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

2016/02/03 by Ben Cox, Kaiming Zhao, Cox, Ben +1
Mathematics · Physics and Astronomy · #17B40 #17B67 #33C47 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT) #math.RT #msc:17B40 #msc:17B67 #msc:33C47

paper · pdf · doi:10.48550/arxiv.1602.01432

arxiv created 2016/02/03 · openalex publication_date 2016/02/03 · arxiv updated 2016/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The associative ring R(P(t))=\mathbb C[t^±1,u | u²=P(t)], where P(t)=∑_i=0ⁿa_itⁱ=∏_k=1ⁿ(t-α_i) with α_i∈\mathbb C pairwise distinct, is the coordinate ring of a hyperelliptic curve. The Lie algebra R(P(t))=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 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)=t⁴-2bt²+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√(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.

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