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Felipe Albino dos Santos

Publications and source records attributed to Felipe Albino dos Santos.

8 recordsLinked to original sources
arXiv2026

Orthogonal polynomials and the centres of the universal central extensions of superelliptic Krichever--Novikov algebras

We study the universal central extensions of the derivation and current algebras of a superelliptic curve $u^m=P(x)$, for $P$ palindromic of degree $2r$. Both centres are finite-dimensional and canonically isomorphic. On the derivation side the dimension is known at every covering degree, while reducing an arbitrary class has been done only for the double cover $m=2$; we do it at every $m$. The spanning classes split into $r(m-1)$ chains, each an associated ultraspherical family whose parameters come from the sector and the residue, and we determine which chains carry a positive orthogonality measure. For quadratic $P$ we evaluate the mixed cocycle in closed form, as a combination of two Legendre antiderivatives, at every pair of generators of total index at least three.

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arXiv2026

A Sugawara--Legendre mechanism for the four-point Heisenberg algebra

This paper provides a representation-theoretic explanation for the recent discovery that families of orthogonal polynomials arise in the centers of universal central extensions of superelliptic affine Lie algebras. Working in the Verma modules of the Heisenberg subalgebra, we compute the canonical contravariant form in closed form on a distinguished family of weight vectors, and show that this family is the Legendre family, forced by the curve itself. The classical squared norms follow by a rescaling available under a positivity condition on the central charge, and the module is irreducible exactly when that charge is non-zero. The main result identifies a Sugawara element carried, by an explicit intertwiner, to the classical Legendre differential operator. The results are proved for the genus-zero, four-point case.

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arXiv2026

Irreducible non-holonomic modules for rational Cherednik algebras

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and let $A_n(\mathbb{K})$ be the $n$-th Weyl algebra. We prove that for every complex reflection group $G$ and every $n \geq 2$ the ring of invariant differential operators $A_n(\mathbb{K})^G$ has irreducible non-holonomic modules of Gelfand-Kirillov dimension $2n-1$, and we exhibit such modules explicitly. Through the Morita equivalence between $A_n(\mathbb{K})^G$ and the rational Cherednik algebra $H_{\mathfrak c}$ at an integral parameter, we deduce that $H_{\mathfrak c}$ has irreducible non-holonomic modules of Gelfand-Kirillov dimension $2n-1$ for $n \geq 2$. In an appendix, following a proof kindly shared by O. Mathieu, we show that $A_n(\mathbb{K})$ has irreducible modules of every Gelfand-Kirillov dimension in the interval $[n, 2n-1]$.

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arXiv2026

Irreducible $φ$-Verma modules for hyperelliptic Heisenberg algebras

We study induced representations of the universal central extension $\widehat{\mathfrak{g}} = (\mathfrak{sl}_2 \otimes R) \oplus Ω_R^1/dR$, where $R = \mathbb{C}[t^{\pm 1}, u]/(u^2 - p(t))$ is a hyperelliptic coordinate ring and $p(t)$ has degree $r+1$. The center of $\widehat{\mathfrak{g}}$ has dimension $r+1$. Inside $\widehat{\mathfrak{g}}$ sits a hyperelliptic Heisenberg subalgebra $\widehat{\mathfrak{h}}$. A sign function $φ\colon \mathbb{Z} \setminus \{0\} \to \{+,-\}$ determines a nonstandard polarization of the imaginary modes, yielding $φ$-Verma modules $M_{\widehat{\mathfrak{h}}, φ}$ and $\mathcal{M}_φ$. Under the specialization $κ_1 = \cdots = κ_r = 0$ and a $p$-admissibility condition on $φ$, we prove: $M_{\widehat{\mathfrak{h}}, φ}$ is irreducible if and only if $κ_0 \neq 0$, and the same criterion governs $\mathcal{M}_φ$ after parabolic induction. For the four-point case $r=1$, we remove the specialization and treat general central characters $(κ_0, κ_1) \in \mathbb{C}^2$: under $p$-admissibility, $M_{\widehat{\mathfrak{h}}, φ}$ is irreducible if and only if $(κ_0, κ_1) \neq (0,0)$ (Theorem A'). A key ingredient is the closed form $ψ_{mn}(a) = δ_{m+n, 0}\, ω_1$ for all $m, n$ when $r=1$, placing the mixed $b^1$-$b$ bracket on the anti-diagonal, independent of the hyperelliptic parameter. We also give a finite checkable criterion for $p$-admissibility via reachable sets in $\mathbb{Z}$. We further describe the weight-space decomposition and formal character of $M_{\widehat{\mathfrak{h}}, φ}$, provide a complete structure theorem for the level-zero case, and prove that $p$-admissibility is sharp by constructing explicit reducible modules at nonzero level for non-admissible polarizations.

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arXiv2026

Free Field Realizations of Superelliptic Affine Lie Algebras

We study Wakimoto-type free field constructions for superelliptic affine Lie algebras associated with coordinate rings $A=\mathbb{C}[t^{\pm1},u \mid u^m = p(t)]$, focusing on $\mathfrak{sl}_2$. We construct explicit operators on a tensor product of $m$ ghost Fock spaces, recovering the standard Wakimoto operator product expansions in the even sector and the correct $h^{(0)}$-charge relations in the odd sector. We then prove that the remaining mixed-sector brackets are obstructed within this class by two independent mechanisms: a charge-residue obstruction, arising from the K"{a}hler differential recurrence, and a Heisenberg branch-cut obstruction, caused by non-integer exponents in vertex operator products. These results yield a unified obstruction theorem for Wakimoto-type constructions in the superelliptic setting, explaining the failure of na"{i}ve free field realizations beyond the classical affine case.

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arXiv2026

Superelliptic Affine Lie algebras and orthogonal polynomials II

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and $r,m\ge 2$. The universal central extension of the superelliptic current algebra $\mathfrak{g}\otimes A$ is $\widehat{\mathfrak{g}\otimes A}\cong\mathfrak{g}\otimes A \oplus(Ω^1_A/dA)$, where $A=\mathbb{C}[t,t^{-1},u]/\langle u^m-(1-2ct^r+t^{2r})\rangle$. We compute the recursion relations governing a natural cocycle basis in $Ω^1_A/dA$ and encode them by generating functions admitting closed integral expressions of superelliptic type. The $2r$ possible choices of initial conditions are classified into four structural types; two canonical choices (types~1 and~2) produce two distinguished polynomial families. We prove that these polynomials satisfy fourth-order linear ordinary differential equations in~$c$, valid for all integers $r,m\ge 2$. For the type~2 family the proof combines the Picard-Fuchs theory of the superelliptic curve $u^m=1-2ct^r+t^{2r}$ with an algebraic identification of the explicit coefficient formulas via a rational-function identity argument. After a parity restriction and a reindexing, the resulting sequences are identified with associated ultraspherical polynomials. We show that, for each admissible~$n$ and $m\ge4$, the corresponding fourth-order equations admit a unique polynomial solution up to scalar multiples.

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arXiv2024

Superelliptic Affine Lie algebras and orthogonal polynomials

We construct two families of orthogonal polynomials associated with the universal central extensions of the superelliptic Lie algebras. These polynomials satisfy certain fourth order linear differential equations, and one of the families is a particular collection of associated ultraspherical polynomials. We show that the generating functions of the polynomials satisfy fourth order linear PDEs. Since these generating function can be represented by superelliptic integrals, we have examples of linear PDEs of fourth order with explicit solutions without complete integrability.

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