Examples of inner linear Hopf algebras

Abstract. The notion of inner linear Hopf algebra is a generalization of the notion of discrete linear group. In this paper, we prove two general results that enable us to enlarge the class of Hopf algebras that are known to be inner linear: the first one is a characterization by using the Hopf dual, while the second one is a stability result under extensions. We also discuss the related notion of inner unitary Hopf ∗-algebra. 1. introduction Throughout the paper, we work over C, the field of complex numbers. The notion of inner linear Hopf algebra was introduced in [4] as a natural generalization of the notion of discrete linear group. The precise definition is the following one. Definition 1.1. A Hopf algebra is said to be inner linear if it contains an ideal of finite codi-mension that does not contain any non-zero Hopf ideal. Indeed, when H = C[Γ] is the group algebra of a discrete group Γ, then H is inner linear if and only if the group Γ is linear in the usual sense, i.e. admits a faithful finite-dimensional linear representation [4]. Also the concept of inner linearity for Hopf algebras generalizes the notion of linear ( = finit