Multiplier Bi- And Hopf Algebras

Abstract

Abstract. We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele in [11] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly non-unital, idempotent, non-degenerate, k-projective) algebra over a commutative ring k is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of k-modules, into a diagram of strict monoidal forgetful functors.

Globalization of Twisted Partial Hopf Actions

“…The map ⊲ : H ⊗ M (B) → M (B) which appears in the third identity requires a more detailed explanation. First, the multiplier algebra M (B) can be viewed as a pull back [32] M (B)…”

Weak multiplier bialgebras

“…If A and B are idempotent and non-degenerate algebras, then so is A ⊗ B with the factorwise multiplication, see e.g. [6,Lemma 1.11]. Let A be a non-unital algebra with a non-degenerate multiplication.…”

“…We show that any regular weak multiplier Hopf algebra obeys these axioms and so does any weak bialgebra (with a unit). By generalizing to the multiplier setting several equivalent properties that distinguish bialgebras among weak bialgebras, we also propose a notion of multiplier bialgebra (which is, however, different from both notions in [6] and [9] occurring under the same name). In Section 3 and Section 4 we study some distinguished subalgebras of the multiplier algebra of a weak multiplier bialgebra.…”

Hopf Algebras—Variant Notions and Reconstruction Theorems

“…The full categorical description of multiplier Hopf algebras is not settled yet. A first attempt was made in [22]. In this paper a reconstruction theorem for multiplier bialgebras was given.…”