Orthogonal polynomials and the centres of the universal central extensions of superelliptic Krichever--Novikov algebras

Abstract

We study the universal central extensions of the derivation and current algebras of a superelliptic curve $u^m=P(x)$, for $P$ palindromic of degree $2r$. Both centres are finite-dimensional and canonically isomorphic. On the derivation side the dimension is known at every covering degree, while reducing an arbitrary class has been done only for the double cover $m=2$; we do it at every $m$. The spanning classes split into $r(m-1)$ chains, each an associated ultraspherical family whose parameters come from the sector and the residue, and we determine which chains carry a positive orthogonality measure. For quadratic $P$ we evaluate the mixed cocycle in closed form, as a combination of two Legendre antiderivatives, at every pair of generators of total index at least three.