A Schur Double Centralizer Theorem for Cotriangular Hopf Algebras and Generalized Lie Algebras

Fischman, Davida; Montgomery, Susan

doi:10.1006/jabr.1994.1246

Cited by 54 publications

(33 citation statements)

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“…On U, we can define the action of the group G by setting a g = λ(f, g)a, a ∈ Λ f , and consider a skew group ring H col = G * U, which has the structure of a Hopf algebra with comultiplication defined on G and Λ via ∆(g) = g ⊗ g and ∆(a) = a ⊗ 1 + f ⊗ a, a ∈ Λ f (and this is exactly the Radford biproduct U(Λ) ⋆ k[G]; see [15]). Now it is easy to see that H col is a character Hopf algebra, for which…”

Section: Definition 12mentioning

confidence: 99%

Algebra of Skew-Primitive Elements

Kharchenko

2015

Lecture Notes in Mathematics

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We study the various term operations on the set of skew primitive elements of Hopf algebras, generated by skew primitive semi-invariants of an Abelian group of grouplike elements. All 1-linear binary operations are described and trilinear and quadrilinear operations are given a detailed treatment. Necessary and sufficient conditions for the existence of multilinear operations are specified in terms of the property of particular noncommutative polynomials being linearly dependent and of one arithmetic condition. We dub the conjecture that this condition implies, in fact, the linear dependence of the polynomials in question and so is itself sufficient ( a proof of this conjecture see in "An existence condition for multilinear quantum operations," Journal of Algebra, 217, 1999, 188 − 228).

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Section: Definition 12mentioning

confidence: 99%

Algebra of Skew-Primitive Elements

Kharchenko

2015

Lecture Notes in Mathematics

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“…To denote the coactions on elements, we use the Sweedler-Heyneman convention, that is for any Hopf algebra H, the coproduct ∆ H (h) of an element h ∈ H is denoted by h 1 ⊗h 2 (notice the omission of the summation sign). If M is an H-comodule with coaction ̺ M,H , for any m ∈ M we set ̺ M,H (m) = m 0 ⊗m 1 . From now on, we implicitely mean that H is an H-module by µ H and an H-comodule by ∆ H .…”

Section: Conventions and Prerequisitesmentioning

confidence: 99%

“…Some definitions. Recall that, for any bialgebra E in YD H H , one may construct the Radford product H⋆E ( [5], [1]). It is a bialgebra and contains H as a sub-bialgebra and E as a subalgebra.…”

Section: Radford Productsmentioning

confidence: 99%

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Non-Abelian Hopf Cohomology of Radford Products

Nuss

Wambst

2011

Algebr Represent Theor

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Abstract. We study the non-abelian Hopf cohomology theory of Radford products with coefficients in a comodule algebra. We show that these sets can be expressed in terms of the non-abelian Hopf cohomology theory of each factor of the Radford product. We write down an exact sequence relating these objects. This allows to compute explicitly the non-abelian Hopf cohomology sets in large classes of examples. Key-words: non-abelian cohomology, Radford products, Hopf comodule algebra, cosimplicial nonabelian groups, Taft algebras. INTRODUCTION.The present article is devoted to the study of the non-abelian Hopf cohomology theory of Radford products. The cohomology theory for Hopf algebras with coefficients in comodule algebras was defined by the authors in [4]. It generalizes the non-abelian cohomology theory for groups ([6], [7]) and is closely related to descent cohomology [3]. Recall that the Radford product of two Hopf algebras (also called in the literature Radford biproducts or bosonization) was introduced in [5] as a generalisation of the semi-direct product of groups (see also Majid's interpretation in terms of braided categories [2]).The Radford product H⋆E, which is a Hopf algebra, is defined for a Hopf algebra H and any algebra E in the braided Yetter-Drinfeld category over H [11]. Our main result allows to describe the cohomology sets of H⋆E in terms of those of H and of a braided version of the cohomology sets of E. This decomposition has a counterpart in the non-abelian cohomology theory of a semi-direct product of groups. We now detail the organisation of the article.

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“…: coassociativity, counity property, and compatibility with the antipode) have the same formal description as in ordinary Hopf algebras. Once again, the abstraction of the representation theory of quasitriangular Hopf algebras provides us with a language in which the above description becomes much more compact: We simply say that A is a Hopf algebra in the braided monoidal category of CZ 2 -modules CZ 2 M or: a braided group where the braiding is given in equation (8). What we actually mean is that A is simultaneously an algebra, a coalgebra and a CZ 2 -module, while all the structure maps of A (multiplication, comultiplication, unity, counity and the antipode) are also CZ 2 -module maps and at the same time the comultiplication ∆ : A → A⊗A and the counit are algebra morphisms in the category CZ 2 M (see also [23,24] or [27] for a more detailed description).…”

Section: Super-hopf Structure Of Parabosons: a Braided Groupmentioning

confidence: 99%

Variants of bosonization in parabosonic algebra: the Hopf and super-Hopf structures in parabosonic algebra

Kanakoglou¹,

Daskaloyannis²

2008

J. Phys. A: Math. Theor.

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Parabosonic algebra in finite or infinite degrees of freedom is considered as a Z 2 -graded associative algebra, and is shown to be a Z 2 -graded (or: super) Hopf algebra. The super-Hopf algebraic structure of the parabosonic algebra is established directly without appealing to its relation to the osp(1/2n) Lie superalgebraic structure. The notion of super-Hopf algebra is equivalently described as a Hopf algebra in the braided monoidal category CZ 2 M. The bosonisation technique for switching a Hopf algebra in the braided monoidal category H M (where H is a quasitriangular Hopf algebra) into an ordinary Hopf algebra is reviewed. In this paper we prove that for the parabosonic algebra P B , beyond the application of the bosonisation technique to the original super-Hopf algebra, a bosonisation-like construction is also achieved using two operators, related to the parabosonic total number operator. Both techniques switch the same super-Hopf algebra P B to an ordinary Hopf algebra, producing thus two different variants of P B , with ordinary Hopf structure.

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A Schur Double Centralizer Theorem for Cotriangular Hopf Algebras and Generalized Lie Algebras

Cited by 54 publications

References 0 publications

Algebra of Skew-Primitive Elements

Algebra of Skew-Primitive Elements

Non-Abelian Hopf Cohomology of Radford Products

Variants of bosonization in parabosonic algebra: the Hopf and super-Hopf structures in parabosonic algebra

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