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if the following conditions are satisfied: (Rib1) [MATH] , where [MATH] denotes the center of [MATH] (Rib2) [MATH] (Rib3) [MATH] |
In the case where [MATH] is a quasitriangular Hopf algebra, the condition (Rib4) [MATH] is also required in addition to the above three conditions. Then, the triplet [MATH] is called a ribbon Hopf algebra By definition any ribbon element [MATH] is invertible, and if [MATH] is of finite dimension, then the condition |
(Rib0) [MATH] is automatically satisfied , where [MATH] is the Drinfeld element of [MATH] A ribbon element is characterized by a special group-like element as follows |
Lemma 2.2 Let [MATH] be a quasitriangular Hopf algebra over [MATH] For an element [MATH] the following conditions (1) and (2) are equivalent. |
[MATH] [MATH] is a ribbon element of [MATH] [MATH] there is an element [MATH] such that [MATH] Here, [MATH] denotes the set of the group-like elements of [MATH] . ∎ |
Although the Drinfeld element is not necessary to be a ribbon element, in the semisimple and cosemisimple case the following holds. |
Proposition 2.3 Gelaki , Lemma 2.1.1] ). Let [MATH] be a quasitriangular Hopf algebra over [MATH] , and [MATH] be its Drinfeld element. If [MATH] is semisimple and cosemisimple, then |
[MATH] and [MATH] Therefore, the Drinfeld element [MATH] of [MATH] is a ribbon element of [MATH] . ∎ Using Lemma 2.2 and Proposition 2.3 we have: |
Proposition 2.4 Let [MATH] be a finite-dimensional quasitriangular Hopf algebra over [MATH] , and [MATH] be its Drinfeld element. If [MATH] is semisimple and cosemisimple, then the set of all ribbon elements [MATH] is given by |
[MATH] . ∎ In order to know the ribbon elements of a finite-dimensional semisimple and cosemisimple quasitriangular Hopf algebra [MATH] it is enough to determine the set |
[MATH] by Proposition 2.4 Let us recall the definitions of braided Hopf algebras and coribbon Hopf algebras that are the dual notions of quasitriangular Hopf algebras and ribbon Hopf algebras, respectively. The former and the letter are introduced by Doi |
and Hayashi , respectively. Let [MATH] be a bialgebra [MATH] over [MATH] A linear functional [MATH] is called a braiding of [MATH] , if it is convolution-invertible, and the following conditions are satisfied: |
(B1) [MATH] (B2) [MATH] (B3) [MATH] for all [MATH] The pair [MATH] is called a braided bialgebra In a braided bialgebra [MATH] the following equation holds: |
(B4) [MATH] for all [MATH] An invertible element [MATH] is said to be a coribbon element of a braided bialgebra [MATH] if the following conditions are satisfied: |
(CR1) [MATH] (CR2) [MATH] (CR3) [MATH] for all [MATH] . The triplet [MATH] is called a coribbon bialgebra Furthermore, if [MATH] is a Hopf algebra and the condition |
(CR4) [MATH] is satisfied, then the triplet [MATH] is called a coribbon Hopf algebra Remark 2.5 If a Hopf algebra [MATH] is of finite dimension, then a braiding [MATH] of [MATH] is a universal [MATH] -matrix of [MATH] via the usual isomorphism [MATH] This construction gives a one-to-one correspondence between the braid... |
Dualizing Proposition 2.4 we have: Corollary 2.6 Let [MATH] be a finite-dimensional braided Hopf algebra over [MATH] , and [MATH] be its Drinfeld element: |
[MATH] If [MATH] is semisimple and cosemisimple, then the Drinfeld element [MATH] is a coribbon element of [MATH] , and the set of all coribbon elements of [MATH] , written by [MATH] , is given by |
[MATH] Remark 2.7 For the dual Hopf algebra [MATH] [EQUATION] 2.2 Polynomial invariants of quasitriangular Hopf algebras In the author introduced some invariant of a finite-dimensional semisimple and cosemisimple Hopf algebra defined by using braiding structures and given as a polynomial. This invariant is a monoidal M... |
Let [MATH] be a finite-dimensional semisimple and cosemisimple Hopf algebra over [MATH] By Etingof and Gelaki , Corollary 1.5] , the set of universal [MATH] -matrices [MATH] is finite. Let us consider a quasitriangular Hopf algebra [MATH] For an element [MATH] and a finite-dimensional left [MATH] -module [MATH] , let [... |
[MATH] is the Drinfeld element of [MATH] . Then, we set [MATH] and call it the categorical dimension of [MATH] We note that if [MATH] is a finite-dimensional semisimple and cosemisimple Hopf algebra over [MATH] , then for any absolutely simple left [MATH] -module [MATH] |
[MATH] by , and the following equation holds 24 , Lemma 3.2] [EQUATION] So, [MATH] is a root of unity in [MATH] Let [MATH] be a positive integer, and [MATH] be a complete system of the absolutely simple left [MATH] -modules of dimension [MATH] Then we define a polynomial [MATH] by |
[EQUATION] If there is no absolutely simple left [MATH] -module of dimension [MATH] , then we define [MATH] For a quasitriangular Hopf algebra [MATH] we denote the braided monoidal category [MATH] by [MATH] Here, [MATH] is the braiding associated to [MATH] , that is, for [MATH] |
[MATH] Two quasitriangular Hopf algebras [MATH] and [MATH] over [MATH] are said to be braided Morita equivalent if the braided monoidal categories |
[MATH] and [MATH] are equivalent as [MATH] -linear braided monoidal categories. By using the same technique in the proof of 24 , Theorem 2.6] |
it can be verified that the above polynomial [MATH] is a braided Morita invariant, that is, [MATH] and [MATH] are braided Morita equivalent, then |
[MATH] for all positive integers [MATH] By using [MATH] the polynomial invariant [MATH] defined in can be written by [MATH] Another braided Morita invariant can be constructed by using ribbon structures. Let [MATH] be a quasitriangular Hopf algebra, and [MATH] be its ribbon element. Then any ribbon element [MATH] of [M... |
[MATH] Suppose that [MATH] is finite-dimensional semisimple and cosemisimple. For a positive integer [MATH] a polynomial [MATH] can be defined as follows. |
[EQUATION] where [MATH] is a complete system of the absolutely simple left [MATH] -modules of dimension [MATH] and [MATH] is a scalar determined by [MATH] This polynomial [MATH] is also a braided Morita invariant. By Proposition 2.3 and Lemma 2.10 given in the next subsection we have: |
Proposition 2.8 Let [MATH] be a semisimple and cosemisimple quasitriangular Hopf algebra of finite dimension. For a positive integer [MATH] |
[MATH] can be divided by [MATH] in [MATH] So, a polynomial [MATH] [MATH] is defined, and it is also a braided Morita invariant. Example 2.9 |
Let [MATH] denote the cyclic group of order [MATH] which is [MATH] , and [MATH] be a primitive [MATH] th root of unity. The universal [MATH] -matrices of the group Hopf algebra [MATH] are |
[MATH] where [MATH] for each [MATH] The Drinfeld element [MATH] of [MATH] is [MATH] We have [MATH] If we set [MATH] , then [MATH] |
forms a complete system of simple [MATH] -modules. Then [MATH] , and if [MATH] is even, then [MATH] for [MATH] by using [MATH] Therefore |
[EQUATION] By comparing [MATH] we see that [MATH] are not mutually braided Morita equivalent for [MATH] In the case of [MATH] [EQUATION] |
So, [MATH] are not mutually braided Morita equivalent. 2.3 Relationship between ribbon and pivotal structures It is known that any ribbon category has a pivotal structure |
In the case where a ribbon category is the module category [MATH] of finite-dimensional left [MATH] -modules over a ribbon Hopf algebra [MATH] the associated pivotal structure [MATH] is given by |
[MATH] for each object [MATH] Therefore the left and right pivotal dimensions of [MATH] in the ribbon category are [EQUATION] Suppose that [MATH] is absolutely simple. Since [MATH] is invertible, it follows that [MATH] , and hence |
[EQUATION] From this, we also have [EQUATION] If [MATH] is semisimple and cosemisimple, then the pivotal structures of [MATH] are uniquely determined by the group [MATH] Thus, [MATH] has finite order, and it follows that [MATH] is a root of unity in [MATH] By ( 2.1 ), [MATH] is also a root of unity, and so by ( 2.5 ), ... |
Lemma 2.10 Let [MATH] be a semisimple and cosemisimple quasitriangular Hopf algebra of finite dimension, and [MATH] be an absolutely simple left [MATH] -module. Then |
[MATH] for the Drinfeld element [MATH] of [MATH] Proof.. The pivotal element [MATH] corresponding to [MATH] is [MATH] Since [MATH] coincides with the trace of [MATH] by ( 2.4 ), we see that [MATH] Now, the desired equation follows from ( 2.5 ). |
Let [MATH] be a coalgebra over [MATH] The dual space [MATH] has a [MATH] -algebra structure, and any finite-dimensional right [MATH] -comodule [MATH] can be regarded as a left [MATH] -module with the action |
[MATH] where we write the right [MATH] -coaction [MATH] in the form [MATH] This construction gives rise to an identical category equivalence between [MATH] -linear monoidal categories of finite-dimensional right [MATH] -comodules and of finite-dimensional left [MATH] -modules. For a finite-dimensional right [MATH] -com... |
[MATH] where [MATH] are mutually dual bases of [MATH] , respectively. The above element does not depend on the choice of bases. Lemma 2.11 |
Let [MATH] be a finite-dimensional Hopf algebra over [MATH] , and [MATH] be a finite-dimensional right [MATH] -comodule. [MATH] [MATH] is a pivotal element of the dual Hopf algebra [MATH] , then |
[MATH] where the left-hand side is the right pivotal dimension of [MATH] viewed as a left [MATH] -module as usual. [MATH] Assume that [MATH] is absolutely simple with [MATH] Then [MATH] for any [MATH] |
Proof.. Let [MATH] and [MATH] be mutually dual bases of [MATH] and [MATH] , respectively, and [MATH] be the right coaction on [MATH] We write [MATH] Then [MATH] , and hence |
[MATH] On the other hand, [MATH] . Thus we have [MATH] Since [MATH] for a pivotal element [MATH] , Part (1) is proved. Assume that [MATH] is absolutely simple with [MATH] Then there is an element [MATH] such that [MATH] Taking the trace of this map we have the formula in Part (2). |
The coribbon elements of Suzuki’s Hopf algebras In this section we review the definition of Suzuki’s Hopf algebras, and describe the braiding structures of them in accordance with Suzuki’s paper |
The correct results on coribbon elements of Suzuki’s braided Hopf algebras described in are also given. Suzuki’s Hopf algebras are given as a family of finite-dimensional cosemisimple Hopf algebras [MATH] -matrices. Suppose that [MATH] is an algebraically closed field whose characteristic is not [MATH] , and let |
[MATH] be the comatrix coalgebra of degree [MATH] over [MATH] , that is, there is a basis [MATH] of [MATH] such that [EQUATION] Let [MATH] be a coideal of the tensor algebra [MATH] defined by |
[EQUATION] We set [MATH] , and denote by [MATH] the image of [MATH] under the natural projection [MATH] For [MATH] we define an element [MATH] in [MATH] by |
[EQUATION] Then we have [MATH] Let [MATH] [MATH] , and consider the following subset of [MATH] [EQUATION] Then [MATH] is a coideal of [MATH] , and |
[MATH] is a bialgebra. We also denote the image of [MATH] by the same symbol. It can be easily shown that [EQUATION] is a basis of [MATH] over [MATH] , and [MATH] The bialgebra [MATH] actually is a cosemisimple Hopf algebra, whose structure maps are given by |
[MATH] Remark 3.1 The description of the antipode of [MATH] in is wrong in the case when [MATH] If [MATH] , then the cosemisimple Hopf algebra [MATH] is also semisimple 21 , Theorem 3.1 vii)] , and |
[MATH] coincide with [MATH] , respectively, which are introduced by Masuoka and generalized in In particular, the Hopf algebra [MATH] is the unique Hopf algebra which is an [MATH] -dimensional non-commutative and non-cocommutative Hopf algebra up to isomorphism. This Hopf algebra is called the Kac-Paljutkin algebra |
By uniqueness we see that [MATH] is self-dual, that is the dual Hopf algebra is isomorphic to itself. By ( 3.2 ), we see that for any integers [MATH] satisfying with [MATH] |
[EQUATION] It is known by Suzuki that the group [MATH] is given by [EQUATION] that is of order [MATH] , and the set [EQUATION] gives a complete system of absolutely simple right [MATH] -comodules, where the coactions of all subspaces above are induced from the comultiplication [MATH] of [MATH] |
Let [MATH] be odd, and set [MATH] or [MATH] if [MATH] is odd or even, respectively. Then, [MATH] is isomorphic to the group algebra of the following finite group |
[MATH] In fact, an algebra isomorphism [MATH] is given by [MATH] Suzuki also determined the all braidings of [MATH] The construction of [MATH] and the method of determination of its braidings are closely related to the universality for quadratic bialgebras (see |
for a detailed statement and also for the above fact). [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] |
[MATH] Theorem 3.2 S.Suzuki [MATH] For [MATH] let [MATH] be a [MATH] -linear map whose values [MATH] are given by the left table above. Then [MATH] is extended to a braiding of [MATH] if and only if |
[MATH] [MATH] Consider the case [MATH] . For [MATH] let [MATH] be a [MATH] -linear map whose values [MATH] are given by the right table above. Then, [MATH] is extended to a braiding of [MATH] if and only if |
[MATH] [MATH] If [MATH] , then the braidings of [MATH] are given by [MATH] If [MATH] , then the braidings of [MATH] are given by |
[EQUATION] We note that there is a natural embedding [MATH] , and therefore [MATH] is [MATH] as an algebra. Thus, a braiding [MATH] of [MATH] is determined by the values on [MATH] by (B2), (B3). |
The following lemma is partially proved in 23 , p.341] The equation [MATH] for an odd integer [MATH] is added. In particular, these values are not equal to [MATH] Hereinafter, we treat the indices of Kronecker’s delta [MATH] as modulo [MATH] |
Lemma 3.3 In the braided Hopf algebra [MATH] the following holds. [EQUATION] Lemma 3.4 In the braided Hopf algebra [MATH] the following holds: |
[EQUATION] The Drinfeld elements of Suzuki’s braided Hopf algebras are given by the following lemma. Lemma 3.5 Suppose that [MATH] contains a [MATH] th root of unity. |
[MATH] The Drinfeld element [MATH] of [MATH] is given by [MATH] [MATH] The Drinfeld element [MATH] of [MATH] is given by [MATH] Proof.. |
(1) [MATH] [MATH] Since [MATH] by Lemma 3.3 it follows that [MATH] (2) By Lemma 3.4 and [MATH] , we have [MATH] The following is the revised version of Lemma 8 in |
(see Appendix for the needed modification). Lemma 3.6 The Yang-Baxter form [MATH] on [MATH] given in Theorem 3.2 (1) can be extended to a braiding of the bialgebra [MATH] We denote it the same symbol [MATH] For an element [MATH] , the [MATH] -linear functional [MATH] defined by |
[MATH] can be extended to a coribbon element of the braided bialgebra [MATH] We denote the coribbon element by the same symbol [MATH] Suppose that [MATH] and [MATH] satisfy [MATH] . Then, |
[MATH] [MATH] induces a coribbon element of the braided bialgebra [MATH] if and only if [MATH] [MATH] for [MATH] with [MATH] [MATH] if and only if [MATH] . ∎ |
Lemma 3.7 The Yang-Baxter form [MATH] on [MATH] given in Theorem 3.2 (2) can be extended to a braiding of the bialgebra [MATH] , where |
[MATH] We denote this braiding of [MATH] by the same symbol [MATH] For an element [MATH] , the [MATH] -linear functional [MATH] by the same formula in Lemma 3.6 can be extended to a coribbon element of the braided bialgebra [MATH] We denote the coribbon element by the same symbol [MATH] Suppose that [MATH] and [MATH] s... |
Combining Lemmas 3.6 and 3.7 we have the following correct version of 23 , Theorem 5] Theorem 3.8 Let [MATH] be an algebraically closed field whose characteristic does not divide [MATH] For each element [MATH] let [MATH] be the [MATH] -linear functional defined by the same formula in Lemma 3.6 Then for a braided Hopf a... |
[MATH] Let [MATH] be elements in [MATH] satisfying [MATH] Then, [MATH] is extended to a coribbon element of the braided bialgebra [MATH] if and only if |
[MATH] and any coribbon element of the braided bialgebra [MATH] is given by the form [MATH] In addition, [MATH] is a coribbon element of the braided Hopf algebra [MATH] |
if and only if [MATH] Therefore, there are exactly two coribbon elements of the braided Hopf algebra [MATH] [MATH] Let [MATH] be elements in [MATH] satisfying [MATH] Then, |
[MATH] is extended to a coribbon element of the braided bialgebra [MATH] if and only if [MATH] and any coribbon element of the braided bialgebra [MATH] is given by the form [MATH] In addition, |
[MATH] is a coribbon element of the braided Hopf algebra [MATH] if and only if [MATH] Therefore, there are exactly two coribbon elements of the braided Hopf algebra [MATH] |
Proof.. (1) Let [MATH] be a coribbon element of the braided bialgebra [MATH] [MATH] By [MATH] for [MATH] and (CR1) we have [MATH] |
Since [MATH] are linearly independent, it follows that [MATH] One can set [MATH] for some [MATH] since [MATH] is convolution-invertible. So, [MATH] is obtained by |
[MATH] where [MATH] is the natural projection, and [MATH] is the coribbon element of the braided bialgebra [MATH] determined by [MATH] for all [MATH] Thus, by Lemma 3.6 (1) it is required that [MATH] for some [MATH] It can be easily shown that the converse is true. By Lemma 3.6 (2) a necessary and sufficient condition ... |
[MATH] is a coribbon element of [MATH] is [MATH] (2) Let [MATH] be a coribbon element of the braided bialgebra [MATH] As the same manner with the proof of Part (1) we see that |
[MATH] and [MATH] is not [MATH] Hence [MATH] is given by [MATH] where [MATH] is the natural projection, [MATH] , and [MATH] is the coribbon element of the braided bialgebra [MATH] Thus, [MATH] by Lemma 3.7 , and [MATH] is needed to be the form in Part (2). The converse is also true. Furthermore, by Lemma 3.7 (2) a nece... |
[MATH] is a coribbon element of [MATH] is [MATH] To determine the coribbon elements of a braided Hopf algebra [MATH] one can apply Corollary 2.6 This fact gives us an alternative proof of Theorem 3.8 as follows. |
Suppose that [MATH] contains a [MATH] th root of unity. If [MATH] is odd, then [MATH] Here, [MATH] are defined by [MATH] and the products between them are given by [MATH] If [MATH] is even, then |
[EQUATION] Here, [MATH] are given by [MATH] and products between them are given by [MATH] [MATH] where the indices of the right-hand sides are treated as modulo [MATH] |
Proposition 3.9 Suppose that [MATH] contains a [MATH] th root of unity. Then, [MATH] where [MATH] is the counit of [MATH] , and [MATH] is the algebra map defined by [MATH] |
Proof.. An element [MATH] belongs to the center of [MATH] if and only if [MATH] Thus, whereas [MATH] [MATH] since [MATH] Furthermore, it follows from [MATH] that |
[MATH] . This implies that [MATH] , that is, [MATH] Since [MATH] , it follows that [MATH] Alternative proof of Theorem 3.8 By Corollary 2.6 and Proposition 3.9 we have |
[MATH] Here, [MATH] is the Drinfeld element of [MATH] , and it is given by [MATH] by Lemma 3.5 (1). Thus, (1) is proved. Similarly, it can be shown that [MATH] , and hence (2) is also proved. |
Polynomial invariants for duals of Suzuki’s braided Hopf algebras In this section we assume that [MATH] [MATH] , and [MATH] is an algebraically closed field which contains a [MATH] th root of unity. We also assume that [MATH] satisfy [MATH] , and [MATH] satisfy [MATH] |
By Lemma 2.11 we have: Lemma 4.1 [MATH] Let us consider the coribbon elements [MATH] and [MATH] of the braided Hopf algebra [MATH] |
[MATH] for the simple right [MATH] -comodule [MATH] [EQUATION] [MATH] for the simple right [MATH] -comodule [MATH] [EQUATION] [MATH] Let us consider the coribbon elements [MATH] and [MATH] of the braided Hopf algebra [MATH] |
[MATH] for the simple right [MATH] -comodule [MATH] [EQUATION] [MATH] for the simple right [MATH] -comodule [MATH] [EQUATION] Proof.. |
In the case of [MATH] and [MATH] by Lemma 2.10 [MATH] [MATH] [MATH] for any absolutely simple right [MATH] -comodule [MATH] The values [MATH] [MATH] have already computed in 24 , Lemma 5.9(1)] although it needs to remove [MATH] from that formula. So, we obtain the formulas for [MATH] and [MATH] in the proposition. Othe... |
(1) (i) First, we note that [MATH] for [MATH] By Lemma 2.11 (2), if [MATH] , then [MATH] and similarly if [MATH] , then [MATH] (ii) Since [MATH] , we have |
[EQUATION] By a similar computation we have the equations of (2). By Lemma 4.1 we have: Theorem 4.2 [MATH] For [MATH] and [MATH] , set |
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