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Journal of Generalized Lie Theory and Applications
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Cohomology of the adjoint of Hopf algebras

J. Scott CARTER1, Alissa S. CRANS2, Mohamed ELHAMDADI3, and Masahico SAITO3

1University of South Alabama, Mobile, AL 36688, USA
E-mail: [email protected]

2Loyola Marymount Unversity, Los Angeles, CA 90045, USA
E-mail: [email protected]

3University of South Florida, Tampa, FL 33620, USA
E-mails: [email protected] and [email protected]

Received May 22, 2007 Revised July 10, 2007

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Abstract

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 image 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] image to the Yang-Baxter equation (YBE): image 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.

Preliminaries

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

generalized-theory-applications-Operations-Hopf

Figure 1: Operations in Hopf algebras

generalized-theory-applications-Axioms-Hopf

Figure 2: Axioms of Hopf algebras

A coalgebra is a vector space C over a field k together with a comultiplication image that is bilinear and coassociative: image. A coalgebra is cocommutative if the comultiplication satisfies image, where image is the transpositionimage A coalgebra with counit is a coalgebra with a linear map called the counit image such thatimage A bialgebra is an algebra A over a field k together with a linear map called the unitimage satisfyingimage where 1 ∈ A is the multiplicative identity and with an associative multiplicationimage 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 image such thatimage 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 image 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 image for any y ∈ H is defined by image where we use the common notationimage andimage 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 image we have image For the adjoint map ad : image we have ad image

generalized-theory-applications-Conditions

Figure 3: Conditions for the YBE for Hopf algebras

Definition 1. Let H be a Hopf algebra and ad be the adjoint map. Then the linear map image defined by image 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:

generalized-theory-applications-adjoint-map

Figure 4: YBE by the adjoint map

image (2.1)

image (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.

generalized-theory-applications-Adjoint-condition

Figure 5: Adjoint condition (1)

generalized-theory-applications-Adjoint

Figure 6: Adjoint condition (2)

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, image for any a, b, c 2 H, where image denotes the right action defined by the adjoint: image The equality (2.2) can be similarly rewritten as

image

Remark 2. It was pointed out to us by Sommerhaeuser that the induced R-matrix Rad is invertible with inverse image

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 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 (Ht, adt) where Ht is a k[[t]]-Hopf algebra given byimage with all Hopf algebra structures inherited by extending those on Ht with the identity on the k[[t]] factor (the trivial deformation as a Hopf algebra), with a deformations of ad given by image where image i = 1, 2, · · · , are maps.

Suppose image satisfies the adjoint conditions (equalities (2.1) and (2.2)) mod tn+1, and suppose that there exist image such that image satisfies the adjoint conditions mod tn+2. Define image by

image

image

For the first adjoint condition (2.1) of image we obtain

image

which is equivalent by degree calculations to

image

For the second adjoint condition (2.2) of image we obtain

image

which is equivalent by degree calculations to

image

In summary we proved the following.

Lemma 2. The mapimage satisfies the adjoint conditions mod tn+2 if and only if

image

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

Differentials and cohomology

Chain groups

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

image

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

image

For n = 1, define

image

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

image

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

image

First differentials

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

Definition 4. The first differential image is defined by

image

Diagrammatically, we represent d1,1 as depicted in Fig. 7, where a 1-cochain is represented by a circle on a string.

generalized-theory-applications-differential

Figure 7: The 1-differential

Second differentials

Definition 5. Define the second differentials by image and

image

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

generalized-theory-applications-cochain

Figure 8: A diagram for a 2-cochain and the 2-cocycle conditions

Theorem 1. D2D1 = 0.

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

generalized-theory-applications-coboundary

Figure 9: The 2-cocycle condition for a 2-coboundary, Part I

generalized-theory-applications-cocycle-condition

Figure 10: The 2-cocycle condition for a 2-coboundary, Part II

Third differentials

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

image

Diagrams for 3-cochains are depicted in Fig. 11. See Fig. 12 (A), (B), and (C) for the diagrammatics for d3,1, d3,2 and d3,3, respectively.

generalized-theory-applications-Diagrams

Figure 11: Diagrams for 3-cochains

generalized-theory-applications-differentials

Figure 12: The 3-differentials

Theorem 2. D3D2 = 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 image This is indicated in Fig. 15, where subscripts ad are suppressed for simplicity. Let image (note that image does not land in the domain of d3,3). The first line of Fig. 15 represents the definition of the differential

generalized-theory-applications

Figure 13: d3,1(d2,1) = 0

generalized-theory-applications

Figure 14: d3,2(d2,1, d2,2) = 0

generalized-theory-applications

Figure 15: d3,3(d2,2) = 0

image

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

image

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

image

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

image

which is equal, by compatibility, to image 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 image to formal deformations in Lemma 2 represent a 3-cocycle:image

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

Cohomology groups

For convenience define image

Then Theorems 1 and 2 are summarized as follows.

Theorem 3.image is a chain complex.

This enables us to propose

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

image for n = 1, 2, 3.

Examples

Group algebras

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

Lemma 3.image

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

image

and the RHS is written as

image

For a given w, fix u and then compare the coefficients of image In the LHS we have image while on the RHS w = u, and furthermore image Thus the diagonal coefficient must satisfy image since char image In the caseimage neither term of imagenor image is equal toimage hence image

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

image

for any x, y, z ∈ G.

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

image

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

image

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

image

and the result follows.

Remark 3. In the preceding proof, since the term imageimage Then the condition stated becomes aimage

Proposition 1. Let G be a group. image where ξ1 is the map that is defined by linearly extending imageimage if and only if ξ2 = 0 and the coefficients satisfy the following properties: imageimage satisfies

image

Proof. Suppose image be the map that is defined by linearly extendingimage Then the third 3-cocycle condition from Definition 6 gives:(abbreviatingimage

image

image

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 image This gives

image

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

image

From the second sum we obtain image In conclusion, if image for kG then ∈2 = 0.

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

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

image

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

image

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

image

If we write image then

image

Hence

image

and in particular for the coefficients cu(x, y, z) from Proposition 1,

image

By setting image we obtain

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

image

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

image

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

Proposition 2. For the symmetric group G = S3 on three letters, we have imageimage

Proof. By Lemma 3, we have imageimage 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) image The set kG of functions from G to k with pointwise addition and multiplication is a unital associative algebra. It has a Hopf algebra structure using image with comultiplication defined through image by image and the antipode by image

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

image

Lemma 6. image

Proof. Recall the defining equality (4.1). Let image be a given finite group and abbreviate image Describeimage Then image is written for basis elements by LHS = f(δiδj) and

image

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

Lemma 7.image

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

By combining the above lemmas, we obtain the following

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

Observe that k(G) and kG are cohomologically distinct.

Bosonization of the superline

Let H be generated by 1, g, x with relations image and Hopf algebra structure image (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, image

image

Remark 5. The induced R-matrix R-matrix Rad has determinant 1, the characteristic polynomial is image and the minimal polynomial isimage

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

Proof. Recall the defining equality (4.1). Let imageAssume that f(x) = a + bx + cg + dxg and image. Applying f to both sides of the equation g2 = 1, one obtains image Similarly evaluating both sides of the equationimageone obtains that f(g) = 0. In a similar way, applying f to the equations x2 = 0 and xg = −gx gives rise to, respectively, a = 0 and c = 0. Also evaluating image at x gives rise to d = 0. We also have f(x) = f(xg)g (since g2 = 1),which implies f(xg) = bxg. In conclusion f satisfies f(1) = 0 = f(g), f(x) = bx, and f(xg) = b(xg). Now consider f in the kernel of D1, that is f satisfies

image

It is directly checked on all the generatorsimage This implies that image

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

Proof. With d1,1 = 0 from the preceding Proposition, we have image Either the direct hand calculations from definitions or the computer calculations give the following general solution for the 2-cocycle image represented in the following table:

image

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

Adjoint, groupoid, and quandle cohomology theories

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 image Let image for non-negative integers i and n. Let Cn(G) be the free abelian group generated by image The boundary map image is defined by by linearly extending

image

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

image

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 image gives a groupoid homomorphism from G to A, which sends Ob(G) to the single object of A, if and only if image defined byimage is a groupoid 1-cocycle. Next we consider extensions of groupoids. Define

image

image

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

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

image

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

image

generalized-theory-applications-Diagrams-groupoid

Figure 16: Diagrams for groupoid 1- and 2-cocycles

Compare with Remark 4.

For image the symmetric group on 3 letters, with coefficient groupimage 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 : G2 → k be adjoint 2-cocycle coefficients that were defined in Remark 3. These satisfy

image

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

image

Proof. By definition

image

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

image

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

image

is a rack 3-cocycle.

Proof. We must show that µ satsifies

image

We compute

image

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 (Ht, adt) where Ht is a k[[t]]-Hopf algebra given by image with all Hopf algebra structures inherited by extending those on Ht. Let image and the Hopf algebra structure mapsimage be inherited on A. As a vector space A can be regarded as image

Recall that a solution to the YBE, R-matrix Rad 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 image be an adjoint 2-cocycle. Then the map image defined by image satisfies the YBE.

Proof. The equalities of Lemma 2 hold in the quotient image where n = 1 and the modulus t2 is considered. These cocycle conditions, on the other hand, were formulated from the motivation from Lemma 1 for the induced R-matrix Rad 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

image

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

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?

Acknowledgements

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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