Let R be an integral domain and A a cellular algebra over R with a cellular basis {CλS,T∣λ∈Λ and S,T∈M(λ)}. Suppose that A is equipped with a family of Jucys–Murphy elements which satisfy the separation condition in the sense of Mathas [‘Seminormal forms and Gram determinants for cellular algebras’, J. reine angew. Math.619 (2008), 141–173, with an appendix by M. Soriano]. Let K be the field of fractions of R and AK=A⨂ RK. We give a necessary and sufficient condition under which the centre of AK consists of the symmetric polynomials in Jucys–Murphy elements. We also give an application of our result to Ariki–Koike algebras.

Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno–Drinfeld Lie algebra . We show that Gaudin subalgebras form a variety isomorphic to the moduli space of stable curves of genus zero with n+1 marked points. In particular, this gives an embedding of in a Grassmannian of (n−1)-planes in an n(n−1)/2-dimensional space. We show that the sheaf of Gaudin subalgebras over is isomorphic to a sheaf of twisted first-order differential operators. For each representation of the Kohno–Drinfeld Lie algebra with fixed central character, we obtain a sheaf of commutative algebras whose spectrum is a coisotropic subscheme of a twisted version of the logarithmic cotangent bundle of .

The Jucys–Murphy elements for wreath products $\Gamma_n = \Gamma \wr S_n $ associated to any finite group $\Gamma$ are introduced and they play an important role in our study of the connections between class algebras of $\Gamma_n$ for all $n$ and vertex algebras. We construct an action of (a variant of) the $\mathcal W_{1 + \infty}$-algebra acting irreducibly on the direct sum ${\mathcal R}_\Gamma$ of the class algebras of $\Gamma_n$ for all $n$ in a group-theoretic manner. We establish various relations between convolution operators using JM elements and Heisenberg algebra operators acting on ${\mathcal R}_\Gamma$. As applications, we obtain two distinct sets of algebra generators for the class algebra of $\Gamma_n$ and establish various stability results concerning products of normalized conjugacy classes of $\Gamma_n$ and the power sums of Jucys--Murphy elements, etc. We introduce a stable algebra which encodes the class algebra structures of $\Gamma_n$ for all $n$, whose structure constants are shown to be non-negative integers. In the symmetric group case (that is, when $\Gamma$ is trivial), we recover and strengthen in a uniform approach various results of Lascoux and Thibon, Kerov and Olshanski, and Farahat and Higman, etc.