Abstract

The commutative ring $$R(P(t))=\mathbb C[t^{\pm 1},u \,|\, u^2=P(t)]$$ , 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 when $$n>4$$ . The Lie algebra $$\mathcal {R}(P(t))={\text {Der}}(R(P(t)))$$ of derivations is called the hyperelliptic Lie algebra associated to P(t) and is a particular type of multipoint Krichever–Novikov algebra. 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 Chebyshev 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 that certain of these polynomials satisfy particular second-order linear differential equations.

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