Abstract
Let (A,Δ) be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra A, with or without identity, and a coproduct Δ:A⟶M(A⊗A), satisfying certain properties. In this paper, we continue the study of these objects and construct new examples. A symmetric pair of the source and target maps εs and εt are studied, and their symmetric pair of images, the source algebra and the target algebra εs(A) and εt(A), are also investigated. We show that the canonical idempotent E (which is eventually Δ(1)) belongs to the multiplier algebra M(B⊗C), where (B=εs(A), C=εt(A)) is the symmetric pair of source algebra and target algebra, and also that E is a separability idempotent (as studied). If the weak multiplier Hopf algebra is regular, then also E is a regular separability idempotent. We also see how, for any weak multiplier Hopf algebra (A,Δ), it is possible to make C⊗B (with B and C as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the ’Hopf algebra part’ of the original weak multiplier Hopf algebra and only remembers symmetric pair of the source and target algebras. It is in turn generalized to the case of any symmetric pair of non-degenerate algebras B and C with a separability idempotent E∈M(B⊗C). We get another example using this theory associated to any discrete quantum group. Finally, we also consider the well-known ’quantization’ of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras introduced).
Highlights
For an associative algebra A with a non-degenerate product, we say that λ ∈ Hom( A, A) is a left multiplier of A if λ( ab) = λ( a)b for all a, b ∈ A
We show that the canonical idempotent E belongs to the multiplier algebra M( B ⊗ C ), where ( B, C ) is the symmetric pair of source algebra and target algebra, and that E is a separability idempotent
In the regular case, we show that the multiplier algebras M (ε s ( A)) and M (ε t ( A)) of the images ε s ( A) and ε t ( A) of the source and target maps can be nicely characterized as certain subalgebras of the multiplier algebra M( A)
Summary
(see [7]), we defined weak multiplier Hopf algebras, by extending the class of weak Hopf algebras It is a pair of a non-degenerate algebra A, with or without identity, and a coproduct ∆ : A −→. If the algebra has an identity, but the coproduct is not unital, we have a weak Hopf algebra (see [5,11]). ∆ to be non-degenerate and so we work with a genuine weak multiplier Hopf algebra (see [6,7,12,13]). The coproduct map ∆ is not necessarily non-degenerate, while the existence of a certain canonical idempotent element E ∈ M( A ⊗ A) is assumed, which coincides with ∆(1) in the unital case.
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