Certain families of Polynomials arising in the study of hyperelliptic Lie algebras

TL;DR

This paper explores the structure of hyperelliptic Lie algebras, describing their universal central extensions and polynomial families related to units in their coordinate rings, with connections to classical polynomials like Legendre and Chebyshev.

Contribution

It provides explicit descriptions of the universal central extension of hyperelliptic Lie algebras and introduces polynomial families linked to units in their coordinate rings, including differential equation characterizations.

Findings

Explicit formulas for the universal central extension in terms of polynomial families.

Identification of polynomial families related to units in specific hyperelliptic rings.

Polynomials satisfying second-order linear differential equations.

Abstract

The associative ring $R(P(t))=\mathbb C[t^{\pm1},u \,|\, u^2=P(t)]$, where $P(t)=\sum_{i=0}^na_it^i=\prod_{k=1}^n(t-\alpha_i)$ with $\alpha_i\in\mathbb C$ pairwise distinct, is the coordinate ring of a hyperelliptic curve. The Lie algebra $\mathcal{R}(P(t))=\text{Der}(R(P(t)))$ of derivations is called the hyperelliptic Lie algebra associated to $P(t)$. In this paper we describe the universal central extension of $\text{Der}(R(P(t)))$ in terms of certain families of polynomials which in a particular case are associated Legendre polynomials. Moreover we describe certain families of polynomials that arise in the study of the group of units for the ring $R(P(t))$ where $P(t)=t^4-2bt^2+1$. In this study pairs of Chebychev polynomials $(U_n,T_n)$ arise as particular cases of a pairs $(r_n,s_n)$ with $r_n+s_n\sqrt{P(t)}$ a unit in $R(P(t))$. We explicitly describe these polynomial pairs as coefficients of certain generating functions and show certain of these polynomials satisfy particular second order linear differential equations.