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1.
We prove a version of the Frobenius–Schur theorem for a finite-dimensional semisimple Hopf algebra H over an algebraically closed field; if the field has characteristic p not 0, H is also assumed to be cosemisimple. Then for each irreducible representation V of H, we define a Schur indicator for V, which reduces to the classical Schur indicator when H is the group algebra of a finite group. We prove that this indicator is 0 if and only if V is not self-dual. If V is self dual, then the indicator is positive (respectively, negative) if and only if V admits a nondegenerate bilinear symmetric (resp., skew-symmetric) H-invariant form. A more general result is proved for algebras with involution.  相似文献   

2.
    
Kok-Ming Teo 《代数通讯》2013,41(9):3027-3035
In their recent paper [13], Tate and Van den Bergh studied certain quadratic algebras, called the “Sklyanin algebras”. They proved that these algebras have the Hilbert series of a polynomial algebra, are Noetherian and Koszul, and satisfy the Auslander-Gorenstein and Cohen-Macaulay conditions. This paper gives an alternative proof of these results, as suggested in [13], and thereby answering a question in their paper.  相似文献   

3.
We introduce a family of braided Hopf algebras that (in characteristic zero) generalizes the rank 1 Hopf algebras introduced by Krop and Radford and we study its cleft extensions.  相似文献   

4.
To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with non-zero modulus and of depth two. This association is shown to yield a bijection between (the isomorphism classes, on both sides, of) such objects.  相似文献   

5.
    
S. Caenepeel 《代数通讯》2013,41(12):4735-4736
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6.
Let k be an algebraically closed field of characteristic zero.This paper proves that semisimple Hopf algebras over k of dimension 66,70 and 78 are of Frobenius type.  相似文献   

7.
    
In this article, we first decompose a cleft extension for X ? Y into two cleft extensions for X and Y respectively, where X and Y are Hopf algebras on a commutative ring R. Conversely, we introduce the concept of consistent cleft Hopf extensions and prove that one can construct a cleft extension for X ? Y by two cleft extensions for X and Y if and only if these two cleft extensions are consistent. An example is also given to show an application of our main results.  相似文献   

8.
张晓辉  吴慧 《数学学报》2019,62(3):373-380
本文研究并刻画了交换环上弱Hopf代数、Yetter-Drinfeld模范畴的一些性质,给出了其能够做成半单范畴的充分条件.  相似文献   

9.
    
Liang-yun Zhang 《代数通讯》2013,41(4):1269-1281
In this article, we mainly give the structure theorem of endomorphism algebras of weak Hopf algebras, and give another structure theorem as well as some applications for weak Doi–Hopf modules.  相似文献   

10.
11.
Rongchuan Xiong 《代数通讯》2020,48(11):4615-4637
Abstract

In this article, we determine cocycle deformations and Galois objects of non-commutative and non-cocommutative semisimple Hopf algebras of dimension 16. We show that these Hopf algebras are pairwise twist inequivalent mainly by calculating their higher Frobenius-Schur indicators, and that except three Hopf algebras which are cocycle deformations of dual group algebras, none of them admit non-trivial cocycle deformations.  相似文献   

12.
Motivated by the construction of new examples of Artin–Schelter regular algebras of global dimension four, Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 26682690 .[Crossref], [Web of Science ®] [Google Scholar]] introduced an algebra extension A P [y 1, y 2; σ, δ, τ] of A, which they called a double Ore extension. This construction seems to be similar to that of a two-step iterated Ore extension over A. The aim of this article is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form A[y 1; σ1, δ1][y 2; σ2, δ2]. We also give partial answers to some questions posed in Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 26682690 .[Crossref], [Web of Science ®] [Google Scholar]].  相似文献   

13.
    
Christian Gottlieb 《代数通讯》2013,41(12):4687-4691
Abstract

Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on rings. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true Hopf algebras over rings. We show that over any commutative ring R that is not a field there exists a Hopf algebra H over R containing a non-zero integral but not being finitely generated as R-module. On the contrary we show that Sweedler's equivalence is still valid for free Hopf algebras or projective Hopf algebras over integral domains. Analogously for a left H-module algebra A we study the influence of non-zero left A#H-linear maps from A to A#H on H being finitely generated as R-module. Examples and application to separability are given.  相似文献   

14.
We conclude the classification of Hopf algebras of dimension 12 over an algebraically closed field of characteristic zero.  相似文献   

15.
郝志峰 《数学季刊》1996,11(3):29-32
In this paper,we show that if H is a finite dimensional Hopf algebra then H is quasitri-angular if and only if H is coquasi-triangular. As a consequentility ,we obtain a generalized result of Sauchenburg.  相似文献   

16.
We study the following question: when is the right adjoint of the forgetful functor from the category of -Doi-Hopf modules to the category of -modules also a left adjoint? We can give some necessary and sufficient conditions; one of the equivalent conditions is that and the smash product are isomorphic as -bimodules. The isomorphism can be described using a generalized type of integral. Our results may be applied to some specific cases. In particular, we study the case , and this leads to the notion of -Frobenius -module coalgebra. In the special case of Yetter-Drinfel'd modules over a field, the right adjoint is also a left adjoint of the forgetful functor if and only if is finite dimensional and unimodular.

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17.
This paper extends classical results in the invariant theory of finite groups and finite group schemes to the actions of finite Hopf algebras on commutative rings. Topics considered include integrality over the invariant rings, properties of the canonical map between the prime spectra, orbital and stabilizer algebras, projectivity over the invariant rings, and descent of Cohen-Macaulayness.  相似文献   

18.
George Szeto 《代数通讯》2013,41(12):3979-3985
Let B be a Galois algebra over a commutative ring R with Galois group G such that B H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 ? e) where e and 1 ? e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V B (A) = ?∑ gG(A) J g , and the centers of A and B G(A) are the same where V B (A) is the commutator subring of A in B, J g  = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.  相似文献   

19.
The Larson–Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra [15 Larson, R. G., Sweedler, M. E. (1969). An associative orthogonal bilinear form for Hopf algebras. Amer. J. Math. 91:7593.[Crossref], [Web of Science ®] [Google Scholar]]. The result has been generalized to finite-dimensional weak Hopf algebras by Vecsernyés [44 Vecsernyés, P. (2003). Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras. J. Algebra 270:471520. See also arXiv: 0111045v3 [math.QA] for an extended version.[Crossref], [Web of Science ®] [Google Scholar]]. In this paper, we show that the result is still true for weak multiplier Hopf algebras. The notion of a weak multiplier bialgebra was introduced by Böhm et al. in [4 Böhm, G., Gómez-Torecillas, J., López-Centella, E. (2015). Weak multiplier bialgebras. Weak multiplier bialgebras. 367(12):86818872. See also arXiv: 1306.1466 [math.QA]. [Google Scholar]]. In this note it is shown that a weak multiplier bialgebra with a regular and full coproduct is a regular weak multiplier Hopf algebra if there is a faithful set of integrals. Weak multiplier Hopf algebras are introduced and studied in [40 Van Daele, A., Wang, S. (2015). Weak multiplier Hopf algebras I. The main theory. J. Ange. Math. (Crelles J.) 705:155209, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crelle-2013-0053, July 2013. See also arXiv:1210.4395v1 [math.RA].[Web of Science ®] [Google Scholar]]. Integrals on (regular) weak multiplier Hopf algebras are treated in [43 Van Daele, A., Wang, S. (2016). Weak multiplier Hopf algebras III. Integrals and duality. Preprint University of Leuven (Belgium) and Southeast University of Nanjing (China), See arXiv: 1701.04951.v3 [math.RA]. [Google Scholar]]. This result is important for the development of the theory of locally compact quantum groupoids in the operator algebra setting, see [13 Kahng, B.-J., Van Daele, A. A class of C*-algebraic locally compact quantum groupoids I. Preprint Canisius College Buffalo (USA) and University of Leuven (Belgium). [Google Scholar]] and [14 Kahng, B.-J., Van Daele, A. A class of C*-algebraic locally compact quantum groupoids II. Preprint Canisius College Buffalo (USA) and University of Leuven (Belgium). [Google Scholar]]. Our treatment of this material is motivated by the prospect of such a theory.  相似文献   

20.
Let G be a group and assume that (A p ) pG is a family of algebras with identity. We have a Hopf G-coalgebra (in the sense of Turaev) if, for each pair p,qG, there is given a unital homomorphism Δ p,q : A pq A p A q satisfying certain properties. Consider now the direct sum A of these algebras. It is an algebra, without identity, except when G is a finite group, but the product is non-degenerate. The maps Δ p,q can be used to define a coproduct Δ on A and the conditions imposed on these maps give that (A,Δ) is a multiplier Hopf algebra. It is G-cograded as explained in this paper. We study these so-called group-cograded multiplier Hopf algebras. They are, as explained above, more general than the Hopf group-coalgebras as introduced by Turaev. Moreover, our point of view makes it possible to use results and techniques from the theory of multiplier Hopf algebras in the study of Hopf group-coalgebras (and generalizations). In a separate paper, we treat the quantum double in this context and we recover, in a simple and natural way (and generalize) results obtained by Zunino. In this paper, we study integrals, in general and in the case where the components are finite-dimensional. Using these ideas, we obtain most of the results of Virelizier on this subject and consider them in the framework of multiplier Hopf algebras. Presented by Ken Goodearl.  相似文献   

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