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ABSTRACT: We study some examples of braided categories and quasitriangular Hopf
algebras and decide which of them is pseudosymmetric, respectively
pseudotriangular. We show also that there exists a universal pseudosymmetric
braided category.
12/2011;
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Florin Panaite
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ABSTRACT: We show that some more results from the literature are particular cases of the so-called "invariance under twisting" for twisted tensor products of algebras, for instance a result of Beattie-Chen-Zhang that implies the Blattner-Montgomery duality theorem. Comment: 8 pages
11/2010;
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Florin Panaite
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ABSTRACT: We prove a result of the type ''invariance under twisting'' for Brzezinski's crossed products, as a common generalization of the invariance under twisting for twisted tensor products of algebras and the invariance under twisting for quasi-Hopf smash products. It turns out that this result contains also as a particular case the equivalence of crossed products by a coalgebra (due to Brzezinski). Comment: 8 pages
07/2010;
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ABSTRACT: Motivated by the recently introduced concept of a pseudosymmetric braided monoidal category, we define the pseudosymmetric group PS_n, as the quotient of the braid group B_n by the relations \sigma_i\sigma_{i+1}^{-1}\sigma_i=\sigma _{i+1}\sigma_i^{-1}\sigma_{i+1}, with 1\leq i\leq n-2. It turns out that PS_n is isomorphic to the quotient of B_n by the commutator subgroup [P_n, P_n] of the pure braid group P_n (which amounts to saying that [P_n, P_n] coincides with the normal subgroup of B_n generated by the elements [\sigma_i^2, \sigma_{i+1}^2], with 1\leq i\leq n-2), and that PS_n is a linear group.
03/2009;
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ABSTRACT: Let H be a Hopf algebra with bijective antipode, let \alpha, \beta be two Hopf algebra automorphisms of H and M a finite dimensional (\alpha, \beta )-Yetter-Drinfeld module. We prove that End(M) endowed with certain structures becomes an H-Azumaya algebra, and the set of H-Azumaya algebras of this type is a subgroup of BQ(k, H), the Brauer group of H.
06/2008;
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ABSTRACT: Let H be a bialgebra and D an H-bimodule algebra H-bicomodule coalgebra. We find sufficient conditions on D for the L-R-smash product algebra and coalgebra structures on D\otimes H to form a bialgebra (in this case we say that (H, D) is an L-R-admissible pair), called L-R-smash biproduct. The Radford biproduct is a particular case, and so is, up to isomorphism, a double biproduct with trivial pairing. We construct a prebraided monoidal category {\cal LR}(H), whose objects are H-bimodules H-bicomodules M endowed with left-left and right-right Yetter-Drinfeld module as well as left-right and right-left Long module structures over H, with the property that, if (H, D) is an L-R-admissible pair, then D is a bialgebra in {\cal LR}(H).
06/2008;
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ABSTRACT: We introduce the concept of pseudotwistor (with particular cases called twistor and braided twistor) for an algebra (A,μ,u) in a monoidal category, as a morphism satisfying a list of axioms ensuring that (A,μ○T,u) is also an algebra in the category. This concept provides a unifying framework for various deformed (or twisted) algebras from the literature, such as twisted tensor products of algebras, twisted bialgebras and algebras endowed with Fedosov products. Pseudotwistors appear also in other topics from the literature, e.g. Durdevich's braided quantum groups and ribbon algebras. We also focus on the effect of twistors on the universal first order differential calculus, as well as on lifting twistors to braided twistors on the algebra of universal differential forms.
Advances in Mathematics. 05/2006;
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ABSTRACT: We prove that a certain bialgebroid introduced recently by Kadison is isomorphic to a bialgebroid introduced earlier by Connes and Moscovici. At the level of total algebras, the isomorphism is a consequence of the general fact that an L-R-smash product over a Hopf algebra is isomorphic to a diagonal crossed product.
10/2005;
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ABSTRACT: If H is a quasi-Hopf algebra and B is a right H-comodule algebra such that there exists v:H\to B a morphism of right H-comodule algebras, we prove that there exists a left H-module algebra A such that B\simeq A# H. The main difference comparing to the Hopf case is that, from the multiplication of B, which is associative, we have to obtain the multiplication of A, which in general is not; for this we use a canonical projection E arising from the fact that B becomes a quasi-Hopf H-bimodule.
07/2005;
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ABSTRACT: In this paper we introduce generalizations of diagonal crossed products, two-sided crossed products and two-sided smash products, for a quasi-Hopf algebra H. The results we obtain may be applied to H^*-Hopf bimodules and generalized Yetter-Drinfeld modules. The generality of our situation entails that the "generating matrix" formalism cannot be used, forcing us to use a different approach. This pays off because as an application we obtain an easy conceptual proof of an important but very technical result of Hausser and Nill concerning iterated two-sided crossed products. Comment: 41 pages, no figures
06/2005;
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ABSTRACT: If H is a Hopf algebra with bijective antipode and \alpha, \beta \in Aut_{Hopf}(H), we introduce a category_H{\cal YD}^H(\alpha, \beta), generalizing both Yetter-Drinfeld and anti-Yetter-Drinfeld modules. We construct a braided T-category {\cal YD}(H) having all these categories as components, which if H is finite dimensional coincides with the representations of a certain quasitriangular T-coalgebra DT(H) that we construct. We also prove that if (\alpha, \beta) admits a so-called pair in involution, then_H{\cal YD}^H(\alpha, \beta) is isomorphic to the category of usual Yetter-Drinfeld modules_H{\cal YD}^H.
04/2005;
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Communications in Algebra. 01/2000; 28(2):585-600.
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Communications in Algebra 01/2000; 28(2):631-651. · 0.35 Impact Factor
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ABSTRACT: We apply to Hopf algebras a construction from graded rings, called the group ring of a graded ring. By using this construction we study the transfer of properties between certain categories of relative Hopf modules. As another application, we obtain a Maschke-type theorem for a Galois extension over a semisimple Hopf algebra.
Algebras and Representation 08/1999; 2(3):211-226. · 0.60 Impact Factor
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Communications in Algebra 01/1999; 27(10):4929-4942. · 0.35 Impact Factor
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ABSTRACT: A laycle is the categorical analogue of a lazy cocycle. Twines (introduced by Bruguières) and strong twines (as introduced by the authors) are laycles satisfying some extra conditions. If c is a braiding, the double braiding c2 is always a twine; we prove that it is a strong twine if and only if c satisfies a sort of modified braid relation (we call such cpseudosymmetric, as any symmetric braiding satisfies this relation). It is known that the category of Yetter–Drinfeld modules over a Hopf algebra H is symmetric if and only if H is trivial; we prove that the Yetter–Drinfeld category HYDH over a Hopf algebra H is pseudosymmetric if and only if H is commutative and cocommutative. We introduce as well the Hopf algebraic counterpart of pseudosymmetric braidings under the name pseudotriangular structures and prove that all quasitriangular structures on the 2n+1-dimensional pointed Hopf algebras E(n) are pseudotriangular. We observe that a laycle on a monoidal category induces a so-called pseudotwistor on every algebra in the category, and we obtain some general results (and give some examples) concerning pseudotwistors, inspired by the properties of laycles and twines.
Journal of Pure and Applied Algebra.
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ABSTRACT: We establish the so-called Invariance under Twisting Theorems, stating that twisted tensor products built up from different algebras that relate in certain ways are canonically iso-morphic. These results generalize (and provide categorical versions of) several well-known results in Hopf algebra theory, such as the invariance of the smash product under Drinfeld twisting, or the isomorphism between the Drinfeld double and a certain smash product for quasitriangular Hopf algebras.
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ABSTRACT: We study some classes of lazy cocycles, called pure (respectively neat), together with their categorical counterparts, entwined (respectively strongly entwined) monoidal categories.
Journal of Pure and Applied Algebra.
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ABSTRACT: We introduce a more general version of the so-called L-R-smash product and study its relations with other kinds of crossed products (two-sided smash and crossed product and diagonal crossed product). We also give an interpretation of the L-R-smash product in terms of an L-R-twisting datum.
Journal of Algebra.
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