Multiplication alteration by two-cocycles - the quantum version

Doi, Yukio; Takeuch, Mitsuhiro

doi:10.1080/00927879408825158

Cited by 128 publications

(128 citation statements)

References 11 publications

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“…Let A and B be two Hopf algebras, σ : A ⊗ B −→ k an invertible skew pair, then we have Doi-Takeuchi double A σ B [4]. In [2] Beattie and Bulacu gave the necessary and sufficient conditions for A σ B to be a bialgebra with a projection generalizing the Majid's result in [8].…”

Section: Introductionmentioning

confidence: 98%

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A method of constructing braided Hopf algebras

Wang

2010

Filomat

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Let A and B be two Hopf algebras and R ∈ Hom(B ⊗ A, A ⊗ B), the twisted tensor product Hopf algebra A# R B was introduced by S. Caenepeel et al in [3] and further studied in our recent work [6]. In this paper we give the necessary and sufficient conditions for A#RB to be a Hopf algebra with a projection. Furthermore, a braided Hopf algebra A is constructed by twisting the multiplication of A through a (γ, R)-pair (A, B). Finally we give a method to construct Radford's biproduct directly by defining the module action and comodule action from the twisted tensor biproduct.

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Section: Introductionmentioning

confidence: 98%

“…Let A and B be two Hopf algebras, σ : A ⊗ B −→ k an invertible skew pair, then we have Doi-Takeuchi double A σ B [4]. If we define the linear map R :…”

Section: The Twisted Tensor Product Hopf Algebrasmentioning

confidence: 99%

A method of constructing braided Hopf algebras

Wang

2010

Filomat

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“…If σ is a convolution invertible left 2-cocycle, then σ −1 is a right 2-cocycle. It is well-known (see for instance [13], [3] or [4]) that the double twist σ B σ −1 with the same coproduct of B is again a bialgebra and if B is a Hopf algebra, then σ B σ −1 is also a Hopf algebra with antipode…”

Section: An Equivalence Of Categoriesmentioning

confidence: 99%

Some Isomorphisms for the Brauer Groups of a Hopf Algebra

Carnovale¹

2001

Communications in Algebra

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Using equivalences of categories we provide isomorphisms between the Brauer groups of different Hopf algebras. As an example, we show that when k is a field of characteristic different from 2 the Brauer groups BC(k, H 4 , r t ) for every dual quasitriangular structure r t on Sweedler's Hopf algebra H 4 are all isomorphic to the direct sum of (k, +) and the Brauer-Wall group of k. We provide an isomorphism between the Brauer group of a Hopf algebra H and the Brauer group of the dual Hopf algebra H * generalizing a result of Tilborghs. Finally we relate the Brauer groups of H and of its opposite and co-opposite Hopf algebras.

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“…In general, there is no analogue to the isomorphism R * cop (l) ∼ = R (r) , but the coquasitriangular map σ induces a skew pairing on H r ⊗ H l . Although the classical Drinfeld double is not available in this setting, there exists a generalized quantum double, in the sense of Majid [13] or Doi and Takeuchi [5], for Hopf algebras in duality, so that the skew pairing between H r and H l gives a generalized quantum double, denoted D(H r , H l ). In general, D(H r , H l ) has no (co)quasitriangular structure (see Proposition 2.8 and Support for the first author's research, and partial support for the second author, came from an NSERC Discovery Grant.…”

Section: Introductionmentioning

confidence: 99%

“…For U, V bialgebras such that there is an invertible pairing from U ⊗ V to k, we show that there is a projection from D(U cop , V ) to U cop covering the natural inclusion if and only if there is a bialgebra morphism γ : V → U cop satisfying a relation analogous to the coquasitriangularity condition for a skew pairing. (See also [5] and [17].) Now if U, V are Hopf algebras with an invertible pairing and γ exists as above, then V has a Hopf algebra structure in the category of left Yetter-Drinfeld modules over U cop .…”

Section: Introductionmentioning

confidence: 99%

Braided Hopf Algebras Obtained from Coquasitriangular Hopf Algebras

Beattie

Bulacu

2008

Commun. Math. Phys.

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Abstract. Let (H, σ) be a coquasitriangular Hopf algebra, not necessarily finite dimensional. Following methods of Doi and Takeuchi, which parallel the constructions of Radford in the case of finite dimensional quasitriangular Hopf algebras, we define Hσ, a sub-Hopf algebra of H 0 , the finite dual of H. Using the generalized quantum double construction and the theory of Hopf algebras with a projection, we associate to H a braided Hopf algebra structure in the category of Yetter-Drinfeld modules over H cop σ . Specializing to H = SLq(N ), we obtain explicit formulas which endow SLq(N ) with a braided Hopf algebra structure within the category of left Yetter-Drinfeld modules over U ext q (sl N ) cop .

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Multiplication alteration by two-cocycles - the quantum version

Cited by 128 publications

References 11 publications

A method of constructing braided Hopf algebras

A method of constructing braided Hopf algebras

Some Isomorphisms for the Brauer Groups of a Hopf Algebra

Braided Hopf Algebras Obtained from Coquasitriangular Hopf Algebras

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