Lie structures of the group of Sheffer operators
Dmitri Finkelshtein, Eugene Lytvynov, Maria Joao Oliveira
TL;DR
The paper develops a rigorous, infinite-dimensional Lie-group framework for the group of Sheffer operators acting on polynomials over the dual of an (LB)-space $\Phi'$. It constructs the full group $\mathbb S(\Phi)$ via a pleasant global parametrization using formal tensor power series, showing it is a regular Lie group with an explicit Lie algebra $\mathfrak s(\Phi)$ and a concrete bracket that mirrors Weyl-algebraic relations. The authors also organize the structure through auxiliary groups $\mathcal F_0(\Phi)$, $\mathcal F_1(\Phi)$, and their semidirect product $\mathcal S(\Phi)$, and connect the Sheffer group to the Riordan group, including higher-dimensional and noncommutative-generalizations. This framework illuminates the algebraic and analytic underpinnings of umbral calculus in infinite dimensions, with implications for combinatorics and operator theory. The results provide explicit, globally defined exponential maps, enabling a Campbell–Baker–Hausdorff perspective on composing Sheffer-type operators.
Abstract
Let $Φ$ be an (LB)-space over $\mathbb F=\mathbb R$ or $\mathbb C$, and let $Φ'$ be the dual space of~$Φ$. We study the set $\mathbb S(Φ)$ of Sheffer operators acting in polynomials on $Φ'$. We prove that $\mathbb S(Φ)$ is a group for the usual product of operators. We equip $\mathbb S(Φ)$ with a natural topology which makes $\mathbb S(Φ)$ into an infinite-dimensional manifold with a global parametrization. We show that $\mathbb S(Φ)$ is an infinite-dimensional, regular Lie group, and provide an explicit description of the Lie algebra of $\mathbb S(Φ)$, including an explicit form of the Lie bracket on it. Our main results are new even in the one-dimensional case, $Φ=\mathbb{F}$. Furthermore, our results lead to improved understanding of the Lie algebra of the Riordan group, cf.\ Cheon, Luzón, Morón, Prieto-Martinez, {\it Adv. Math.} 319 (2017) 522--566.