K. A. Brown, J. J. Zhang
This paper constitutes the first part of a program to classify all affine prime regular Hopf algebras $H$ of Gelfand-Kirillov dimension one over an algebraically closed field of characteristic zero. We prove a number of properties of such an algebra, list some classes of examples, and then prove that - when the PI-degree of $H$ is prime - our list contains all such algebras.
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The classification of affine prime regular Hopf algebras of GK-dimension one is completed. As consequences, 1) we give a negative answer to an open question posed by Brown-Zhang and 2) we show that there do exist prime regular Hopf algebras of GK-dimension one which are not pointed.
This paper is an attempt to construct a special kind of Hopf pairing $\langle-,-\rangle:H^\bullet\otimes H\rightarrow\Bbbk$. Specifically, $H^\bullet$ and $H$ should be both affine, noetherian and of the same GK-dimension. In addition, some properties of them would be dual to each other. We test the ideas in two steps for all the affine prime regular Hopf algebras $H$ of GK-dimension one: 1) We compute the finite duals $H^\circ$ of them, which are given by generators and relations; 2) the Hopf pairings desired are determined by choosing certain Hopf subalgebras $H^\bullet$ of $H^\circ$, where …
We prove a structure theorem for Yetter-Drinfel'd Hopf algebras over groups of prime order that are nontrivial, cocommutative, and cosemisimple: Under certain assumptions on the base field, these algebras can be decomposed into a tensor product of the dual group ring of the group of prime order and an ordinary group ring of some other group. This tensor product is a crossed product as an algebra and an ordinary tensor product as a coalgebra. In particular, the dimension of such a Yetter-Drinfel'd Hopf algebra is divisible by the prime under consideration. We also find explicit examples of such…
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 $5\neq q$.
We determine the Hopf Galois structures on a Galois field extension of degree twice an odd prime square and classify the corresponding left braces. Besides we determine the separable field extensions of degree twice an odd prime square allowing a cyclic Hopf Galois structure and the number of these structures.
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.
Let $H$ be a connected Hopf algebra acting on an algebra $A$. Working over a base field having characteristic $0$, we show that for a given prime (semi-prime, completely prime) ideal $I$ of $A$, the largest $H$-stable ideal of A contained in $I$ is also prime (semi-prime, completely prime). We also prove a similar result for certain subrings of convolution algebras.
The Hopf algebra $\mathcal{D}$ which is the subject of this paper can be viewed as a Drinfeld double of the bosonisation of the Jordan plane. Its prime and primitive spectra are completely determined. As a corollary of this analysis it is shown that $\mathcal{D}$ satisfies the Dixmier-Moeglin Equivalence, leading to the formulation of a conjecture on the validity of this equivalence for pointed Noetherian Hopf algebras.
The characterization of $H$-prime radical is given in many ways. Meantime, the relations between the radical of smash product $R # H$ and the $H$-radical of Hopf module algebra $R$ are obtained.
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.
Since the discovery of quantum groups (Drinfeld, Jimbo) and finite dimensional variations thereof (Lusztig, Manin), these objects were studied from different points of view and had many applications. The present paper is part of a series where we intend to show that important classes of Hopf algebras are quantum groups and therefore belong to Lie theory. One of our main results is the explicit construction of a general family of pointed Hopf algebras from Dynkin diagrams. All the Frobenius-Lusztig kernels and their parabolic subalgebras belong to this family, but in addition we get many new ex…
A Hopf Galois structure on a finite field extension $L/K$ is a pair $(H,μ)$, where $H$ is a finite cocommutative $K$-Hopf algebra and $μ$ a Hopf action. In this paper we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree and several properties of those. We show a table which summarizes the program results. Besides, for separable field extensions of degree $2p^n$, with $p$ an odd prime number, we prove that the occurrence of some type of Hopf Galois structure may either imply or exclude the o…
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 $pq^3$.
In formulating a generalized framework to study certain noncommutative algebras naturally arising in representation theory, K. A. Brown asked if every finitely generated Hopf algebra satisfying a polynomial identity was finite over a normal commutative Hopf subalgebra. In this note we show that Radford's biproduct, applied to the enveloping algebra of the Lie superalgebra pl(1,1), provides a noetherian prime counterexample.
The aim of this work is to discuss the concepts of degeneration, deformation and rigidity, and to apply them to the geometric study of the varieties of Hopf algebras. The main result is the description of the n-dimensional rigid Hopf algebras and the irreducible components for n=p^2, p prime and for n<14.
We completely describe by generators and relations and classify all Hopf algebras which factorize through the Taft algebra $T_{m^{2}}(q)$ and the group Hopf algebra $K[C_{n}]$: they are $nm^{2}$-dimensional quantum groups $T_{nm^{2}}^ ω(q)$ associated to an $n$-th root of unity $ω$. Furthermore, using Dirichlet's prime number theorem we are able to count the number of isomorphism types of such Hopf algebras. More precisely, if $d = {\rm gcd}(m,ν(n))$ and $\frac{ν(n)}{d} = p_1^{α_1} \cdots p_r^{α_r}$ is the prime decomposition of $\frac{ν(n)}{d}$ then the number of types of Hopf algebras that f…
We prove that any action of a finite dimensional Hopf algebra H on a Weyl algebra A over an algebraically closed field of characteristic zero factors through a group action. In other words, Weyl algebras do not admit genuine finite quantum symmetries. This improves a previous result by the authors, where the statement was established for semisimple H. The proof relies on a refinement of the method previously used: namely, considering reductions of the action of H on A modulo prime powers rather than primes. We also show that the result holds, more generally, for algebras of differential operat…
This note discusses a framework for the investigation of the prime spectrum of an associative algebra A that is equipped with an action of a Hopf algebra H. In particular, we study a notion of H-rationality for ideals of A and comment on a possible Dixmier-Moeglin equivalence for H-prime ideals of A.
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.
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 $0$.
We study finite dimensional Hopf algebra actions on so-called filtered Artin-Schelter regular algebras of dimension n, particularly on those of dimension 2. The first Weyl algebra is an example of such on algebra with n=2, for instance. Results on the Gorenstein condition and on the global dimension of the corresponding fixed subrings are also provided.
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.
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.
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 consider Hopf-Galois structures on separable (but not necessarily normal) field extensions $L/K$ of squarefree degree $n$. If $E/K$ is the normal closure of $L/K$ then $G=\mathrm{Gal}(E/K)$ can be viewed as a permutation group of degree $n$. We show that $G$ has derived length at most $4$, but that many permutation groups of squarefree degree and of derived length $2$ cannot occur. We then investigate in detail the case where $n=pq$ where $q \geq 3$ and $p=2q+1$ are both prime. (Thus $q$ is a Sophie Germain prime and $p$ is a safeprime). We list the permutation groups $G$ which can arise, a…
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 $q>2$ 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 investigate Hopf-Galois structures on a cyclic field extension $L/K$ of squarefree degree $n$. By a result of Greither and Pareigis, each such Hopf-Galois structure corresponds to a group of order $n$, whose isomorphism class we call the type of the Hopf-Galois structure. We show that every group of order $n$ can occur, and we determine the number of Hopf-Galois structures of each type. We then express the total number of Hopf-Galois structures on $L/K$ as a sum over factorisations of $n$ into three parts. As examples, we give closed expressions for the number of Hopf-Galois structures on a…
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.
Let H be a Hopf algebra with antipode S, and let A be an N-Koszul Artin-Schelter regular algebra. We study connections between the Nakayama automorphism of A and S^2 of H when H coacts on A inner-faithfully. Several applications pertaining to Hopf actions on Artin-Schelter regular algebras are given.
We investigate a generalization of Hopf algebra $\mathfrak{sl}_{q}(2)$ by weakening the invertibility of the generator $K$, i.e. exchanging its invertibility $KK^{-1}=1$ to the regularity $K\overline{K}K=K$. This leads to a weak Hopf algebra $w\mathfrak{sl}_{q}(2)$ and a $J$-weak Hopf algebra $v\mathfrak{sl}_{q}(2)$ which are studied in detail. It is shown that the monoids of group-like elements of $w\mathfrak{sl}_{q}(2)$ and $v\mathfrak{sl}_{q}(2)$ are regular monoids, which supports the general conjucture on the connection betweek weak Hopf algebras and regular monoids. Moreover, from $w\mat…
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.
Hopf--Galois structures of cyclic type on parallel extensions of prime power degree
本文研究了当扩展L/K的度数为素数幂且类型为循环时，平行扩展是否也具有相同Hopf-Galois结构的问题，采用群论方法并结合Greither-Pareigis和Byott的工作予以解决。
We show that every central simple algebra A over a field k is Brauer equivalent to a quotient of a finite dimensional Hopf algebra over the same field (that is- A is Hopf Schur). If the characteristic of the field is zero, or if the algebra has a Galois splitting field of degree prime to the characteristic of k, we can take this Hopf algebra to be semisimple. We also show that if F is any finite extension of k, then F is a quotient of a finite dimensional Hopf algebra over k. We use it in order to show why the algebric closeness assumption is necessary in a weak form of Kaplansky's tenth conje…
In this paper we classify the finite-dimensional pointed rank one Hopf algebras which are generated as algebras by the first element of the coradical filtration over a field of prime characteristic.
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.
There is no systematic general procedure by which isomorphism classes of Hopf algebras that are extensions of $\k F$ by ${\k}^G$ can be found. We develop the general procedure for classification of isomorphism classes of Hopf algebras which are extensions of the group algebra $\k C_p$ by ${\k}^G$ where $C_p$ is a cyclic group of prime order $p$ and ${\k}^G$ is the Hopf algebra dual of $\k G$, $G$ a finite abelian $p$-group and $\k$ is an algebraically closed field of characteristic $0$. We apply the method to calculate the number of isoclasses of commutative extensions and certain extensions o…
Let $H$ be a cocommutative Hopf algebra acting on an algebra $A$. Assuming the base field to be algebraically closed and the $H$-action on $A$ to be integral, that is, it is given by a coaction of some Hopf subalgebra of the finite dual $H^\circ$ that is an integral domain, we stratify the prime spectrum $\mbox{Spec}\, A$ in terms of the prime spectra of certain commutative algebras. For arbitrary $H$-actions in characteristic $0$, we show that the largest $H$-stable ideal of $A$ that is contained in a given semiprime ideal of $A$ is semiprime as well.
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.
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.
For any Koszul Artin-Schelter regular algebra A, we consider a version of the universal Hopf algebra aut(A) coacting on A, introduced by Manin. To study the representations (i.e. finite dimensional comodules) of this Hopf algebra, we use the Tannaka-Krein formalism. Specifically, we construct an explicit combinatorial rigid monoidal category U, equipped with a functor M to finite dimensional vector spaces such that aut(A)= coend_U(M). Using this pair (U,M) we show that aut(A) is quasi-hereditary as a coalgebra and in addition is derived equivalent to the representation category of U.
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.
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.
The left and right homological integrals are introduced for a large class of infinite dimensional Hopf algebras. Using the homological integrals we prove a version of Maschke's theorem for infinite dimensional Hopf algebras. The generalization of Maschke's theorem and homological integrals are the keys to study noetherian regular Hopf algebras of Gelfand-Kirillov dimension one.
We study semisimple Hopf algebra actions on Artin-Schelter regular algebras and prove several upper bounds on the degrees of the minimal generators of the invariant subring, and on the degrees of syzygies of modules over the invariant subring. These results are analogues of results for group actions on commutative polynomial rings proved by Noether, Fogarty, Fleischmann, Derksen, Sidman, Chardin, and Symonds.
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.
Actions of Hopf-Ore Extensions of Group Algebras on Path Algebras of Quivers
本文分类群代数 Hopf-Ore 扩张在箭图路代数上的滤过 Hopf 作用，并将结果应用于 GK 维数一和二的 Noether 素 Hopf 代数等特例。
We study algebraic and homological properties of two classes of infinite dimensional Hopf algebras over an algebraically closed field k of characteristic zero. The first class consists of those Hopf k-algebras that are connected graded as algebras, and the second class are those Hopf k-algebras that are connected as coalgebras. For many but not all of the results presented here, the Hopf algebras are assumed to have finite Gel'fand-Kirillov dimension. It is shown that if the Hopf algebra H is a connected graded algebra of finite Gel'fand-Kirillov dimension n, then H is a noetherian domain whic…
Based on a pairing of two regular multiplier Hopf algebras $A$ and $B$, Heisenberg double $\mathscr{H}$ is the smash product $A \# B$ with respect to the left regular action of $B$ on $A$. Let $\mathscr{D}=A\bowtie B$ be the Drinfel'd double, then Heisenberg double $\mathscr{H}$ is a Yetter-Drinfel'd $\mathscr{D}$-module algebra, and it is also braided commutative by the braiding of Yetter-Drinfel'd module, which generalizes the results in [10] to some infinite dimensional cases.
Let $p$ be prime, let $K$ be a non-archimedean local field of characteristic $p$ and let $C_p^3$ denote the elementary abelian group of order $p^3$. We give a complete classification of Hopf orders in the $K$-Hopf algebra $(K[C_p^3])^*$ and under a mild condition compute their dual Hopf orders in the group ring $K[C_p^3]$.
Let $(A,Δ)$ be a regular weak multiplier Hopf algebra. Denote by $E$ the canonical idempotent of $(A,Δ)$ and by $B$ the image of the source map. Recall that $B$ is a non-degenerate algebra, sitting nicely in the multiplier algebra $M(A)$ of $A$ so that also $M(B)$ can be viewed as a subalgebra of $M(A)$. Assume that $u,v$ are invertible elements in $M(B)$ so that $E(vu\otimes 1)E=E$. This last condition is obviously fulfilled if $u$ and $v$ are each other inverses, but there are also other cases. Now modify $Δ$ and define $Δ'(a)=(u\otimes 1)Δ(a)(v\otimes 1)$ for all $a \in A$. We show in this …
It is well-known that any weak Hopf algebra gives rise to a Hopf algebroid. Moreover it is possible to characterize those Hopf algebroids that arise in this way. Recently, the notion of a weak Hopf algebra has been extended to the case of algebras without identity. This led to the theory of weak multiplier Hopf algebras. Similarly also the theory of Hopf algebroids was recently developed for algebras without identity. They are called multiplier Hopf algebroids. Then it is quite natural to investigate the expected link between weak multiplier Hopf algebras and multiplier Hopf algebroids. This r…
In this paper, we study weak bialgebras and weak Hopf algebras. These algebras form a class wider than bialgebras respectively Hopf algebras. The main results of this paper are Kaplansky's constructions type which lead to weak bialgebras or weak Hopf algebras starting from a regular algebra or a bialgebra. Also we provide a classification of 2-dimensional and 3-dimensional weak bialgebras and weak Hopf algebras. We determine then the stabilizer group and the representative of these classes, the action being that of the linear group.
Prime and semiprime quantum linear space smash products
本文研究量子线性空间玻色子化与结合代数的 smash 积，给出其为（半）素的条件，并应用于量子仿射空间，推广了 Bergen 的结果。
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 $p$ be a prime. We complete the classification on pointed Hopf algebras of dimension $p^2$ over an algebraically closed field $k$. When $\text{char}k \neq p$, our result is the same as the well-known result for $\text{char}k=0$. When $\text{char}k=p$, we obtain 14 types of pointed Hopf algebras of dimension $p^2$, including a unique noncommutative and noncocommutative type.
Multiplier Hopf coquasigroup: Definition and Coactions
该研究利用Galois映射定义广义multiplier Hopf余拟群，给出其充要条件，并探讨正则情形下的余作用与Yetter-Drinfeld拟模，丰富相关代数结构理论。
In this paper, we generalize Majid's bicrossproduct construction. We start with a pair (A,B) of two regular multiplier Hopf algebras. We assume that B is a right A-module algebra and that A is a left B-comodule coalgebra. We recall and discuss the two notions in the first sections of the paper. The right action of A on B gives rise to the smash product A # B. The left coaction of B on A gives a possible coproduct on A # B. We will discuss in detail the necessary compatibility conditions between the action and the coaction for this to be a proper coproduct on A # B. The result is again a regula…
For a regular multiplier Hopf algebra $A$, the Yetter-Drinfel'd module category ${}_{A}\mathcal{YD}^{A}$ is equivalent to the centre $Z({}_{A}\mathcal{M})$ of the unital left $A$-module category ${}_{A}\mathcal{M}$. Then we introduce the generalized $(α, β)$-Yetter-Drinfel'd module categories ${}_{A}\mathcal{GYD}^{A}(α, β)$, which are treated as components of a braided $T$-category. Especially when $A$ is a coFrobenius Hopf algebra, ${}_{A}\mathcal{YD}^{A}(α, β)$ is isomorphic to the unital $\hat{A} \bowtie A(α, β)$-module category ${}_{\hat{A} \bowtie A(α, β)}\mathcal{M}$. Finally for a Yette…
We introduce the notion of the Frobenius-Perron dimension of an integral $\Bbb Z_+$-ring and give some applications of this notion to classification of finite dimensional quasi-Hopf algebras with a unique nontrivial simple module, and of quasi-Hopf and Hopf algebras of prime dimension $p$.
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