共查询到20条相似文献,搜索用时 15 毫秒
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We classify all finite-dimensional connected Hopf algebras with large abelian primitive spaces. We show that they are Hopf algebra extensions of restricted enveloping algebras of certain restricted Lie algebras. For any abelian matched pair associated with these extensions, we construct a cohomology group, which classifies all the extensions up to equivalence. Moreover, we present a 1–1 correspondence between the isomorphism classes and a group quotient of the cohomology group deleting some exceptional points, where the group respects the automorphisms of the abelian matched pair and the exceptional points represent those restricted Lie algebra extensions. 相似文献
3.
Fernando Fantino Gastón Andrés García Mitja Mastnak 《Journal of Pure and Applied Algebra》2019,223(8):3611-3634
We classify all finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero such that its coradical is isomorphic to the algebra of functions over a dihedral group , with . We obtain this classification by means of the lifting method, where we use cohomology theory to determine all possible deformations. Our result provides an infinite family of new examples of finite-dimensional copointed Hopf algebras over dihedral groups. 相似文献
4.
《Quaestiones Mathematicae》2013,36(5):683-708
AbstractThe category HopfR of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category HopfR is locally presentable, it is coreflective in the category of bialgebras over R, over every R-algebra there exists a cofree Hopf algebra. (2) If, in addition, R is absoluty flat, then HopfR is reflective in the category of bialgebras as well, and there exists a free Hopf algebra over every R-coalgebra. Similar results are obtained for relevant subcategories of HopfR. Moreover it is shown that, for every commutative unital ring R, the so-called “dual algebra functor” has a left adjoint and that, more generally, universal measuring coalgebras exist. 相似文献
5.
Stefan Ufer 《Journal of Pure and Applied Algebra》2007,210(2):307-320
We consider an interesting class of braidings defined in [S. Ufer, PBW bases for a class of braided Hopf algebras, J. Algebra 280 (2004) 84-119] by a combinatorial property. We show that it consists exactly of those braidings that come from certain Yetter-Drinfeld module structures over pointed Hopf algebras with abelian coradical.As a tool we define a reduced version of the FRT construction. For braidings induced by Uq(g)-modules the reduced FRT construction is calculated explicitly. 相似文献
6.
We describe a method for constructing characters of combinatorial Hopf algebras by means of integrals over certain polyhedral cones. This is based on ideas from resurgence theory, in particular on the construction of well-behaved averages induced by diffusion processes on the real line. We give several interpretations and proofs of the main result in terms of noncommutative symmetric and quasi-symmetric functions, as well as generalizations involving matrix quasi-symmetric functions. The interpretation of noncommutative symmetric functions as alien operators in resurgence theory is also discussed, and a new family of Lie idempotents of descent algebras is derived from this interpretation. 相似文献
7.
Alexandru Chirv?situ 《Journal of Pure and Applied Algebra》2011,215(2):101-107
For a matrix coalgebra C over some field, we determine all small subcoalgebras of the free Hopf algebra on C, the free Hopf algebra with a bijective antipode on C, and the free Hopf algebra with antipode S satisfying on C for some fixed d. We use this information to find the endomorphisms of these free Hopf algebras, and to determine the centers of the categories of Hopf algebras, Hopf algebras with bijective antipode, and Hopf algebras with antipode of order dividing 2d. 相似文献
8.
《Quaestiones Mathematicae》2013,36(5):631-682
AbstractThe category Hopf ? of Hopf monoids in a symmetric monoidal category ?, assumed to be locally finitely presentable as a category, is analyzed with respect to its categorical properties. Assuming that the functors “tensor squaring” and “tensor cubing” on ? preserve directed colimits one has the following results: (1) If, in ?, extremal epimorphisms are stable under tensor squaring, then Hopf C is locally presentable, coreflective in the category of bimonoids in ? and comonadic over the category of monoids in C. (2) If, in ?, extremal monomorphisms are stable under tensor squaring, then Hopf ? is locally presentable as well, reflective in the category of bimonoids in C and monadic over the category of comonoids in ?. 相似文献
9.
Hui-Xiang Chen Hassen Suleman Esmael Mohammed Hua Sun 《Journal of Pure and Applied Algebra》2017,221(11):2752-2790
In this paper, we study the tensor product structure of the category of finite dimensional modules over Drinfeld doubles of Taft Hopf algebras. Tensor product decomposition rules for all finite dimensional indecomposable modules are explicitly given. 相似文献
10.
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. 相似文献
11.
Alan Koch Timothy Kohl Paul J. Truman Robert Underwood 《Journal of Pure and Applied Algebra》2019,223(5):2230-2245
Let be a finite separable extension of fields whose Galois closure has Galois group G. Greither and Pareigis use Galois descent to show that a Hopf algebra giving a Hopf–Galois structure on has the form for some group N of order . We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as K-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and K-algebras that appear in the classification of Hopf–Galois structures on a cyclic extension of degree , for p an odd prime number. 相似文献
12.
Edward L. Green Gregory Hartman Eduardo N. Marcos Øyvind Solberg 《Archiv der Mathematik》2005,85(2):118-127
In this paper we show that if
is a Koszul algebra with Λ0 isomorphic to a product of copies of a field, then the minimal projective resolution of Λ0 as a right Λ-module provides all the information necessary to construct both a minimal projective resolution of Λ0 as a left Λ-module and a minimal projective resolution of Λ as a right module over the enveloping algebra of Λ. The main tool for this is showing that there is a comultiplicative structure on a minimal projective resolution of Λ0 as a right Λ-module.Received: 14 September 2004 相似文献
13.
Pu Zhang 《Advances in Mathematics》2004,183(1):80-126
By introducing a twisted Hopf algebra we unify several important objects of study. Skew derivations of such an algebra are defined and the corresponding skew differential operator algebras are studied. This generalizes results in the Weyl algebra. Applying this investigation to the twisted Ringel-Hall algebra we get, in particular, a natural realization of the non-positive part of a quantized generalized Kac-Moody algebra, by identifying the canonical generators with some linear, skew differential operators. This also induces some algebras which are quantum-group like. 相似文献
14.
Guangbin Zhuang 《Journal of Pure and Applied Algebra》2011,215(12):2912-2922
Let H be a pointed Hopf algebra over an algebraically closed field of characteristic zero. If H is a domain with finite Gelfand-Kirillov dimension greater than or equal to two, then H contains a Hopf subalgebra of Gelfand-Kirillov dimension two. 相似文献
15.
A.L. Agore 《Journal of Pure and Applied Algebra》2018,222(4):914-930
We classify all Hopf algebras which factor through two Taft algebras and respectively . To start with, all possible matched pairs between the two Taft algebras are described: if then the matched pairs are in bijection with the group of d-th roots of unity in k, where while if then besides the matched pairs above we obtain an additional family of matched pairs indexed by . The corresponding bicrossed products (double cross product in Majid's terminology) are explicitly described by generators and relations and classified. As a consequence of our approach, we are able to compute the number of isomorphism types of these bicrossed products as well as to describe their automorphism groups. 相似文献
16.
J. CuadraJ.M. García-Rubira J.A. López-Ramos 《Journal of Computational and Applied Mathematics》2011,235(7):1833-1839
We determine all indecomposable codes over a class of Hopf algebras named Taft Algebras. We calculate dual codes and tensor products of these indecomposable codes and give applications of them. 相似文献
17.
Adriana Mejía Castaño Susan Montgomery Sonia Natale Maria D. Vega Chelsea Walton 《Journal of Pure and Applied Algebra》2018,222(7):1643-1669
Let p and q be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension and of dimension . We obtain that the non-isomorphic self-dual semisimple Hopf algebras of dimension classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8-dimensional Kac–Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita equivalence classes among semisimple Hopf algebras of odd dimension , established by the third-named author in an appendix. 相似文献
18.
G. C. Wraith 《Annali di Matematica Pura ed Applicata》1967,76(1):149-163
Summary In any category with products and a terminal object one may define the notions of group, module over a group etc. if f: R′→R
is a homomorphism of groups, and M an R-module, then one has an induced R′-module f*(M). If one is working in the category
of sets, one may define a functor left adjoint to f* by N→R⊗R′ N, where N is an R′-module. In this paper we show that f* has a left adjoint when one is working in the category of graded
connected coalgebras over a field. 相似文献
19.
Sarah Scherotzke 《Journal of Pure and Applied Algebra》2011,215(5):829-838
We construct rank varieties for the Drinfeld double of the Taft algebra Λn and for uq(sl2). For the Drinfeld double when n=2 this uses a result which identifies a family of subalgebras that control projectivity of Λ-modules whenever Λ is a Hopf algebra satisfying a certain homological condition. In this case we show that our rank variety is homeomorphic to the cohomological support variety. We also show that Ext∗(M,M) is finitely generated over the cohomology ring of the Drinfeld double for any finitely generated module M. 相似文献
20.
Wenhua Zhao 《Journal of Pure and Applied Algebra》2007,210(2):363-382
This paper is the first of a sequence of papers [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134; W. Zhao, Noncommutative symmetric functions and the inversion problem (submitted for publication). math.CV/0509135; W. Zhao, A system over the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509136; W. Zhao, systems over differential operator algebras and the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509138. preprint] on the (noncommutative symmetric) systems over differential operator algebras in commutative or noncommutative variables [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134]; the systems over the Grossman-Larson Hopf algebras [R. Grossman, R.G. Larson, Hopf-algebraic structure of families of trees, J. Algebra 126 (1) (1989) 184-210. [MR1023294]; L. Foissy, Les algèbres de Hopf des arbres enracinés décorés I, II, Bull. Sci. Math. 126 (3) (2002) 193-239; (4) 249-288. See also math.QA/0105212. [MR1909461]] of labeled rooted trees [W. Zhao, A system over the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509136]; as well as their connections and applications to the inversion problem [H. Bass, E. Connell, D. Wright, The Jacobian conjecture, reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. 7 (1982) 287-330. [MR 83k:14028]; A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, in: Progress in Mathematics, vol. 190, Birkhäuser Verlag, Basel, 2000. [MR1790619]] and specializations of NCSFs [W. Zhao, Noncommutative symmetric functions and the inversion problem (submitted for publication). math.CV/0509135; W. Zhao, systems over differential operator algebras and the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509138. preprint]. In this paper, inspired by the seminal work [I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (2) (1995) 218-348. See also hep-th/9407124. [MR1327096]] on NCSFs (noncommutative symmetric functions), we first formulate the notion of systems over associative Q-algebras. We then prove some results for systems in general; the systems over bialgebras or Hopf algebras; and the universal system formed by the generating functions of certain NCSFs in [I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (2) (1995) 218-348. See also hep-th/9407124. [MR1327096]]. Finally, we review some of the main results that will be proved in the following papers [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134; W. Zhao, A system over the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509136; W. Zhao, systems over differential operator algebras and the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509138. preprint] as some supporting examples for the general discussions given in this paper. 相似文献