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 a natural Hopf algebra structure on the space of noncommutative symmetric functions. The bases for this algebra are indexed by set partitions. We show that there exists a natural inclusion of the Hopf algebra of noncommutative symmetric functions in this larger space. We also consider this algebra as a subspace of noncommutative polynomials and use it to understand the structure of the spaces of harmonics and coinvariants with respect to this collection of noncommutative polynomials and conclude two analogues of Chevalley's theorem in the noncommutative setting.

A coplactic class in the symmetric group $\Sym_n$ consists of all permutations in $\Sym_n$ with a given Schensted $Q$-symbol, and may be described in terms of local relations introduced by Knuth. Any Lie element in the group algebra of $\Sym_n$ which is constant on coplactic classes is already constant on descent classes. As a consequence, the intersection of the Lie convolution algebra introduced by Patras and Reutenauer and the coplactic algebra introduced by Poirier and Reutenauer is the direct sum of all Solomon descent algebras.

We develop an explicit skein-theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the homology of $U(1)$-representation varieties on the one side and the combinatorially constructed Hennings TQFT based on the quasitriangular Hopf algebra $\mathcal{N} = \mathbb{Z}/2 \ltimes \bigwedge^* \mathbb{R}^2$ on the other side. We find that both TQFT's are $\SL (2,\mathbb{R})$-equivariant functors and, as such, are isomorphic. The $\SL (2,\mathbb{R})$-action in the Hennings construction comes from the natural action on $\mathcal{N}$ and in the case of the Frohman--Nicas theory from the Hard--Lefschetz decomposition of the $U(1)$-moduli spaces given that they are naturally K\"ahler. The irreducible components of this TQFT, corresponding to simple representations of $\SL(2,\mathbb{Z})$ and $\Sp(2g,\mathbb{Z})$, thus yield a large family of homological TQFT's by taking sums and products. We give several examples of TQFT's and invariants that appear to fit into this family, such as Milnor and Reidemeister Torsion, Seiberg--Witten theories, Casson type theories for homology circles {\it \`a la} Donaldson, higher rank gauge theories following Frohman and Nicas, and the $\mathbb{Z}/p\mathbb{Z}$ reductions of Reshetikhin--Turaev theories over the cyclotomic integers $\mathbb{Z} [\zeta_p]$. We also conjecture that the Hennings TQFT for quantum-$\mathfrak{sl}_2$ is the product of the Reshetikhin--Turaev TQFT and such a homological TQFT.

When $H$ is a finite dimensional, semisimple, almost cocommutative Hopf algebra, we examine a table of characters which extends the notion of the character table for a finite group. We obtain a formula for the structure constants of the representation ring in terms of values in the character table, and give the example of the quantum double of a finite group. We give a basis of the centre of $H$ which generalizes the conjugacy class sums of a finite group, and express the class equation of $H$ in terms of this basis. We show that the representation ring and the centre of $H$ are dual character algebras (or signed hypergroups).