Cohomology of the adjoint of Hopf algebras

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.


Introduction
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 ad(x ⊗ y) = S(y (1) )xy (2) 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] R = (1 ⊗ ad)(τ ⊗ 1)(1 ⊗ ∆) to the Yang-Baxter equation (YBE): (R⊗1)(1⊗R)(R⊗1) = (1⊗R)(R⊗1)(1⊗R). 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.

Acknowledgements
JSC and MS gratefully acknowledge the support of the NSF without which substantial portions of the work would not have been possible. JSC, ME, and MS have benefited from several detailed presentations on deformation theory that have been given by Jörg Feldvoss. AC acknowledges useful and on-going conversations with John Baez.

Preliminaries
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 ∆ : C → C ⊗ C that is bilinear 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 ǫ : C → k such that (ǫ ⊗ 1)∆ = 1 = (1 ⊗ ǫ)∆ via k ⊗ C ∼ = C. 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)∆ = ηǫ = µ(1 ⊗ S)∆, where ǫ 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 A ⊗ A 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 Ad y : H → H for any y ∈ H is defined by Ad y (x) = S(y (1) )xy (2) , where we use the common notation ∆(x) = x (1) ⊗ x (2) and µ(x ⊗ y) = xy.

Unit
Counit Counit is an algebra hom

Compatibility Coassociativity
Antipode condition Unit is a coalgebra hom S S Figure 2: Axioms of Hopf algebras Its diagram is depicted in Fig. 3. Notice the analogy with group conjugation: in a group ring H = kG over a field k, where ∆(y) = y ⊗ y and S(y) = y −1 , we have Ad y (x) = y −1 xy. When we view the adjoint map as a map from H ⊗ H to H, we use the notation is said to be the R-matrix induced from ad.

Lemma 2.2
The R-matrix induced from ad satisfies the YBE.
Proof. In Fig. 5, it is indicated that the YBE follows from two properties of the adjoint map: 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. 6 and Fig. 7, respectively. Definition 2.3 We call the above equalities (1) and (2)

Remark 2.4
The equality (1) is equivalent to the fact that the adjoint map defines an algebra action of H on itself (see [14]). Specifically, (a ⊳ b) ⊳ c = a ⊳ (bc) for any a, b, c ∈ H, where ⊳ denotes the right action defined by the adjoint: a⊳b = ad(a⊗ b). The equality (2) can be similarly rewritten as: Remark 2.5 It was pointed out to us by Sommerhaeuser that the induced R-matrix R ad is invertible with inverse antip. compat.

Deformations of the Adjoint Map
We follow the exposition in [16] for deformation of bialgebras to propose a similar deformation theory for the adjoint map. In light of Lemma 2.2, we deform the two equalities (1) and (2). Let H be a Hopf algebra and ad its adjoint map.
] with all Hopf algebra structures inherited by extending those on H t with the identity on the k[[t]] factor (the trivial deformation as a Hopf algebra), with a deformations of ad given by ad t = ad + tad 1 + · · · + t n ad n + · · · : H t ⊗ H t → H t where ad i : H ⊗ H → H, i = 1, 2, · · ·, are maps. Supposeād = ad + · · · + t n ad n satisfies the adjoint conditions (equalities (1) and (2)) mod t n+1 , and suppose that there exist ad n+1 : H ⊗ H → H such thatād + t n+1 ad n+1 satisfies the adjoint conditions mod t n+2 . Define ξ 1 ∈ Hom(H ⊗3 , H) and ξ 2 ∈ Hom(H ⊗2 , H ⊗2 ) bȳ For the first adjoint condition (1) ofād + t n+1 ad n+1 mod t n+2 we obtain: which is equivalent by degree calculations to: For the second adjoint condition (2) ofād + t n+1 ad n+1 mod t n+2 we obtain: which is equivalent by degree calculations to: In summary we proved the following: 4 Differentials and Cohomology

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 4.1 The first differential is defined by ( d 1,1 ) Figure 8: The 1-differential Diagrammatically, we represent d 1,1 as depicted in Fig. 8, where a 1-cochain is represented by a circle on a string.
Proof. This follows from direct calculations, and can be seen from diagrams in Figs. 12 and 13.

Third Differentials
Definition 4. 4 We define 3-differentials as follows. Let ξ i ∈ C 3,i (H; H) for i = 1, 2. Then  Figure 12: The 2-cocycle condition for a 2-coboundary, Part I Diagrams for 3-cochains are depicted in Fig. 14 Proof. The proof follows from direct calculations that are indicated in Figs. 18, 19 and 20. We demonstrate how to recover algebraic calculations from these diagrams for the part (d 3,3 d 2,2 )(η 1 ) = 0 for any η 1 ∈ C 2 (H; H). This is indicated in Fig. 20, where subscripts ad are suppressed for simplicity. Let ξ 2 = d 2,2 (η 1 ) ∈ C 3,2 (H; H) (note that ξ 1 = d 2,1 (η 1 ) ∈ C 3,1 (H; H) does not land in the domain of d 3,3 ). The first line of Fig. 20 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 x ⊗ y ∈ H ⊗ H, the results are computed as (2) . By coassociativity applied to y and η 1 (x ⊗ y (2)(2) ), 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. 20, and therefore is cancelled. The map represented by the second term in the second line of Fig. 20 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. 18,19) are computed similarly. This enables us to define: 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 k = 2). Then H has a Hopf algebra structure induced from the group operation as multiplication, ∆(x) = x ⊗ x for basis elements x ∈ G, and the antipode induced from S(x) = x −1 for x ∈ G. Here and below, we denote the conjugation action on a group G by x ⊳ y := y −1 xy. Note that this defines a quandle structure on G; see [12].
The LHS of the second condition is written as For a given w, fix u and then compare the coefficients of u ⊗ u. In the LHS we have a u (w), while on the RHS w = u, and furthermore w = h = v for u ⊗ u. Thus the diagonal coefficient must satisfy for any x, y, z ∈ G.
Proof. The first 2-cocycle condition for φ : kG ⊗ kG → kG is written by: for basis elements x, y, z ∈ G. The second is formulated by They have the common term u ⊗ xy for w = y −1 xy = u, and otherwise they are different terms.
Thus we obtain a w (x, y) = 0 unless w = y −1 xy. For these terms, the first condition becomes and the result follows.

Remark 5.3
In the preceding proof, since the term a w (x, y) = 0 unless w = x ⊳ y, let a x⊳y (x, y) = a(x, y). Then the condition stated becomes a(x, y) + a(x ⊳ y, z) − a(x, yz) = 0.
Proof. Suppose (ξ 1 , ξ 2 ) ∈ Z 3 ad (kG; kG). Let ξ 2 be the map that is defined by linearly extending ξ 2 (x ⊗ y) = u,v∈G a u,v (x, y)u ⊗ v. Then the third 3-cocycle condition from Definition 4.4 gives: (abbreviating a u,v (x, y) = a u,v ) 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 (u ⊗ xy ⊗ xy). This gives: u (a u,xy + a u,xy − a u,xy )(u ⊗ xy ⊗ xy) = 0, which implies a u,xy = 0 for all u ∈ G. The remaining terms are From the second sum we obtain a u,v (x, y) = 0 for v = xy. In conclusion, if d 3,3 (ξ 1 , ξ 2 ) = 0 for kG then ξ 2 = 0.
Finally we consider the first 3-cocycle condition from Definition 4.4, which is formulated for basis elements by Substituting in the formula for c(x, y, z) which we found above, we obtain c(x, y, z) + c(x, yz, w) = c(y −1 xy, z, w) + c(x, y, zw). This is a group 3-cocycle condition with the first term x · c(y, z, w) omitted. This is expected from Fig. 15. Constant functions, for example, satisfy this condition.

Function Algebras on Groups
Let G be a finite group and k a field with char(k) = 2. 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 k G×G ∼ = k G ⊗ k G with comultiplication defined through ∆ : k G → k G×G by ∆(f )(u ⊗ v) = f (uv) and the antipode by S(f )(x) = f (x −1 ). Now k G has basis (the characteristic function) δ g : G → k defined by δ g (x) = 1 if x = g and zero otherwise. Since S(δ g ) = δ g −1 and ∆(δ h ) = uv=h δ u ⊗ δ v , the adjoint map becomes Proof. Recall that Let G = {g 1 , . . . , g n } be a given finite group and abbreviate δ g i = δ i for i = 1, . . . , n. Describe

is written for basis elements by
For i = j we obtain LHS= n w=1 s w i δ w and RHS = 2s i i δ i so that s j i = 0 for all i, j as desired.
Observe that k(G) and k G are cohomologically distinct.
Proposition 5.12 The first cohomology of H is given by H 1 ad (H, H) ∼ = k.
The first 2-differential is written as Take b = c = 1, then since ad(a ⊗ 1) = a for any a ∈ H, all three terms are the same and gives that φ(a ⊗ 1) = 0 for any a.
Take a = g and b = c = x, then the third term vanishes and we obtain ad(φ(g ⊗ x) ⊗ x) + φ(2x ⊗ x) = 0. For any possible value of φ(g ⊗ x), the value of the first term is written as hx for some h ∈ k from the table of ad above. Since φ is bilinear, constants can be renamed to obtain The second differential is written as (2) . (2) , and using that φ(x ⊗ x) = hx, we obtain h(x ⊗ g + g ⊗ gx) for the RHS.
Let a = g and b = x in the second differential. Then the LHS = φ(g ⊗ x) ⊗ g, and the RHS = φ(g ⊗ x) (1) ⊗ gφ(g ⊗ x) (2) . For the RHS to have terms ending in ⊗g only, φ(g ⊗ x) can have neither g nor gx terms since they would result in a ( ⊗1) term, so let φ(g ⊗ x) = h g,x 1 + h ′ g,x x. Then one computes RHS = (h g, . Equating this with LHS, we obtain h ′ g,x = 0. Thus we obtained φ(g ⊗ x) = h g,x 1 = −φ(x ⊗ g) . In the first differential, take a = b = g and c = x to obtain φ(g ⊗ gx) = φ(g ⊗ x) = h g,x 1 .
Let a = 1 and b = x in the second differential. Then the (2) . For the RHS to have terms ending in ⊗1 or ⊗x only, φ(1 ⊗ x) can have neither 1 nor x terms since they would result in a ( ⊗g) term, so let φ(1 ⊗ x) = h 1,x g + h 1,g gx . Then one computes RHS = h 1,x g ⊗ 1 + h 1,g (gx ⊗ 1 + 1 ⊗ x). Comparing with the LHS, we obtain φ(1 ⊗ g) = h 1,g 1 . With a = 1, b = x and c = g in the first differential, we also obtain φ(1 ⊗ gx) = −h 1,x g + h 1,g gx .
Recall that φ(x ⊗ gx) = h ′ x. For a = x and b = gx in the second differential gives which implies φ(x ⊗ gx) = 0 . In the second differential, take a = gx and b = x. Then we obtain The LHS has only ⊗g and ⊗gx terms, so that φ(gx⊗x) does not have g or gx terms, and we can write φ(gx⊗x) = h gx,x 1+h ′ gx,x x and compute RHS = h gx,x (1⊗g)+h ′ gx,x (x⊗g +g ⊗gx). Comparing with the LHS we obtain h ′ gx,x g = φ(gx⊗g)+h 1,x g+h 1,g gx, so that φ(gx ⊗ g) = (h ′ gx,x − h 1,x )g − h 1,g gx . By the first differential with (a, b, c) = (gx, g, x), we obtain By the first differential with (a, b, c) = (gx, gx, g), we obtain gx,x x . By the second differential with a = b = gx, we obtain and comparing the terms we obtain 2h 1,g = 0. In summary, resetting free variables by h g,x = α, h gx,x = β and h ′ gx,x = γ, we obtained a general solution represented by the following table.
1 g x gx It is checked, either by hand, or computer guided calculations, that these are indeed solutions.

Adjoint, Groupoid, and Quandle Cohomology Theories
From Remark 5.6, 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 G(x, y) for x, y ∈ Ob(G). Let f i ∈ G(x i , x i+1 ), 0 ≤ i < n, for non-negative integers i and n. Let C n (G) be the free abelian group generated by The boundary map ∂ : C n+1 (G) → C n (G) is defined by by linearly extending . . , f n ) 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 6.1 Let G be a groupoid and A be an abelian group regarded as a one-object groupoid. Then α : hom(x 0 , x 1 ) → A gives a groupoid homomorphism from G to A, which sends Ob(G) to the single object of A, if and only if a : C 1 (G) → A, defined by a(x 0 , f 0 ) = α(f 0 ), is a groupoid 1-cocycle. Next we consider extensions of groupoids. Define • : (hom(x 0 , where c(x 0 , f 0 , f 1 ) ∈ hom(C 2 (G), A). If G×A is a groupoid, the function c with the value c(x 0 , f 0 , f 1 ) is a groupoid 2-cocycle.
Let G be a finite group, and c : G 3 → k be a coefficient of the adjoint 3-cocycle defined in Proposition 5.4. This satisfies c(x, y, z) + c(x, yz, w) = c(x ⊳ y, z, w) + c(x, y, zw). Proposition 6.4 Let G be a group that is considered as a quandle under conjugation. Then θ : G 3 → k defined by θ(x, y, z) = c(x, y, z) − c(x, z, z −1 yz) is a rack 3-cocycle.
We compute

Deformations of R-matrices by adjoint 2-cocycles
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 t ) where H t is a k[[t]]-Hopf algebra given by H t = H ⊗ k[[t]] with all Hopf algebra structures inherited by extending those on H t . Let A = (H ⊗ k[[t]])/(t 2 )) and the Hopf algebra structure maps µ, ∆, ǫ, η, S be inherited on A. As a vector space A can be regarded as H ⊕ tH 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. Proof. The equalities of Lemma 3.2 hold in the quotient A = (H ⊗ k[[t]])/(t 2 ), where n = 1 and the modulus t 2 is considered. These cocycle conditions, on the other hand, were formulated from the motivation from Lemma 2.2 for the induced R-matrix R ad to satisfy the YBE. Hence these two lemmas imply the theorem.

Concluding Remarks
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?