2014/07/02 by Alfons Van Daele, Van Daele, Alfons
Mathematics · #16T05 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:16T05
paper · pdf · doi:10.48550/arxiv.1407.0513
arxiv created 2014/07/02 · openalex publication_date 2014/07/02 · arxiv updated 2014/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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⊗ 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⊗ 1)Δ(a)(v⊗ 1) for all a ∈ 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.