Lie structures of the group of Sheffer operators

Abstract

Let $\Phi$ be an (LB)-space over $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$, and let ${\Phi }^{\prime }$ be the dual space of~$\Phi$. We study the set $\mathbb{S}(\Phi )$ of Sheffer operators acting in polynomials on ${\Phi }^{\prime }$. We prove that $\mathbb{S}(\Phi )$ is a group for the usual product of operators. We equip $\mathbb{S}(\Phi )$ with a natural topology which makes $\mathbb{S}(\Phi )$ into an infinite-dimensional manifold with a global parametrization. We show that $\mathbb{S}(\Phi )$ is an infinite-dimensional, regular Lie group, and provide an explicit description of the Lie algebra of $\mathbb{S}(\Phi )$, including an explicit form of the Lie bracket on it. Our main results are new even in the one-dimensional case, $\Phi =\mathbb{F}$. 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.