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Weak multiplier Hopf algebras. Preliminaries, motivation and basic examples

100%

Van Daele A., Wang S.

Banach Center Publications

2012

tom 98

nr 1

367-415

Let G be a finite group. Consider the algebra A of all complex functions on G (with pointwise product). Define a coproduct Δ on A by Δ(f)(p,q) = f(pq) where f ∈ A and p,q ∈ G. Then (A,Δ) is a Hopf algebra. If G is only a groupoid, so that the product of two elements is not always defined, one still can consider A and define Δ(f)(p,q) as above when pq is defined. If we let Δ(f)(p,q) = 0 otherwise, we still get a coproduct on A, but Δ(1) will no longer be the identity in A ⊗ A. The pair (A,Δ) is not a Hopf algebra but a weak Hopf algebra. If G is a group, but no longer finite, one takes for A the algebra of functions with finite support. Then A has no identity and (A,Δ) is not a Hopf algebra but a multiplier Hopf algebra. Finally, if G is a groupoid, but not necessarily finite, the standard construction above will give what we call in this paper a weak multiplier Hopf algebra. Indeed, this paper is devoted to the development of this 'missing link': weak multiplier Hopf algebras. We denote a great part of this paper to the motivation of our notion and to explaining where the various assumptions come from. The goal is to obtain a good definition of a weak multiplier Hopf algebra. Throughout the paper, we consider the basic examples and use them, as far as possible, to illustrate what we do. In particular, we think of the finite-dimensional weak Hopf algebras. On the other hand however, we are also inspired by the far more complicated existing analytical theory. In our forthcoming papers on the subject, we develop the theory further. In [VD-W2] we start from the definition of a weak multiplier Hopf algebra as it is obtained and motivated in this paper and we prove the main properties. In [VD-W3] we continue with the study of the source and target algebras and the corresponding source and target maps. In that paper, we also give more examples. Finally, in [VD-W4] we study integrals on weak multiplier Hopf algebras and duality. Other aspects of the theory will be considered later.

Multiplier Hopf algebroids arising from weak multiplier Hopf algebras

100%

Timmermann T., Van Daele A.

Banach Center Publications

2015

tom 106

nr 1

73-110

It is well-known that any weak Hopf algebra gives rise to a Hopf algebroid. Moreover it is possible to characterize those Hopf algebroids that arise in this way. Recently, the notion of a weak Hopf algebra has been extended to the case of algebras without identity. This led to the theory of weak multiplier Hopf algebras. Similarly also the theory of Hopf algebroids was recently developed for algebras without identity. They are called multiplier Hopf algebroids. Then it is quite natural to investigate the expected link between weak multiplier Hopf algebras and multiplier Hopf algebroids. This relation has been considered already in the original paper on multiplier Hopf algebroids. In this note, we investigate the connection further. First we show that any regular weak multiplier Hopf algebra gives rise, in a natural way, to a regular multiplier Hopf algebroid. Secondly we give a characterization, mainly in terms of the base algebra, for a regular multiplier Hopf algebroid to have an underlying weak multiplier Hopf algebra. We illustrate this result with some examples. In particular, we give examples of multiplier Hopf algebroids that do not arise from a weak multiplier Hopf algebra.

Ograniczanie wyników

2 Banach Center Publications

2 Van Daele A.

1 Timmermann T.

1 Wang S.

1 2015

1 2012

Wyniki wyszukiwania

Weak multiplier Hopf algebras. Preliminaries, motivation and basic examples

Multiplier Hopf algebroids arising from weak multiplier Hopf algebras