A partial dictionary between universal central extensions and orthogonal polynomials in the superelliptic Krichever--Novikov setting
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2026
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| author | Santos, Felipe Albino dos |
| author_facet | Santos, Felipe Albino dos |
| contents | Let $m \geq 2$, let $P(x) \in \mathbb{C}[x]$ have simple roots, and let $A = \mathbb{C}[x^{\pm 1},\,u \mid u^m = P(x)]$ be the coordinate ring of the associated superelliptic curve. The derivation algebra $\mathrm{Der}(A)$ and the current algebra $\mathfrak{g}\otimes A$ (for $\mathfrak{g}$ a simple Lie algebra) each admit a universal central extension whose center is multi-dimensional and carries linear algebraic relations among its basis elements. We establish a systematic dictionary between these relations and families of orthogonal polynomials in the parameter $a$ encoding the branch locus of $P$.
The dictionary has three canonical entries: (1)~the basis reduction relations in the center of $\widehat{\mathrm{Der}(A)}$ are exactly the three-term recurrence of an orthogonal polynomial family; (2)~the generating function of the center satisfies the Sturm--Liouville ODE of that family; (3)~the mixed-sector $2$-cocycle equals the Legendre antiderivative $(P_{n-1}(a)-P_{n+1}(a))/(2n+1)$ in the quadratic case.
We prove the dictionary completely for $P(x)=x^2-2ax+1$ (Legendre polynomials) and for the quartic palindromic case $P(x)=x^4-2ax^2+1$. In the quadratic case, palindromic symmetry forces the recurrence to be the Legendre three-term recurrence; in the quartic case, the odd sector is Legendre and the even sector satisfies a two-component recurrence with palindromic coefficients. We conjecture this pattern -- palindromic $P$ forcing symmetric recurrence coefficients -- holds in all even degrees. The same dictionary governs the Kähler side $Ω^1_A/dA$: all sectors reduce to the sector-$1$ family at a rescaled parameter, and the recurrence and ODE entries are canonical while the mixed-cocycle entry is partially choice-dependent. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_02530 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A partial dictionary between universal central extensions and orthogonal polynomials in the superelliptic Krichever--Novikov setting Santos, Felipe Albino dos Representation Theory 17B65, 17B56, 33C45, 14H99, 17B10 Let $m \geq 2$, let $P(x) \in \mathbb{C}[x]$ have simple roots, and let $A = \mathbb{C}[x^{\pm 1},\,u \mid u^m = P(x)]$ be the coordinate ring of the associated superelliptic curve. The derivation algebra $\mathrm{Der}(A)$ and the current algebra $\mathfrak{g}\otimes A$ (for $\mathfrak{g}$ a simple Lie algebra) each admit a universal central extension whose center is multi-dimensional and carries linear algebraic relations among its basis elements. We establish a systematic dictionary between these relations and families of orthogonal polynomials in the parameter $a$ encoding the branch locus of $P$. The dictionary has three canonical entries: (1)~the basis reduction relations in the center of $\widehat{\mathrm{Der}(A)}$ are exactly the three-term recurrence of an orthogonal polynomial family; (2)~the generating function of the center satisfies the Sturm--Liouville ODE of that family; (3)~the mixed-sector $2$-cocycle equals the Legendre antiderivative $(P_{n-1}(a)-P_{n+1}(a))/(2n+1)$ in the quadratic case. We prove the dictionary completely for $P(x)=x^2-2ax+1$ (Legendre polynomials) and for the quartic palindromic case $P(x)=x^4-2ax^2+1$. In the quadratic case, palindromic symmetry forces the recurrence to be the Legendre three-term recurrence; in the quartic case, the odd sector is Legendre and the even sector satisfies a two-component recurrence with palindromic coefficients. We conjecture this pattern -- palindromic $P$ forcing symmetric recurrence coefficients -- holds in all even degrees. The same dictionary governs the Kähler side $Ω^1_A/dA$: all sectors reduce to the sector-$1$ family at a rescaled parameter, and the recurrence and ODE entries are canonical while the mixed-cocycle entry is partially choice-dependent. |
| title | A partial dictionary between universal central extensions and orthogonal polynomials in the superelliptic Krichever--Novikov setting |
| topic | Representation Theory 17B65, 17B56, 33C45, 14H99, 17B10 |
| url | https://arxiv.org/abs/2605.02530 |