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13. 8 i). 7]. 14. G/ ! 4. 1]. 1/ D 0g [1, p. 198]. 6. Chapter 2 Multiplier Hopf algebras and their duality A multiplier Hopf algebra is a non-unital generalization of a Hopf algebra, where the target of the comultiplication is no longer the twofold tensor product of the underlying algebra, but an enlarged multiplier algebra. 1. 4. This duality is based on left- and right-invariant linear functionals called integrals, which are analogues of the Haar measures of a group. 3. The theory of multiplier Hopf algebras was developed by Van Daele; all results presented in this chapter are taken from the articles [174], [177].

13. 8 i). 7]. 14. G/ ! 4. 1]. 1/ D 0g [1, p. 198]. 6. Chapter 2 Multiplier Hopf algebras and their duality A multiplier Hopf algebra is a non-unital generalization of a Hopf algebra, where the target of the comultiplication is no longer the twofold tensor product of the underlying algebra, but an enlarged multiplier algebra. 1. 4. This duality is based on left- and right-invariant linear functionals called integrals, which are analogues of the Haar measures of a group. 3. The theory of multiplier Hopf algebras was developed by Van Daele; all results presented in this chapter are taken from the articles [174], [177].

A0 , b 7! jb/; are isomorphisms of Hopf algebras. B; B / be Hopf -algebras and . j / W A B ! k a dual pairing of Hopf algebras. 4. b/ / for all a 2 A and b 2 B. b/ / for all a 2 A; b 2 B, and the reverse implication follows similarly. Thus, one of these conditions may be omitted in the definition of a dual pairing of Hopf -algebras. Let us consider several examples of dual pairings. 6. C// ! C/. C/ ! C/ is identified with the space of n n-matrices with vanishing trace. 14. 7. Let G be a compact connected Lie group with Lie algebra g.