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 $\Phi'$ be the dual space of~$\Phi$. We study the set $\mathbb S(\Phi)$ of Sheffer operators acting in polynomials on $\Phi'$. 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, Luz\'on, Mor\'on, Prieto-Martinez, {\it Adv. Math.} 319 (2017) 522--566.