Weak multiplier Hopf algebras I. The main theory

Abstract

Abstract 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 T 1 and T 2 defined from A ⊗ A to M(A ⊗ A) by T 1(a ⊗ b) = Δ(a)(1 ⊗ b) and T 2(a ⊗ 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 ⊗ A) = E(A ⊗ A) and (A ⊗ 1)Δ(A) = (A ⊗ A)E. This element is unique if it exists. Then it is possible to extend the coproduct in a unique way to a homomorphism Δ˜: M(A) → M(A ⊗ A) such that Δ˜(1) = E. In the case of a multiplier Hopf algebra we have E = 1 ⊗ 1 but this is no longer assumed for weak multiplier Hopf algebras. 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 ⊗ 1) = (E ⊗ 1)(1 ⊗ E). Finally, the last condition determines the kernels of the canonical maps T 1 and T 2 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. We will give more and interesting examples of weak multiplier Hopf algebras in parts II and III of this series of papers.