1,721,023 research outputs found
On Primitively Generated Braided Bialgebras
Full text linkThe main aim of this paper is to investigate the structure of primitively generated braided bialgebras A with respect to the braided vector space P consisting of their primitive elements. When the Nichols algebra of P is obtained dividing out the tensor algebra T(P) by the two-sided ideal generated by its primitive elements of degree at least two, we show that A can be recovered as a sort of universal enveloping algebra of P. One of the main applications of our
construction is the description, in terms of universal enveloping algebras, of braided bialgebras whose associated graded coalgebra is a quadratic algebra
Universal Enveloping Algebras of PBW Type
No full textWe continue our investigation of the general notion of universal enveloping algebra introduced in [A. Ardizzoni, "A Milnor-Moore Type Theorem for Primitively Generated Braided Bialgebras", J. Algebra 327 (2011), no. 1, 337--365]. Namely we study when such an algebra is of PBW type, meaning that a suitable PBW type
theorem holds. We discuss the problem of finding a basis for a universal enveloping algebra of PBW type: As an application we recover the PBW basis both of an ordinary universal enveloping algebra and of a restricted enveloping algebra. We prove that a universal enveloping algebra is of PBW type if and only if it is cosymmetric. We characterize braided bialgebra liftings
of Nichols algebras as universal enveloping algebras of PBW type
The Category of Modules over a Monoidal Category: Abelian or not?
No full textLet A be an algebra in an abelian monoidal category M. We prove that the category of left A-modules is abelian, whenever A is right coflat
A Milnor–Moore type theorem for primitively generated braided bialgebras
No full textAbstractA braided bialgebra is called primitively generated if it is generated as an algebra by its space of primitive elements. We prove that any primitively generated braided bialgebra is isomorphic to the universal enveloping algebra of its infinitesimal braided Lie algebra, notions hereby introduced. This result can be regarded as a Milnor–Moore type theorem for primitively generated braided bialgebras and leads to the introduction of a concept of braided Lie algebra for an arbitrary braided vector space
Braided Bialgebras of Type One
No full textBraided bialgebras of type one in abelian braided monoidal categories are characterized as braided graded bialgebras which are strongly -graded both as an algebra and as a coalgebra
Bosonization for dual quasi-bialgebras and preantipode
No full textTo every dual quasi-bialgebra H and every bialgebra R in the category of Yetter-Drinfeld modules over H, one can associate a dual quasi-bialgebra, called bosonization. In this paper, using the fundamental theorem, we characterize as bosonizations the dual quasi-bialgebras with a projection onto a dual quasi-bialgebra with a preantipode. As an application we investigate the structure of the graded coalgebra gr. A associated to a dual quasi-bialgebra A with the dual Chevalley property (e.g. A is pointed)
Associated Graded Algebras and Coalgebras
No full textWe investigate the notion of associated graded coalgebra (algebra) of a bialgebra with respect to a subbialgebra (quotient bialgebra) and characterize those which are bialgebras of type one in the framework of abelian braided monoidal categories
Preantipodes for Dual Quasi-Bialgebras
No full textIt is known that a dual quasi-bialgebra with antipode H, i.e. a dual quasi-Hopf algebra, fulfils a fundamental theorem for right dual quasi-Hopf H-bicomodules. The converse in general is not true. We prove that, for a dual quasi-bialgebra H, the structure theorem amounts to the existence of a suitable map S : H --> H that we call a preantipode of H
Semiseparable Functors and Conditions up to Retracts
Full text linkIn a previous paper we introduced the concept of semiseparable functor. Here we continue our study of these functors in connection with idempotent (Cauchy) completion. To this aim, we introduce and investigate the notions of (co)reflection and bireflection up to retracts. We show that the (co)comparison functor attached to an adjunction whose associated (co)monad is separable is a coreflection (reflection) up to retracts. This fact allows us to prove that a right (left) adjoint functor is semiseparable if and only if the associated (co)monad is separable and the (co)comparison functor is a bireflection up to retracts, extending a characterization pursued by X.-W. Chen in the separable case. Finally, we provide a semi-analogue of a result obtained by P. Balmer in the framework of pre-triangulated categories
Small Bialgebras with a Projection: Applications
No full textIn this paper we continue the investigation started in \cite{A.M.St.-Small}, dealing with bialgebras with an -bilinear coalgebra projection over an arbitrary subbialgebra with antipode. These bialgebras can be described as deformed bosonizations of a pre-bialgebra by with a cocycle . Here we describe the behavior of in the case when is f.d. and thin i.e. it is connected with one dimensional space of primitive elements. This is used to analyze the arithmetic properties of . Meaningful results are obtained when is cosemisimple. By means of Ore extension construction, we provide some examples of atypical situations
(e.g. the multiplication of is not -colinear or is non-trivial)
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