共查询到20条相似文献,搜索用时 15 毫秒
1.
D. Stefan 《Proceedings of the American Mathematical Society》1997,125(11):3191-3193
In this paper we construct certain Hopf subalgebras of a pointed Hopf algebra over a field of characteristic 0. Some applications are given in the case of Hopf algebras of dimension 6, and , where and are different prime numbers.
2.
Using descent theory, we study Hopf algebra forms of pointed Hopf algebras. It turns out that the set of isomorphism classes
of such forms are in one-to-one correspondence to other known invariants, for example the set of isomorphism classes of Galois
extensions with a certain group F, or the set of isometry classes of m-ary quadratic forms. Our theory leads to a classification of all Hopf algebras over a field of characteristic zero that become
pointed after a base extension, in dimension p, p
2 and p
3, with p odd.
Received: 22 November 1998 相似文献
3.
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
4.
Shilin Yang 《Frontiers of Mathematics in China》2007,2(2):305-316
The group of Hopf algebra automorphisms for a finite-dimensional semisimple cosemisimple Hopf algebra over a field k was considered by Radford and Waterhouse. In this paper, the groups of Hopf algebra automorphisms for two classes of pointed
Hopf algebras are determined. Note that the Hopf algebras we consider are not semisimple Hopf algebras.
相似文献
5.
Martín Mombelli 《Mathematische Zeitschrift》2010,266(2):319-344
We develop some techniques for studying exact module categories over some families of pointed finite-dimensional Hopf algebras.
As an application we classify exact module categories over the tensor category of representations of the small quantum groups
uq(\mathfraksl2){u_q(\mathfrak{sl}_2)}. 相似文献
6.
《Advances in Mathematics》2003,178(2):177-243
A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of braided vector spaces , where X is a rack and q is a 2-cocycle on X with values in . Racks and cohomology of racks appeared also in the work of topologists. This leads us to the study of the structure of racks, their cohomology groups and the corresponding Nichols algebras. We will show advances in these three directions. We classify simple racks in group-theoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks containing properly the existing ones. We introduce a “Fourier transform” on racks of certain type; finally, we compute some new examples of finite-dimensional Nichols algebras. 相似文献
7.
Sarah Scherotzke 《Journal of Algebra》2008,319(7):2889-2912
We classify pointed rank one Hopf algebras over fields of prime characteristic which are generated as algebras by the first term of the coradical filtration. We obtain three types of Hopf algebras presented by generators and relations. For Hopf algebras with semi-simple coradical only the first and second type appears. We determine the indecomposable projective modules for certain classes of pointed rank one Hopf algebras. 相似文献
8.
9.
Mitja Mastnak 《Journal of Pure and Applied Algebra》2009,213(7):1399-1417
We give explicit formulas for maps in a long exact sequence connecting bialgebra cohomology to Hochschild cohomology. We give a sufficient condition for the connecting homomorphism to be surjective. We apply these results to compute all bialgebra two-cocycles of certain Radford biproducts (bosonizations). These two-cocycles are precisely those associated to the finite dimensional pointed Hopf algebras in the recent classification of Andruskiewitsch and Schneider, in an interpretation of these Hopf algebras as graded bialgebra deformations of Radford biproducts. 相似文献
10.
I. Heckenberger 《Journal of Algebra》2010,323(8):2130-2182
In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by Lusztig led to the first general construction of PBW bases of these algebras. Also, they have important applications to the representation theory of these and related algebras. In the present paper the Drinfel'd double for a class of graded Hopf algebras is investigated. Various quantum algebras, including small quantum groups and multiparameter quantizations of semisimple Lie algebras and of Lie superalgebras, are covered by the given definition. For these Drinfel'd doubles Lusztig maps are defined. It is shown that these maps induce isomorphisms between doubles of bosonizations of Nichols algebras of diagonal type. Further, the obtained isomorphisms satisfy Coxeter type relations in a generalized sense. As an application, the Lusztig isomorphisms are used to give a characterization of Nichols algebras of diagonal type with finite root system. 相似文献
11.
Representations of quantum algebras 总被引:3,自引:0,他引:3
12.
Inventiones mathematicae - 相似文献
13.
We study the representations of two types of pointed Hopf algebras: restricted two-parameter quantum groups, and the Drinfel'd doubles of rank one pointed Hopf algebras of nilpotent type. We study, in particular, under what conditions a simple module can be factored as the tensor product of a one-dimensional module with a module that is naturally a module for the quotient by central group-like elements. For restricted two-parameter quantum groups, given θ a primitive ℓth root of unity, the factorization of simple -modules is possible, if and only if gcd((y−z)n,ℓ)=1. For rank one pointed Hopf algebras, given the data , the factorization of simple -modules is possible if and only if |χ(a)| is odd and |χ(a)|=|a|=|χ|. Under this condition, the tensor product of two simple -modules is completely reducible, if and only if the sum of their dimensions is less than or equal to |χ(a)|+1. 相似文献
14.
Piotr Grzeszczuk Malgorzata Hryniewicka 《Proceedings of the American Mathematical Society》2007,135(8):2381-2389
Let be an -module algebra, where is a pointed Hopf algebra acting on finitely of dimension . Suppose that for every nonzero -stable left ideal of . It is proved that if satisfies a polynomial identity of degree , then satisfies a polynomial identity of degree provided at least one of the following additional conditions is fulfilled:
- is semiprime and is almost central in ,
- is reduced.
15.
Nicolás Andruskiewitsch Fernando Fantino Shouchuan Zhang 《manuscripta mathematica》2009,128(3):359-371
Let G be the symmetric group . It is an important open problem whether the dimension of the Nichols algebra is finite when is the class of the transpositions and ρ is the sign representation, with m ≥ 6. In the present paper, we discard most of the other conjugacy classes showing that very few pairs might give rise to finite-dimensional Nichols algebras.
This work was partially supported by CONICET, ANPCyT and Secyt (UNC). 相似文献
16.
N. Andruskiewitsch F. Fantino M. Graña L. Vendramin 《Annali di Matematica Pura ed Applicata》2011,190(2):225-245
It is shown that Nichols algebras over alternating groups
\mathbb Am{\mathbb A_m} (m ≥ 5) are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes
isomorphic to
\mathbb Am{\mathbb A_m} is isomorphic to the group algebra. In a similar fashion, it is shown that the Nichols algebras over the symmetric groups
\mathbb Sm{\mathbb S_m} are all infinite-dimensional, except maybe those related to the transpositions considered in Fomin and Kirillov (Progr Math
172:146–182, 1999), and the class of type (2, 3) in
\mathbb S5{\mathbb S_5}. We also show that any simple rack X arising from a symmetric group, with the exception of a small list, collapse, in the sense that the Nichols algebra
\mathfrak B(X, q){\mathfrak B(X, \bf q)} is infinite dimensional, q an arbitrary cocycle. 相似文献
17.
18.
Teodor Banica Julien Bichon Jean-Marc Schlenker 《Journal of Functional Analysis》2009,257(9):2864-2910
We develop a combinatorial approach to the quantum permutation algebras, as Hopf images of representations of type π:As(n)→B(H). We discuss several general problems, including the commutativity and cocommutativity ones, the existence of tensor product or free wreath product decompositions, and the Tannakian aspects of the construction. The main motivation comes from the quantum invariants of the complex Hadamard matrices: we show here that, under suitable regularity assumptions, the computations can be performed up to n=6. 相似文献
19.
We study the connections between one-sided Hopf algebras and one-sided quantum quasigroups, tracking the four possible invertibility conditions for the left and right composite morphisms that combine comultiplications and multiplications in these structures. The genuinely one-sided structures exhibit precisely two of the invertibilities, while it emerges that imposing one more condition often entails the validity of all four. A main result shows that under appropriate conditions, just one of the invertibility conditions is su?cient for the existence of a one-sided antipode. In the left Hopf algebra which is a variant of the quantum special linear group of two-dimensional matrices, it is shown explicitly that the right composite is not injective, and the left composite is not surjective. 相似文献
20.
Basic Hopf algebras and quantum groups 总被引:10,自引:0,他引:10
This paper investigates the structure of basic finite dimensional Hopf algebras H over an algebraically closed field k. The algebra H is basic provided H modulo its Jacobson radical is a product of the field k. In this case H is isomorphic to a path algebra given by a finite quiver with relations. Necessary conditions on the quiver and on the coalgebra
structure are found. In particular, it is shown that only the quivers given in terms of a finite group G and sequence of elements of G in the following way can occur. The quiver has vertices and arrows , where the set is closed under conjugation with elements in G and for each g in G, the sequences W and are the same up to a permutation. We show how is a kG-bimodule and study properties of the left and right actions of G on the path algebra. Furthermore, it is shown that the conditions we find can be used to give the path algebras themselves a Hopf algebra structure (for an arbitrary field k). The results are also translated into the language of coverings. Finally, a new class of finite dimensional basic Hopf algebras
are constructed over a not necessarily algebraically closed field, most of which are quantum groups. The construction is not
characteristic free. All the quivers , where the elements of W generates an abelian subgroup of G, are shown to occur for finite dimensional Hopf algebras. The existence of such algebras is shown by explicit construction.
For closely related results of Cibils and Rosso see [Ci-R].
Received August 15, 1994; in final form May 16, 1997 相似文献