Modified weak multiplier Hopf algebras

Abstract

Let $(A,Δ)$ be a regular weak multiplier Hopf algebra. Denote by $E$ the canonical idempotent of $(A,Δ)$ and by $B$ the image of the source map. Recall that $B$ is a non-degenerate algebra, sitting nicely in the multiplier algebra $M(A)$ of $A$ so that also $M(B)$ can be viewed as a subalgebra of $M(A)$. Assume that $u,v$ are invertible elements in $M(B)$ so that $E(vu\otimes 1)E=E$. This last condition is obviously fulfilled if $u$ and $v$ are each other inverses, but there are also other cases. Now modify $Δ$ and define $Δ'(a)=(u\otimes 1)Δ(a)(v\otimes 1)$ for all $a \in A$. We show in this paper that $(A,Δ')$ is again a regular weak multiplier Hopf algebra and we obtain formulas for the various data of $(A,Δ')$ in terms of the data associated with the original pair $(A,Δ)$. In the case of a finite-dimensional weak Hopf algebra, the above deformation is a special case of the twists as studied by Nikshych and Vainerman. It is known that any regular weak multiplier Hopf algebra gives rise in a natural way to a regular multiplier Hopf algebroid. This result applies to both the original weak multiplier Hopf algebra $(A,Δ)$ and the modified version $(A,Δ')$. However, the same method can be used to associate another regular multiplier Hopf algebroid to the triple $(A,Δ,Δ')$. This turns out to give an example of a regular multiplier Hopf algebroid that does not arise from a regular weak multiplier Hopf algebra although the base algebra is separable Frobenius.