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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.14898 |
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| _version_ | 1866912722978865152 |
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| author | Finkelshtein, Dmitri Lytvynov, Eugene Oliveira, Maria Joao |
| author_facet | Finkelshtein, Dmitri Lytvynov, Eugene Oliveira, Maria Joao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_14898 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lie structures of the group of Sheffer operators Finkelshtein, Dmitri Lytvynov, Eugene Oliveira, Maria Joao Functional Analysis 22E66, 05A40, 46G20, 47D03, 46A13, 46M05, 46M40 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. |
| title | Lie structures of the group of Sheffer operators |
| topic | Functional Analysis 22E66, 05A40, 46G20, 47D03, 46A13, 46M05, 46M40 |
| url | https://arxiv.org/abs/2511.14898 |