We introduce and study several notions of amenability for unitary corepresentations and $*$-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantum groups. As a background for this study, we investigate the associated tensor C$^{*}$-categories.

Keywords:

quantum group, amenability quantum group, amenability

MSC Classifications:

46L05, 46L65, 22D10, 22D25, 43A07, 43A65, 58B32 show english descriptions General theory of $C^*$-algebras
Quantizations, deformations
Unitary representations of locally compact groups
$C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx]
Means on groups, semigroups, etc.; amenable groups
Representations of groups, semigroups, etc. [See also 22A10, 22A20, 22Dxx, 22E45]
Geometry of quantum groups 46L05 - General theory of $C^*$-algebras
46L65 - Quantizations, deformations
22D10 - Unitary representations of locally compact groups
22D25 - $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx]
43A07 - Means on groups, semigroups, etc.; amenable groups
43A65 - Representations of groups, semigroups, etc. [See also 22A10, 22A20, 22Dxx, 22E45]
58B32 - Geometry of quantum groups

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