We find an analytic formulation of the notion of Hopf image, in terms of the associated idempotent state. More precisely, if π:A → Mₙ(ℂ) is a finite-dimensional representation of a Hopf C*-algebra, we prove that the idempotent state associated to its Hopf image A' must be the convolution Cesàro limit of the linear functional φ = tr ∘ π. We then discuss some consequences of this result, notably to inner linearity questions.

The incidence coalgebras $K^{\square }I$ of interval finite posets I and their comodules are studied by means of the reduced Euler integral quadratic form $q^{•}:{\mathbb{Z}}^{\left(I\right)}\to \mathbb{Z}$, where K is an algebraically closed field. It is shown that for any such coalgebra the tameness of the category $K^{\square }I-comod$ of finite-dimensional left $K^{\square }I$-modules is equivalent to the tameness of the category $K^{\square }I-Como{d}_{fc}$ of finitely copresented left $K^{\square }I$-modules. Hence, the tame-wild dichotomy for the coalgebras $K^{\square }I$ is deduced. Moreover, we prove that for an interval finite ̃ *ₘ-free...

The half-liberated orthogonal group $O_n^{*}$ appears as intermediate quantum group between the orthogonal group $O_n$, and its free version $O_n^{+}$. We discuss here its basic algebraic properties, and we classify its irreducible representations. The classification of representations is done by using a certain twisting-type relation between $O_n^{*}$ and $U_n$, a non abelian discrete group playing the role of weight lattice, and a number of methods inspired from the theory of Lie algebras. We use these results for showing that...