In this paper we use the recent developments in the representation theory of locally compact quantum groups, to assign, to each locally compact quantum group $\mathbb{G}$, a locally compact group $\tilde {\mathbb{G}}$ which is the quantum version of point-masses, and is an invariant for the latter. We show that ``quantum point-masses" can be identified with several other locally compact groups that can be naturally assigned to the quantum group $\mathbb{G}$. This assignment preserves compactness as well as discreteness (hence also finiteness), and for large classes of quantum groups, amenability. We calculate this invariant for some of the most well-known examples of non-classical quantum groups. Also, we show that several structural properties of $\mathbb{G}$ are encoded by $\tilde {\mathbb{G}}$: the latter, despite being a simpler object, can carry very important information about $\mathbb{G}$.

We study locally compact quantum groups $\mathbb{G}$ through the convolution algebras $L_1(\mathbb{G})$ and $(T(L_2(\mathbb{G})), \triangleright)$. We prove that the reduced quantum group $C^*$-algebra $C_0(\mathbb{G})$ can be recovered from the convolution $\triangleright$ by showing that the right $T(L_2(\mathbb{G}))$-module $\langle K(L_2(\mathbb{G}) \triangleright T(L_2(\mathbb{G}))\rangle$ is equal to $C_0(\mathbb{G})$. On the other hand, we show that the left $T(L_2(\mathbb{G}))$-module $\langle T(L_2(\mathbb{G}))\triangleright K(L_2(\mathbb{G})\rangle$ is isomorphic to the reduced crossed product $C_0(\widehat{\mathbb{G}}) \,_r\!\ltimes C_0(\mathbb{G})$, and hence is a much larger $C^*$-subalgebra of $B(L_2(\mathbb{G}))$. We establish a natural isomorphism between the completely bounded right multiplier algebras of $L_1(\mathbb{G})$ and $(T(L_2(\mathbb{G})), \triangleright)$, and settle two invariance problems associated with the representation theorem of Junge-Neufang-Ruan (2009). We characterize regularity and discreteness of the quantum group $\mathbb{G}$ in terms of continuity properties of the convolution $\triangleright$ on $T(L_2(\mathbb{G}))$. We prove that if $\mathbb{G}$ is semi-regular, then the space $\langle T(L_2(\mathbb{G}))\triangleright B(L_2(\mathbb{G}))\rangle$ of right $\mathbb{G}$-continuous operators on $L_2(\mathbb{G})$, which was introduced by Bekka (1990) for $L_{\infty}(G)$, is a unital $C^*$-subalgebra of $B(L_2(\mathbb{G}))$. In the representation framework formulated by Neufang-Ruan-Spronk (2008) and Junge-Neufang-Ruan, we show that the dual properties of compactness and discreteness can be characterized simultaneously via automatic normality of quantum group bimodule maps on $B(L_2(\mathbb{G}))$. We also characterize some commutation relations of completely bounded multipliers of $(T(L_2(\mathbb{G})), \triangleright)$ over $B(L_2(\mathbb{G}))$.

We introduce and study a remarkable family of real probability measures $\pi_{st}$ that we call free Bessel laws. These are related to the free Poisson law $\pi$ via the formulae $\pi_{s1}=\pi^{\boxtimes s}$ and ${\pi_{1t}=\pi^{\boxplus t}}$. Our study includes definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.

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.