Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2021; 58(4): 865-876

Online first article June 25, 2021      Printed July 31, 2021

https://doi.org/10.4134/BKMS.b200590

Copyright © The Korean Mathematical Society.

Restricted polynomial extensions

No-Ho Myung, Sei-Qwon Oh

Chungnam National University; Chungnam National University

Abstract

Let $\mathbb F$ be a commutative ring. A restricted skew polynomial extension over $\mathbb F$ is a class of iterated skew polynomial $\mathbb F$-algebras which include well-known quantized algebras such as the quantum algebra $U_q(\mathfrak{sl}_2)$, Weyl algebra, etc. Here we obtain a necessary and sufficient condition in order to be restricted skew polynomial extensions over $\mathbb F$. We also introduce a restricted Poisson polynomial extension which is a class of iterated Poisson polynomial algebras and observe that a restricted Poisson polynomial extension appears as semiclassical limits of restricted skew polynomial extensions. Moreover, we obtain usual as well as unusual quantized algebras of the same Poisson algebra as applications.

Keywords: Skew polynomial algebra (Ore extension), Poisson algebra, quantization, semiclassical limit

MSC numbers: 16S36, 17B63

Supported by: The second author is supported by Chungnam Nationality University Grant.

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