A sample computation with a 3-cocycle yzyzzwwwyzwzwwyzzwwzzywwywzzzwwwzyz w1111111q(y, z,w)q(y, z,w)10 Journal of Homotopy and Related Structures, vol.
3(1), 2008 63 This article may be accessed via WWW at http://jhrs.rmi.acnet.ge J. Scott Carter carter@jaguar1.usouthal.edu Department of Mathematics and Statistics ILB 325, University of South Alabama Mobile, AL 36688-0002 Alissa S. Crans acrans@lmu.edu Department of Mathematics Loyola Marymount University One LMU Drive, Suite 2700 Los Angeles, CA 90045 Mohamed Elhamdadi emohamed@math.usf.edu Department of Mathematics and Statistics University of South Florida 4202 East Fowler Ave., PHY114 Tampa, FL 33620-5700 Masahico Saito saito@math.usf.edu Department of Mathematics and Statistics University of South Florida 4202 East Fowler Ave., PHY114 Tampa, FL 33620-5700