We construct two families of orthogonal polynomials associated with the universal central extensions of the superelliptic Lie algebras. These polynomials satisfy certain fourth-order linear differential equations, and one of the families is a particular collection of associated ultraspherical polynomials. We show that the generating functions of the polynomials satisfy fourth-order linear PDEs. Since these generating functions can be represented by superelliptic integrals, we have examples of linear PDEs of fourth order with explicit solutions without complete integrability.

We describe (braided-) commutative algebras with non-degenerate multiplicative form in certain braided monoidal categories, corresponding to abelian metric Lie algebras (so-called Drinfeld categories). We also describe local modules over these algebras and classify commutative algebras with a finite number of simple local modules.

A famous result of Kostant's states that the universal enveloping algebra of a semisimple complex Lie algebra is a free module over its center. An analogue of this result is proved for the class of special filtered algebras. This is then applied to show that the restricted Yangian and the universal enveloping algebra of the restricted current algebra, associated with the general linear Lie algebra, are both free over their centers.

We show that a quantum Verma-type module for a quantum group associated to an affine Kac-Moody algebra is characterized by its subspace of finite-dimensional weight spaces. In order to do this we prove an explicit equivalence of categories between a certain category containing the quantum Verma modules and a category of modules for a subalgebra of the quantum group for which the finite part of the Verma module is itself a module.