2012/10/16 by Alfons Van Daele, Van Daele, Alfons, Shuanhong Wang +1 · 2 citations
Mathematics · #16T05 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1210.4395
openalex publication_date 2012/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A weak multiplier Hopf algebra is a pair (A,Δ) of a non-degenerate idempotent algebra A and a coproduct Δ on A. The coproduct is a coassociative homomorphism from A to the multiplier algebra M(A⊗ A) with some natural extra properties (like the existence of a counit). Further we impose extra but natural conditions on the ranges and the kernels of the canonical maps T1 and T2 defined from A⊗ A to M(A⊗ A) by T1(a⊗ b)=Δ(a)(1⊗ b) and T2(a\ot b)=(a⊗ 1)Δ(b). The first condition is about the ranges of these maps. It is assumed that there exists an idempotent element E∈ M(A⊗ A) such that Δ(A)(1\ot A)=E(A\ot A) and (A⊗ 1)Δ(A)=(A⊗ A)E. The second condition determines the behavior of the coproduct on the legs of E. We require (Δ⊗ ι)(E)=(ι⊗Δ)(E)=(1⊗ E)(E\ot 1)=(E⊗ 1)(1⊗ E) where ι is the identity map and where Δ⊗ ι and ι⊗Δ are extensions to the multipier algebra M(A⊗ A). Finally, the last condition determines the kernels of the canonical maps T1 and T2 in terms of this idempotent E by a very specific relation. From these conditions we develop the theory. In particular, we construct a unique antipode satisfying the expected properties and various other data. Special attention is given to the regular case (that is when the antipode is bijective) and the case of a *-algebra (where regularity is automatic). Weak Hopf algebras are special cases of such weak multiplier Hopf algebras. Conversely, if the underlying algebra of a (regular) weak multiplier Hopf algebra has an identity, it is a weak Hopf algebra. Also any groupoid, finite or not, yields two weak multiplier Hopf algebras in duality.