Yevgenia Kashina, Yorck Sommerhaeuser, Yongchang Zhu
We prove that, over an algebraically closed field of characteristic zero, a semisimple Hopf algebra that has a nontrivial self-dual simple module must have even dimension. This generalizes a classical result of W. Burnside. As an application, we show under the same assumptions that a semisimple Hopf algebra that has a simple module of even dimension must itself have even dimension.
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We show that there is a family of complex semisimple Hopf algebras that do not admit a Hopf order over any number ring. They are Drinfel'd twists of certain group algebras. The twist contains a scalar fraction which makes impossible the definability of such Hopf algebras over number rings. We also prove that a complex semisimple Hopf algebra satisfies Kaplansky's sixth conjecture if and only if it admits a weak order, in the sense of Rumynin and Lorenz, over the integers.
A general Mackey type decomposition for representations of semisimple Hopf algebras is investigated. We show that such a decomposition occurs in the case that the module is induced from an arbitrary Hopf subalgebra and it is restricted back to a group subalgebra. Some other examples when such a decomposition occurs are also constructed. They arise from gradings on the category of corepresentations of a semisimple Hopf algebra and provide new examples of Green functors in the literature.
A description of all normal Hopf subalgebras of a semisimple Drinfeld double is given. This is obtained by considering an analogue of Goursat's lemma concerning fusion subcategories of Deligne products of two fusion categories. As an application we show that the Drinfeld double of any abelian extension is also an abelian extension.
In this note the notion of kernel of a representation of a semisimple Hopf algebra is introduced. Similar properties to the kernel of a group representation are proved in some special cases. In particular, every normal Hopf subalgebra of a semisimple Hopf algebra H is the kernel of a representation of H.
In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16. We also compute all the possible structures of the Grothendieck ring of semisimple non-commutative Hopf algebras of dimension 16. Moreover, we prove that non-commutative semisimple Hopf algebras of dimension p^n, p is prime, cannot have a cyclic group of grouplikes.
Masuoka proved (2009) that a finite-dimensional irreducible Hopf algebra $H$ in positive characteristic is semisimple if and only if it is commutative semisimple if and only if the Hopf subalgebra generated by all primitives is semisimple. In this paper, we give another proof of this result by using Hochschild cohomology of coalgebras.
We construct a class of non-commutative, non-cocommutative, semisimple Hopf algebras of dimension $2n^2$ and present conditions to define an inner faithful action of these Hopf algebras on quantum polynomial algebras, providing, in this way, more examples of semisimple Hopf actions which do not factor through group actions. Also, under certain condition, we classify the inner faithful Hopf actions of the Kac-Paljutkin Hopf algebra of dimension $8$, $H_8$, on the quantum plane.
Let $k$ be an algebraically closed field of characteristic $0$. In this paper, we obtain the structure theorems for semisimple Hopf algebras of dimension $p^2q^2$ over $k$, where $p,q$ are prime numbers with $p^2<q$. As an application, we also obtain the structure theorems for semisimple Hopf algebras of dimension $9p^2$ and $25q^2$ for all primes $3\neq p$ and $5\neq q$.
Let $p,q$ be prime numbers with $p>q^3$, and $k$ an algebraically closed field of characteristic 0. In this paper, we obtain the structure theorems for semisimple Hopf algebras of dimension $pq^3$.
We study actions of semisimple Hopf algebras H on Weyl algebras A over a field of characteristic zero. We show that the action of H on A must factor through a group algebra; in other words, if H acts inner faithfully on A, then H is cocommutative. The techniques used include reduction modulo a prime number and the study of semisimple cosemisimple Hopf actions on division algebras.
Let $p,q$ be prime numbers with $p^4<q$, and $k$ an algebraically closed field of characteristic 0. We show that semisimple Hopf algebras of dimension $p^2q^2$ can be constructed either from group algebras and their duals by means of extensions, or from Radford biproduct $R#kG$, where $kG$ is the group algebra of group $G$ of order $p^2$, $R$ is a semisimple Yetter-Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^2$. As an application, the special case that the structure of semisimple Hopf algebras of dimension $4q^2$ is given.
An extension of $k$-algebras $B \subset A$ is said to have depth one if there exists a positive integer $n$ such that $ A$ is a direct summand of $ B^n$ in $_B\mtr{Mod}_B$. Depth one extensions of semisimple algebras are completely characterized in terms of their centers. For extensions of semisimple Hopf algebras our results are similar to those obtained for finite group algebra extensions in \cite{BKone}.
Let $k$ be an algebraically closed field of characteristic 0. In this paper, we obtain the structure theorems for semisimple Hopf algebras of dimension $9q^2$ over $k$, where $q$ is a prime number. We also prove that odd-dimensional semisimple Hopf algebras over $k$ of dimension less than 600 are of Frobenius type.
We describe the Galois objects and biGalois groups of monomial nonsemisimple Hopf algebras. The main feature of our description is the use of modified versions of the second cohomology group of the grouplike elements. These computations generalize the previous ones of Masuoka and Schauenburg for the Taft algebras.
We develop the theory of semisimple weak Hopf algebras and obtain analogues of a number of classical results for ordinary semisimple Hopf algebras. We prove a criterion for semisimplicity and analyze the square of the antipode S^2 of a semisimple weak Hopf algebra A. We explain how the Frobenius-Perron dimensions of irreducible A-modules and eigenvalues of S^2 can be computed using the inclusion matrix associated to A. A trace formula of Larson and Radford is extended to a relation between the global and Frobenius-Perron dimensions of A. Finally, an analogue of the Class Equation of Kac and Zh…
On $4n$-dimensional neither pointed nor semisimple Hopf algebras and the associated weak Hopf algebras
本文研究了 $4n$ 维非指针非半单霍普夫代数及其相关弱霍普夫代数的结构，揭示了其准三角性、 $R$-矩阵描述及绿色环的非可交换性。
Let $q>2$ be a prime number, $d$ be an odd square-free natural number, and $n$ be a non-negative integer. We prove that a semisimple quasitriangular Hopf algebra of dimension $dq^n$ is solvable in the sense of Etingof, Nikshych and Ostrik. In particular, if $n\leq 3$ then it is either isomorphic to $k^G$ for some abelian group $G$, or twist equivalent to a Hopf algebra which fits into a cocentral abelian exact sequence.
We continue our study of the McKay Correspondence for grading preserving actions of semisimple Hopf algebras H on (noncommutative) Artin-Schelter regular algebras A. Here, we establish correspondences between module categories over A^H, over A\#H, and over \End_{A^H} A. We also study homological properties of (endomorphism rings of) maximal Cohen-Macaulay modules over A^H.
We prove that a depth two Hopf subalgebra K of a semisimple Hopf algebra H is normal (where the ground field $k$ is algebraically closed of characteristic zero). This means on the one hand that a Hopf subalgebra is normal when inducing (then restricting) modules several times as opposed to one time creates no new simple constituents. This point of view was taken in the paper http://arxiv.org/abs/math/0409346 which established a normality result in case H and K are finite group algebras. On the other hand this means that K is normal in H when H | K is a Galois extension with respect to action o…
Let H be a finite dimensional semisimple Hopf algebra over an algebraically closed field of characteristic zero. In this note we give a short proof of the fact that a Hopf subalgebra of H is a depth two subalgebra if and only if it is normal Hopf subalgebra.
Let H be a Hopf algebra of dimension pq over an algebraically closed field of characteristic 0, where p <= q are odd primes. Suppose that S is the antipode of H. If H is not semisimple, then S^{4p}=id_H and Tr(S^{2p}) is an integer divisible by p^2. In particular, if dim H = p^2, we prove that H is isomorphic to a Taft algebra. We then complete the classification for the Hopf algebras of dimension p^2.
In establishing a more general version of the McKay correspondence, we prove Auslander's theorem for actions of semisimple Hopf algebras H on noncommutative Artin-Schelter regular algebras A of global dimension two, where A is a graded H-module algebra, and the Hopf action on A is inner faithful with trivial homological determinant. We also show that each fixed ring A^H under such an action arises an analogue of a coordinate ring of a Kleinian singularity.
Let $q$ be a prime number, $k$ an algebraically closed field of characteristic 0, and $H$ a semisimple Hopf algebra of dimension $2q^3$. This paper proves that $H$ is always semisolvable. That is, such Hopf algebras can be obtained by (a number of) extensions from group algebras or duals of group algebras.
We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field by using the braiding structures of A. We investigate basic properties of the polynomial invariants including stability under extension of the base field. Furthermore, we show that our polynomial invariants are indeed tensor invariants of the representation category of A, and recognize the difference of the representation category and the representation ring of A. Actually, by computing and comparing polynomial invariants, we find new examples of pairs of Hopf algebras whose r…
The notion of double coset for semisimple finite dimensional Hopf algebras is introduced. This is done by considering an equivalence relation on the set of irreducible characters of the dual Hopf algebra. As an application formulae for the restriction of the irreducible characters to normal Hopf subalgebras are given.
Let H be a semisimple (so, finite dimensional) Hopf algebra over an algebraically closed field k of characteristic zero and let A be a commutative domain over k. We show that if A arises as an H-module algebra via an inner faithful H-action, then H must be a group algebra. This answers a question of E. Kirkman and J. Kuzmanovich and partially answers a question of M. Cohen. The main results of this article extend to working over k of positive characteristic. On the other hand, we obtain results on Hopf actions on Weyl algebras as a consequence of the main theorem.
A criterion for subcoalgebras to be invariant under the adjoint action is given generalizing Masuoka's criterion for normal Hopf subalgebras. At the level of characters, the image of the induction functor from a normal Hopf subalgebra is isomorphic to the image of the restriction functor.
In this paper, we prove that a non-semisimple Hopf algebra H of dimension 4p with p an odd prime over an algebraically closed field of characteristic zero is pointed provided H contains more than two group-like elements. In particular, we prove that non-semisimple Hopf algebras of dimensions 20, 28 and 44 are pointed or their duals are pointed, and this completes the classification of Hopf algebras in these dimensions.
In this paper, we define the higher Frobenius-Schur (FS-)indicators for finite-dimensional modules $V$ of a semisimple quasi-Hopf algebra $H$ via the categorical counterpart developed in \cite{NS05}. We prove that this definition of higher FS-indicators coincides with the higher indicators introduced by Kashina, Sommerhäuser, and Zhu when $H$ is a Hopf algebra. We also obtain a sequence of canonical central elements of $H$, which is invariant under gauge transformations, whose values, when evaluated by the character of an $H$-module $V$, are the higher Frobenius-Schur indicators of $V$. As an …
Two new results concerning complements in a semisimple Hopf algebra are proved. They extend some well known results from group theory. The uniqueness of Krull Schmidt Remak type decomposition is proved for semisimple completely reducible Hopf algebras.
We consider the action of a semisimple Hopf algebra $H$ on an $m$-Koszul Artin-Schelter regular algebra $A$. Such an algebra $A$ is a derivation-quotient algebra for some twisted superpotential $\mathsf{w}$, and we show that the homological determinant of the action of $H$ on $A$ can be easily calculated using $\mathsf{w}$. Using this, we show that the smash product $A\,\#\,H$ is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra $Λ$ to which $A\,\#\,H$ is Morita equivalent, generalising a result of Bocklandt-Schedler-Wemyss. We also show how $Λ$ can be u…
For each nontrivial semisimple Hopf algebra $H$ of dimension sixteen over $\mathbb{C}$, the smallest dimension inner-faithful representation of $H$ acting on a quadratic AS regular algebra $A$ of dimension 2 or 3, homogeneously and preserving the grading, is determined. Each invariant subring $A^H$ is determined. When $A^H$ is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that $H$ is a reflection Hopf algebra for $A$.
Frobenius-Schur Indicators of Dual Fusion Categories and Semisimple Partially Dualized Quasi-Hopf Algebras
本文研究球面融合范畴的对偶范畴中对象的Frobenius-Schur指标，建立特定对象指标间的关系，并推导出半单Hopf代数及其部分对偶化quasi-Hopf代数的指标公式与指数等式。
In this paper we classify, up to equivalence, all semisimple nontrivial Hopf algebras of dimension $2^{2n+1}$ for $n\geq 2$ over an algebraically closed field of characteristic $0$ with the group of group-like elements isomorphic to $\mathbb{Z}_{2^{n}}\times \mathbb{Z}_{2^{n}}$. Moreover we classify all such nonisomorphic Hopf algebras of dimension $32$ and show that they are not twist-equivalent to each other. More generally, given an abelian group of order $2^{m-1}$ we give an upper bound for the number of nonisomorphic nontrivial Hopf algebras of dimension $2^{m}$ which have this group as t…
In this note we prove a generalization of the Frobenius-Schur theorem for finite groups for the case of semisimple Hopf algebra over an algebraically closed field of characteristic 0. A similar result holds in characteristic $p > 2$ if the Hopf algebra is also cosemisimple. In fact we show a more general version for any finite-dimensional semisimple algebra with an involution.
Let $q$ be a prime number, $k$ an algebraically closed field of characteristic 0, and $H$ a non-trivial semisimple Hopf algebra of dimension $2q^3$. This paper proves that $H$ can be constructed either from group algebras and their duals by means of extensions, or from Radford's biproduct $H\cong R#kG$, where $kG$ is the group algebra of $G$ of order 2, $R$ is a semisimple Yetter-Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^3$.
S. Montgomery and S. Witherspoon proved that upper and lower semisolvable, semisimple, finite dimensional Hopf algebras are of Froebenius type when their dimensions are not divisible by the characteristic of the base field. In this note we show that a finite dimensional, semisimple, lower solvable Hopf algebra is always of Froebenius type, in arbitrary characteristic.
Let $\Bbbk$ be a base field of characteristic $p>0$ and let $U$ be the restricted enveloping algebra of a 2-dimensional nonabelian restricted Lie algebra. We classify all inner-faithful $U$-actions on noetherian Koszul Artin-Schelter regular algebras of global dimension up to three.
We develop a mechanism for classication of isomorphism types of non-trivial semisimple Hopf algebras whose group of grouplikes $G(H)$ is abelian of prime index $p$ which is the smallest prime divisor of $|G(H)|$. We describe structure of the second cohomology group of extensions of $\k C_p$ by $\k^G$ where $C_p$ is a cyclic group of order $p$ and $G$ a finite abelian group. We carry out an explicit classification for Hopf algebras of this kind of dimension $p^4$ for any odd prime $p$. The ground field is algebraically closed of characteristic $0$.
Let $H$ be a finite-dimensional Hopf algebra over an algebraically closed field of characteristic 0. If $H$ is not semisimple and $\dim(H)=2n$ for some odd integer $n$, then $H$ or $H^*$ is not unimodular. Using this result, we prove that if $\dim(H)=2p$ for some odd prime $p$, then $H$ is semisimple. This completes the classification of Hopf algebras of dimension $2p$.
We prove the non-existence of Hopf orders over number rings for two families of complex semisimple Hopf algebras. They are constructed as Drinfel'd twists of group algebras for the following groups: $A_n$, the alternating group on $n$ elements, with $n \geq 5$; and $S_{2m}$, the symmetric group on $2m$ elements, with $m \geq 4$ even. The twist for $A_n$ arises from a $2$-cocycle on the Klein four-group contained in $A_4$. The twist for $S_{2m}$ arises from a $2$-cocycle on a subgroup generated by certain transpositions which is isomorphic to $\mathbb{Z}_2^m$. This provides more examples of com…
Let $H$ be a non-semisimple Hopf algebra with antipode $S$ of dimension $pq$ over an algebraically closed field of characteristic 0 where $p \le q$ are odd primes. We prove that $\Tr(S^{2p})=p^2d$ where $d \equiv pq \pmod{4}$. As a consequence, if $p,q$ are twin primes, then any Hopf algebra of dimension $pq$ is semisimple.
We investigate when a weak Hopf algebra H is Frobenius; we show this is not always true, but it is true if the semisimple base algebra A has all its matrix blocks of the same dimension. However, if A is a semisimple algebra not having this property, there is a weak Hopf algebra H with base A which is not Frobenius (and consequently, it is not Frobenius "over" A either). We give, moreover, a categorical counterpart of the result that a Hopf algebra is a Frobenius algebra for a noncoassociative generalization of weak Hopf algebra.
We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z_2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a strengthening of a result of Nichols and Richmond on cosemisimple Hopf algebras with a 2-dimensional irreducible comodule in the finite dimensional context. Finally, we give some applications to the classification of certain classes of semisimple Hopf alge…
Study of the quotient module of a finite-dimensional Hopf subalgebra pair in order to compute its depth yields a relative Maschke Theorem, in which semisimple extension is characterized as being separable, and is therefore an ordinary Frobenius extension. We study the core Hopf ideal of a Hopf subalgebra, noting that the length of the annihilator chain of tensor powers of the quotient module is linearly related to the depth, if the Hopf algebra is semisimple. A tensor categorical definition of depth is introduced, and a summary from this new point of view of previous results are included. It i…
It is an open question whether the smash product of a semisimple Hopf algebra and a semiprime module algebra is semiprime. In this paper we show that the smash product of a commutative semiprime module algebra over a semisimple cosemisimple Hopf algebra is semiprime. In particular we show that the central $H$-invariant elements of the Martindale ring of quotients of a module algebra form a von Neumann regular and self-injective ring whenever $A$ is semiprime. For a semiprime Goldie PI $H$-module algebra $A$ with central invariants we show that $\AH$ is semiprime if and only if the $H$-action c…
Actions of semisimple Hopf algebras H over an algebraically closed field of characteristic zero on commutative domains were classified recently by the authors. The answer turns out to be very simple- if the action is inner faithful, then H has to be a group algebra. The present article contributes to the non-semisimple case, which is much more complicated. Namely, we study actions of finite dimensional (not necessarily semisimple) Hopf algebras on commutative domains, particularly when H is pointed of finite Cartan type. The work begins by reducing to the case where H acts inner faithfully on …
We call a monoidal category ${\mathcal C}$ a Serre category if for any $C$, $D \in {\mathcal C}$ such that $C\ot D$ is semisimple, $C$ and $D$ are semisimple objects in ${\mathcal C}$. Let $H$ be an involutory Hopf algebra, $M$, $N$ two $H$-(co)modules such that $M \otimes N$ is (co)semisimple as a $H$-(co)module. If $N$ (resp. $M$) is a finitely generated projective $k$-module with invertible Hattory-Stallings rank in $k$ then $M$ (resp. $N$) is (co)semisimple as a $H$-(co)module. In particular, the full subcategory of all finite dimensional modules, comodules or Yetter-Drinfel'd modules over…
Let H be a non-semisimple Hopf algebra of dimension 2p^2 over an algebraically closed field of characteristic zero, where p is an odd prime. We prove that H or H^* is pointed, which completes the classification for Hopf algebras of these dimensions.
We introduce and study new families of finite-dimensional Hopf algebras with the Chevalley property that are not pointed nor semisimple arising as twistings of quantum linear spaces. These Hopf algebras generalize the examples introduced in [AEG], [EG1], [EG2].
In this paper we introduce a trace-like invariant for the irreducible representations of a finite dimensional complex Hopf algebra H. We do so by considering the trace of the map induced by the antipode S on the endomorphisms End(V) of a self-dual module V. We also compute the values of this trace for the representations of two non-semisimple Hopf algebras: u_q(sl_2) and D(H_n(q)), the Drinfeld double of the Taft algebra.
We complete the classification of Hopf algebras of dimension 16 over an algebraically closed field of characteristic zero. We show that a non-semisimple Hopf algebra of dimension 16, has either the Chevalley property or its dual is pointed.
Under suitable assumptions on the base field, we prove that a commutative semisimple Yetter-Drinfel'd Hopf algebra over a finite abelian group is trivial, i.e., is an ordinary Hopf algebra, if its dimension is relatively prime to the order of the finite abelian group. Furthermore, we prove that a finite-dimensional cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra contains a trivial Yetter-Drinfel'd Hopf subalgebra of dimension greater than one, at least if the Yetter-Drinfel'd Hopf algebra itself has dimension greater than one.
We study the higher Frobenius-Schur indicators of modules over semisimple Hopf algebras, and relate them to other invariants as the exponent, the order, and the index. We prove various divisibility and integrality results for these invariants. In particular, we prove a version of Cauchy's theorem for semisimple Hopf algebras. Furthermore, we give some examples that illustrate the general theory.
About $39$ years ago, Kaplansky conjectured that the dimension of a semisimple Hopf algebra over an algebraically closed field of characteristic zero is divisible by the dimensions of its simple modules. Although it still remains open, some partial answers to this conjecture play an important role in classifying semisimple Hopf algebras. This paper focuses on the recent development of Kaplansky's sixth conjecture and its applications in classifying semisimple Hopf algebras.
We prove that a finite-dimensional irreducible Hopf algebra $H$ in positive characteristic is semisimple, if and only if it is commutative and semisimple, if and only if the restricted Lie algebra $P(H)$ of the primitives is a torus. This generalizes Hochschild's theorem on restricted Lie algebras, and also generalizes Demazure and Gabriel's and Sweedler's results on group schemes, in the special but essential situation with finiteness assumption added.
Let $H$ be a semisimple Hopf algebra acting on an Artin-Schelter regular algebra $A$, homogeneously, inner-faithfully, preserving the grading on $A$, and so that $A$ is an $H$-module algebra. When the fixed subring $A^H$ is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that $H$ is a reflection Hopf algebra for $A$. We show that each of the semisimple Hopf algebras $H_{2n^2}$ of Pansera, and $\mathcal{A}_{4m}$ and $\mathcal{B}_{4m}$ of Masuoka is a reflection Hopf algebra for an AS regular algebra of dimension 2 or 3.
Let H be a Hopf algebra of dimension pq over an algebraically closed field of characteristic zero, where p, q are odd primes with p < q < 4p+12. We prove that H is semisimple and thus isomorphic to a group algebra, or the dual of a group algebra.
The objects of study in this paper are Hopf algebras $H$ which are finitely generated algebras over an algebraically closed field and are extensions of a commutative Hopf algebra by a finite dimensional Hopf algebra. Basic structural and homological properties are recalled and classes of examples are listed. Bounds are obtained on the dimensions of simple $H$-modules, and the structure of $H$ is shown to be severely constrained when the finite dimensional extension is semisimple and cosemisimple.
We revisit a class of examples described in the original paper on biproducts, expand the class, and provide a detailed analysis of the coalgebra and algebra structures of many of these examples. Connections with the semisimple Hopf algebras of dimension a power of two determined by Kashina are examined. The finite-dimensional non-trivial semisimple cosemisimple Hopf algebras we construct are shown to be lower cosolvable. Some of these have one proper normal Hopf subalgebra and are not lower solvable.
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