# Weak Multiplier Hopf Algebras. The main theory

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Weak Multiplier Hopf Algebras. The main theory**

A weak multiplier Hopf algebra is a pair A,\Delta of a non-degenerate idempotent algebra A and a coproduct $\Delta$ on A. The coproduct is a coassociative homomorphism from A to the multiplier algebra MA\otimes 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\otimes A to MA\otimes A by T 1a\otimes b=\Deltaa1\otimes b and T 2a\ot b=a\otimes 1\Deltab. The first condition is about the ranges of these maps. It is assumed that there exists an idempotent element E\in MA\otimes A such that \DeltaA1\ot A=EA\ot A and A\otimes 1\DeltaA=A\otimes AE. The second condition determines the behavior of the coproduct on the legs of E. We require \Delta\otimes \iotaE=\iota\otimes\DeltaE=1\otimes EE\ot 1=E\otimes 11\otimes E where $\iota$ is the identity map and where $\Delta\otimes \iota$ and $\iota\otimes\Delta$ are extensions to the multipier algebra MA\otimes A. 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.

Author: **Alfons Van Daele; Shuanhong Wang**

Source: https://archive.org/