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^{1}University of South Alabama, Mobile, AL 36688, USA

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^{2}Loyola Marymount Unversity, Los Angeles, CA 90045, USA

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^{3}University of South Florida, Tampa, FL 33620, USA

**E-mails:** [email protected] and [email protected]

**Received** May 22, 2007 **Revised** July 10, 2007

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

A cohomology theory of the adjoint of Hopf algebras, via deformations, is presented by means of diagrammatic techniques. Explicit calculations are provided in the cases of group algebras, function algebras on groups, and the bosonization of the super line. As applications, solutions to the YBE are given and quandle cocycles are constructed from groupoid cocycles.

Algebraic deformation theory [10] can be used to define 2-dimensional cohomology in a wide variety of contexts. This theory has also been understood diagrammatically [7, 16, 17] via PROPs, for example. In this paper, we use diagrammatic techniques to define a cohomological deformation of the adjoint map in an arbitrary Hopf algebra. We have concentrated on the diagrammatic versions here because diagrammatics have led to topological invariants [6, 13, 19], diagrammatic methodology is prevalent in understanding particle interactions and scattering in the physics literature, and most importantly kinesthetic intuition can be used to prove algebraic identities.

The starting point for this calculation is a pair of identities that the adjoint map satisfies and that are sufficient to construct Woronowicz’s solution [22] to the Yang-Baxter equation (YBE): We use deformation theory to define an extension 2-cocycle. Then we show that the resulting 2-coboundary map, when composed with the Hochschild 1-coboundary map is trivial. A 3-coboundary is defined via the “movie move” technology. Applications of this cohomology theory include constructing new solutions to the YBE by deformations and constructing quandle cocycles from groupoid cocycles that arise from this theory.

The paper is organized as follows. Section 2 reviews the definition of Hopf algebras, defines the adjoint map, and illustrates Woronowicz’s solution to the YBE. Section 3 contains the deformation theory. Section 4 defines the chain groups and differentials in general. Example calculations in the case of a group algebra, the function algebra on a group, and a calculation of the 1- and 2-dimensional cohomology of the bosonization of the superline are presented in Section 5. Interestingly, the group algebra and the function algebra on a group are cohomologically different. Moreover, the conditions that result when a function on the group algebra satisfies the cocycle condition coincide with the definition of groupoid cohomology. This relationship is given in Section 6, along with a construction of quandle 3-cocycles from groupoid 3-cocycles. In Section 7, we use the deformation cocycles to construct solutions to the Yang-Baxter equation.

We begin by recalling the operations and axioms in Hopf algebras, and their diagrammatic
conventions depicted in **Figures 1 and 2**.

A coalgebra is a vector space *C* over a field *k* together with a *comultiplication* that is bilinear and *coassociative*: . A coalgebra is cocommutative if the
comultiplication satisfies , where is the transposition A *coalgebra with counit* is a coalgebra with a linear map called the counit such that A *bialgebra* is an algebra *A* over a field *k* together with
a linear map called the *unit* satisfying where 1 ∈ A is the multiplicative identity and with an associative multiplication that is also a coalgebra such that
the comultiplication Δ is an algebra homomorphism. A Hopf algebra is a bialgebra *C* together
with a map called the antipode such that where ∈ is
the counit.

In diagrams, the compositions of maps are depicted from bottom to top. Thus a multiplication *μ* is represented by a trivalent vertex with two bottom edges representing and one top
edge representing A. Other maps in the definition are depicted in **Fig. 1** and axioms are depicted
in **Fig. 2.**

Let *H* be a Hopf algebra. The adjoint map for any y ∈ H is defined by where we use the common notation and Its diagram is depicted in **Fig. 3 (A)**. Notice the analogy with group conjugation: in a group
ring* H = kG* over a field k, where we have For
the adjoint map ad : we have ad

**Definition 1.** Let *H* be a Hopf algebra and ad be the adjoint map. Then the linear map defined by is said to be the *R*-matrix *induced from* ad.

**Lemma 1.** The *R-matrix induced from* ad *satisfies the YBE*.

**Proof.** In **Fig. 4**, it is indicated that the YBE follows from two properties of the adjoint map:

(2.1)

(2.2)

It is known that these properties are satisfied, and proofs are found in [11, 22]. Here we
include diagrammatic proofs for reader’s convenience in **Fig. 5 and Fig. 6**, respectively.

**Definition 2.** We call the equalities in equations (2.1) and (2.2) the *adjoint conditions*.

**Remark 1.** The equality (2.1) is equivalent to the fact that the adjoint map defines an algebra
action of *H* on itself (see [14]). Specifically, for any a, b, c 2 H, where denotes the right action defined by the adjoint: The equality (2.2) can be
similarly rewritten as

**Remark 2.** It was pointed out to us by Sommerhaeuser that the induced R-matrix R_{ad} is
invertible with inverse

We follow the exposition in [16] for deformation of bialgebras to propose a similar deformation
theory for the adjoint map. In light of Lemma 1, we deform the two equalities (2.1) and (2.2).
Let *H* be a Hopf algebra and ad its adjoint map.

**Definition 3**. A deformation of (*H*, ad) is a pair (*H _{t}*, ad

Suppose satisfies the adjoint conditions (equalities (2.1) and (2.2)) mod *t*^{n+1}, and suppose that there exist such that satisfies the
adjoint conditions mod t^{n+2}. Define by

For the first adjoint condition (2.1) of we obtain

which is equivalent by degree calculations to

For the second adjoint condition (2.2) of we obtain

which is equivalent by degree calculations to

In summary we proved the following.

**Lemma 2.** *The map* *satisfies the adjoint conditions mod t ^{n+2} if and only if*

We interpret this lemma as being *the primary obstructions* to formal deformation
of the adjoint map. In the next section we will define coboundary operators, and show that = (0, 0) gives that ad_{n+1} is a 2-cocycle, and that satisfies the 3-cocycle condition,
just as in the case of deformations and Hochshild cohomology for bialgebras.

**Chain groups**

We define chain groups, for positive integers *n*, *n* > 1, and *i* = 1, . . . , n by

Specifically, chain groups in low dimensions of our concern are:

For n = 1, define

In the remaining sections we will define differentials that are homomorphisms between the chain groups:

that will be defined individually for *n* = 1, 2, 3 and for *i* with *2i* ≤ *n* + 1, and

**First differentials**

By analogy with the differential for multiplication, we make the following definition.

**Definition 4.** The first differential is defined by

Diagrammatically, we represent *d*^{1,1} as depicted in **Fig. 7**, where a 1-cochain is represented
by a circle on a string.

**Second differentials**

**Definition 5.** Define the second differentials by and

Diagrams for 2-cochain and 2-differentials are depicted in **Fig. 8.**

**Theorem 1.** *D _{2}D_{1}* = 0.

**Proof.** This follows from direct calculations, and can be seen from diagrams in **Figs. 9 and
10.**

**Third differentials**

**Definition 6.** We define 3-differentials as follows. Let Then

Diagrams for 3-cochains are depicted in **Fig. 11**. See **Fig. 12 (A), (B), and (C)** for the
diagrammatics for d^{3,1}, d^{3,2} and d^{3,3}, respectively.

**Theorem 2.** *D*_{3}*D*_{2} = 0.

**Proof.** The proof follows from direct calculations that are indicated in **Figs. 13, 14 and 15**. We
demonstrate how to recover algebraic calculations from these diagrams for the part This is indicated in **Fig. 15**, where subscripts ad are
suppressed for simplicity. Let (note that does not land in the domain of *d*^{3,3}). The first line of **Fig. 15** represents the definition of the differential

where each term represents each connected diagram. The first parenthesis of the second line represents that

is substituted in the first term When these two maps are applied to a general element the results are computed as

By coassociativity applied to *y* and the second term is equal to

which is equal, by compatibility, to This last term is represented exactly by the last term in the second line of **Fig. 15**, and therefore
is cancelled. The map represented by the second term in the second line of **Fig. 15** cancels with
the third term by coassociativity, and the fourth term cancels with the sixth by coassociativity
applied twice and compatibility once. Other cases (**Figs. 13, 14**) are computed similarly.

From point of view of Lemma 2, we state the relation between deformations and the third differential map as follows.

**Corollary 1.** *The primary obstructions* *to formal deformations in Lemma 2 represent
a 3-cocycle*:

Furthermore, in Lemma 2 we see that so we regard that the obstructions represent a cohomology class after the definition of the cohomology groups in the next section.

**Cohomology groups**

For convenience define

Then Theorems 1 and 2 are summarized as follows.

**Theorem 3.** *is a chain complex.*

This enables us to propose

**Definition 7.** The *adjoint* n-coboundary, cocycle and cohomology groups are defined by:

for *n* = 1, 2, 3.

**Group algebras**

Let G be a group and *H = kG* be its group algebra with the coefficient field k (char Then H has a Hopf algebra structure induced from the group operation as multiplication, for basis elements x ∈ G, and the antipode induced from Here and below, we denote the conjugation action on a group *G* by Note that
this defines a quandle structure on G; see [12].

**Lemma 3.**

**Proof.** For any given w ∈ G write where a : G → k is a function. Recall
the defining equality (4.1). The LHS of the second condition is written as

and the RHS is written as

For a given w, fix u and then compare the coefficients of In the LHS we have while
on the RHS *w* = *u*, and furthermore Thus the diagonal coefficient must
satisfy since char In the case neither
term of nor is equal to hence

**Lemma 4.** *For x, y ∈ G, write* *where a : G×G → k. Then the induced
linear map* *if and only if a satisfies*

*for any x, y, z ∈ G.*

**Proof.** The first 2-cocycle condition for is written by

for basis elements *x, y, z ∈ G*. The second is formulated by

They have the common term and otherwise they are different terms. Thus we obtain unless For these terms, the first condition becomes

and the result follows.

**Remark 3.** In the preceding proof, since the term Then the condition stated becomes a

**Proposition 1.** Let G be a group. where ξ_{1} is the map that is
defined by linearly extending *if and only if* ξ_{2} = 0 and the coefficients satisfy the following properties: satisfies

**Proof.** Suppose be the map that is defined by linearly extending Then the third 3-cocycle condition from Definition 6 gives:(abbreviating

We first consider terms in which the third tensorand is xy. From the third summand, this forces the second tensorand to be xy, so we collect the terms of the form This gives

which implies for all u ∈ G. The remaining terms are

From the second sum we obtain In conclusion, if for *kG* then ∈_{2} = 0.

We now consider Let ∈_{1} be the map that is defined by linearly
extending The second 3-cocycle condition
from Definition 6, with In order to combine like
terms, we need *yzu* = *xyz*, meaning except in the case
when In this case, we obtain where

Finally we consider the first 3-cocycle condition from Definition 6, which is formulated for basis elements by

Substituting in the formula for *c(x, y, z)* which we found above, we obtain

This is a group 3-cocycle condition with the first term *x · c(y, z,w)* omitted. This is expected
from **Fig. 12 (A)**. Constant functions, for example, satisfy this condition.

Next we look at a coboundary condition. A 3-coboundary is written as

If we write then

Hence

and in particular for the coefficients* c _{u}(x, y, z)* from Proposition 1,

By setting we obtain

**Lemma 5.** *A 3-cocycle c(x, y, z) is a coboundary if for some a(x, y),*

Remark 4. From Remark 3, Proposition 1, and Lemma 5, we have the following situation. The 2-cocycle condition, the 3-cocycle condition, and the 3-coboundary condition, respectively, gives rise to the equations

This suggests a cohomology theory, which we investigate in Section 6.

**Proposition 2.** For the symmetric group G = S_{3} on three letters, we have

**Proof.** By Lemma 3, we have which is computed by solving the system of equations stated in Lemma 4 and
Remark 3. Computations by Maple and Mathematica shows that the solution set is of dimension
3 and generated by (a((1 2 3), (1 2)), a((2 3), (1 3 2)), and a((1 3), (1 2)) for the above mentioned
coefficient fields.

**Function algebras on groups**

Let G be a finite group and k a field with char(k) The set k^{G} of functions from G to k
with pointwise addition and multiplication is a unital associative algebra. It has a Hopf algebra
structure using with comultiplication defined through by and the antipode by

Now kG has basis (the characteristic function) defined by and zero otherwise. Since the adjoint map becomes

**Lemma 6. **

**Proof.** Recall the defining equality (4.1). Let be a given finite group and
abbreviate Describe Then is written for basis elements by *LHS = f(δ _{i}δ_{j})* and

For *i *= *j* we obtain LHS for all *i, j* as desired.

**Lemma 7.**

**Proof.** Recall that Describe
a general element If then the first term is zero by the definition of ad. If and b = 1, then the third
term is also zero, and we obtain that the second term is zero. Hence unless c = 1. Next, set b = c = 1 in the general form. Then all three terms equal and we obtainand the result follows.

By combining the above lemmas, we obtain the following

**Theorem 4.** *For any finite group G and a field k, we have*

Observe that *k(G)* and k^{G} are cohomologically distinct.

**Bosonization of the superline**

Let *H* be generated by 1, *g, x* with relations and Hopf algebra
structure (this
Hopf algebra is called the bosonization of the superline [15], page 39, Example 2.1.7).

The operation ad is represented by the following table, where, for example,

Remark 5. The induced R-matrix R-matrix R_{ad} has determinant 1, the characteristic polynomial is and the minimal polynomial is

**Proposition 3.** *The first cohomology of H is given by *

**Proof.** Recall the defining equality (4.1). Let Assume that f(x) = a + bx +
cg + dxg and . Applying f to both sides of the equation g^{2} = 1, one obtains Similarly evaluating both sides of the equationone obtains that *f(g) *= 0. In a similar way,
applying *f* to the equations *x ^{2}* = 0 and

It is directly checked on all the generators This implies that

**Proposition 4.** *For any field k of characteristic not 2,*

**Proof. **With d^{1,1} = 0 from the preceding Proposition, we have Either
the direct hand calculations from definitions or the computer calculations give the following
general solution for the 2-cocycle represented in the following table:

Here, for example, where *α, β, γ* are free variables.

From Remark 4, the adjoint cohomology leads us to cohomology, especially for conjugate groupoids of groups as defined below. Through the relation between Reidemeister moves for knots and the adjoint, groupoid cohomology, we obtain a new construction of quandle cocycles. In this section we investigate these relations. First we formulate a general definition. Many formulations of groupoid cohomology can be found in literature, and relations of the following formulation to previously known theories are not clear. See [20], for example.

Let G be a groupoid with objects Ob(G) and morphisms Let for non-negative integers *i* and *n*. Let C_{n}(*G*) be the free abelian
group generated by The boundary
map is defined by by linearly extending

Then it is easily seen that this differential defines a chain complex. The corresponding groupoid 1- and 2-cocycle conditions are written as

The general cohomological theory of homomorphisms and extensions applies, such as:

**Remark 6.** Let *G* be a groupoid and *A* be an abelian group regarded as a one-object groupoid.
Then gives a groupoid homomorphism from *G* to *A*, which sends Ob(*G*) to
the single object of A, if and only if defined by is a groupoid
1-cocycle. Next we consider extensions of groupoids. Define

by where is a groupoid, the function c with the value is a groupoid 2-cocycle.

**Example.** Let G be a group. Define the conjugate groupoid of G, denoted and where the source of the morphism and its
target is Composition is defined by For this
example, the groupoid 1- and 2-cocycle conditions are

Diagrammatic representations of these equations are depicted in **Fig. 16 (A) and (B),** respectively.
Furthermore, c is a coboundary if

Compare with Remark 4.

For the symmetric group on 3 letters, with coefficient group respectively, the dimensions of the conjugation groupoid 2-cocycles are 3, 5, 4, 3 and 3.

For the rest of the section, we present new constructions of quandle cocycles from groupoid
cocycles of conjugate groupoids of groups. Let *G* be a finite group, and a : G^{2} → k be adjoint
2-cocycle coefficients that were defined in Remark 3. These satisfy

**Proposition 5.** *Let* *satisfies the rack 2-cocycle condition*

**Proof.** By definition

Let G be a finite group, and be a coefficient of the adjoint 3-cocycle defined in Proposition 1. This satisfies

**Proposition 6.** *Let G be a group that is considered as a quandle under conjugation. Then defined by*

*is a rack 3-cocycle.*

Proof. We must show that µ satsifies

We compute

In this section we give, in an explicit form, deformations of R-matrices by 2-cocycles of the
adjoint cohomology theory we developed in this paper. Let *H* be a Hopf algebra and ad its
adjoint map. In Section 3 a deformation of (H, ad) was defined to be a pair (*H _{t}*, ad

Recall that a solution to the YBE, *R*-matrix *R*_{ad} is induced from the adjoint map. Then
from the constructions of the adjoint cohomology from the point of view of the deformation
theory, we obtain the following deformation of this R-matrix induced from the adjoint map.

**Theorem 5.** *Let* *be an adjoint 2-cocycle. Then the map * defined by *satisfies the YBE.*

Proof. The equalities of Lemma 2 hold in the quotient where *n* = 1 and
the modulus *t*^{2} is considered. These cocycle conditions, on the other hand, were formulated
from the motivation from Lemma 1 for the induced R-matrix R_{ad} to satisfy the YBE. Hence
these two lemmas imply the theorem.

Example. In Subsection 5.3, the adjoint map ad was computed for the bosonization H of the
superline, with basis {1, g, x, gx}, as well as a general 2-cocycle Á with three free variables *α, β, γ* written by

and zero otherwise. Thus we obtain the deformed solution to the on A with three variables tα, tβ, tγ of degree one.

In [7] we concluded with A *Compendium of Questions* regarding our discoveries. Here we attempt
to address some of these questions by providing relationships between this paper and [7], and
offer further questions for our future consideration.

It was pointed out in [7] that there was a clear distinction between the Hopf algebra case and the cocommutative coalgebra case as to why self-adjoint maps satisfy the YBE. In [7] a cohomology theory was constructed for the coalgebra case. In this paper, many of the same ideas and techniques, in particular deformations and diagrams, were used to construct a cohomology theory in the Hopf algebra case, with applications to the YBE and quandle cohomology.

The aspects that unify these two theories are deformations and a systematic process we call “diagrammatic infiltration.” So far, these techniques have only been successful in defining coboundaries up through dimension 3. This is a deficit of the diagrammatic approach, but diagrams give direct applications to other algebraic problems such as the YBE and quandle cohomology, and suggest further applications to knot theory. By taking the trace as in Turaev’s [21], for example, a new deformed version of a given invariant is expected to be obtained.

Many questions remain: Can 3-cocycles be used for solving the tetrahedral equation? Can they be used for knotted surface invariants? Can the coboundary maps be expressed skein theoretically? How are the deformations of R-matrices related to deformations of underlying Hopf algebras? When a Hopf algebra contains a coalgebra, such as the universal enveloping algebra and its Lie algebra together with the ground field of degree-zero part, what is the relation between the two theories developed in this paper and in [7]? How these theories, other than the same diagrammatic techniques, can be uniformly formulated, and to higher dimensions?

JSC (NSF Grant DMS #0301095, #0603926) and MS (NSF Grant DMS #0301089, #0603876) gratefully acknowledge the support of the NSF without which substantial portions of the work would not have been possible. The opinions expressed in this paper do not reflect the opinions of the National Science Foundation or the Federal Government. JSC, ME, and MS have benefited from several detailed presentations on deformation theory that have been given by J¨org Feldvoss. AC acknowledges useful and on-going conversations with John Baez.

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