# Special Cases: Unimodular, Compact, Discrete and Finite-Dimensional Kac Algebras

## Abstract

Let \(\Bbbk = (M,\Gamma ,k,\varphi )\) be a Kac algebra, \(\hat \Bbbk = (\hat M,\hat \Gamma ,\hat k,\hat \varphi )\) the dual Kac algebra. We have seen that the modular operator \(\hat \Delta = {\Delta _{\hat \varphi }}\) is the RadonNikodym derivative of the weight \(\varphi \) with respect to the weight \(\varphi ok\) (*3.6.7*).So, it is just a straightforward remark to notice that *so* is invariant under is if and only if \(\hat \varphi \) is a trace. Moreover, the class of Kac algebras whose Haar weight is a trace invariant under *k* is closed under duality (*6.1..4*). These Kac algebras are called “unimodular” because, for any locally compact group *G*,the Kac algebra \({\Bbbk _a}\left( G \right)\) is unimodular if and only if the group *G* is unimodular. Unimodular Kac algebras are the objects studied by Kac in 1961 ([66], [70]).

## Keywords

Irreducible Representation Hopf Algebra Compact Group Banach Algebra Fourier Representation## Preview

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