Multiplier Hopf Algebras in Categories and the Biproduct Construction

Abstract

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.

Heisenberg double as braided commutative Yetter–Drinfel’d module algebra over Drinfel’d double in multiplier Hopf algebra case

“…, then equation (3.5) defines a twisted tensor product algebra in [1]. By Proposition 2.3 in [4], we can easily get X ∝ Y is an A-module and A-comodule algebra satisfying the compatibility condition of Yetter-Drinfel'd module, i.e., a Yetter-Drinfel'd A-module algebra.…”

“…For any two (left-left) Yetter-Drinfel'd A-module algebras X and Y , their braided product (shown in the proof of Theorem 4.1 in[4]) X ∝ Y is defined as follows…”

Characterization of automorphisms of Hom-biproducts

“…In the theory of the classical Hopf algebras, Radford's biproducts are very important Hopf algebras, which play a central role in the theory of classification of pointed Hopf algebra [1] and account for many examples of semisimple Hopf algebra. There has been many generalizations of Radford's biproducts such as [2] for quasi-Hopf algebra case, [9] for multiplier Hopf algebra case and [13] for monoidal Hom-Hopf algebra.…”