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

Abstract

The associative ring $R(P(t))=\mathbb{C}[t^{\pm 1},u|u^2=P(t)]$, where $P(t)={\sum }_{i=0}^n{a}_i{t}^i={\prod }_{k=1}^n(t-{\alpha }_i)$ 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 $P(t)=t^4-2b{t}^2+1$. 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.

References in corpus (2)

• The Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras, I
• Determining the automorphism group of a hyperelliptic curve

Cited by in corpus (1)

• On the module structure of the center of hyperelliptic Krichever-Novikov algebras