Twisted tensor product of multiplier Hopf (∗-)algebras

Delvaux, Lydia

doi:10.1016/s0021-8693(03)00467-8

Cited by 15 publications

(40 citation statements)

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“…To see that the expressions for S # ((a#b )(x#y)) and S # (x#y)S # (a#b ) are equal, we use that S A is a braided anti-homomorphism, in the sense of Lemma 2.6. To finish the proof, we observe that for all a, x in A and b , y in B, we have # (a#b )((x#y) ⊗ (1#1)) = (ι ⊗ T ⊗ ι)( (a)((b (1) x)⊗1) ⊗ b (2) y ⊗ b (3) ) in (A#B) ⊗ (A#B).…”

Section: Radford's Biproduct For Multiplier Hopf Algebrasmentioning

confidence: 94%

“…This investigation for multiplier Hopf algebras is already done in special cases. More precisely, in [3], we take T equal to the usual flip map and in [4], we take R equal to the usual flip map. For a ∈ A and b ∈ B, the coproduct # (a#b ) will be introduced (as usual) via two linear maps T # 1 and T # 2 on (A#B) ⊗ (A#B).…”

Section: The Simultaneous Twisting Of Multiplications and Comultiplicmentioning

confidence: 99%

“…Therefore the results can be applied in several concrete realizations for R and T. However, to find appropriate conditions so that # is an algebra homomorphism on the algebra A#B, one has to use more explicit expressions for the twist maps R and T. In [3,Theorem 3.7], we have investigated the situation for an arbitrary twist map R, satisfying the conditions (I), but we have restricted T to be the trivial flip map. In [4,Theorem 3.8], we have assumed that T is an isomorphism satisfying the conditions (II), but R equals the trivial flip map.…”

Section: Remark 38mentioning

confidence: 99%

“…Next, consider the candidate for the antipode S # = R • (S B ⊗ S A ) • T, see Remarks 3.8 (3). Observe that S # is a linear bijection on A#B.…”

Section: Radford's Biproduct For Multiplier Hopf Algebrasmentioning

confidence: 99%

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Multiplier Hopf Algebras in Categories and the Biproduct Construction

Delvaux

2007

Algebr Represent Theor

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Let B be a regular multiplier Hopf algebra. Let A be an algebra with a non-degenerate multiplication such that A is a left B-module algebra and a left B-comodule algebra. By the use of the left action and the left coaction of B on A, we determine when a comultiplication on A makes A into a "B-admissible regular multiplier Hopf algebra." If A is a B-admissible regular multiplier Hopf algebra, we prove that the smash product A#B is again a regular multiplier Hopf algebra. The comultiplication on A#B is a cotwisting (induced by the left coaction of B on A) of the given comultiplications on A and B. When we restrict to the framework of ordinary Hopf algebra theory, we recover Majid's braided interpretation of Radford's biproduct.

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Section: Radford's Biproduct For Multiplier Hopf Algebrasmentioning

confidence: 94%

Section: The Simultaneous Twisting Of Multiplications and Comultiplicmentioning

confidence: 99%