SOME ALGEBRAS IN TERMS OF DIFFERENTIAL OPERATORS

Year 2021, Volume: 29 Issue: 29, 120 - 133, 05.01.2021

Ghorban Soleymanpour Ali S. Janfada

https://doi.org/10.24330/ieja.772801

Abstract

Let $C$ be a commutative ring and $C[x_1,x_2,\ldots]$ the polynomial ring in a countable number of variables $x_i$ of degree 1. Suppose that the differential operator $d^1=\sum_i x_{i} \partial_{i} $ acts on $C[x_1,x_2,\ldots]$. Let $\mathbb{Z}_p$ be the $p$--adic integers, $K$ the extension field of the $p$--adic numbers $\mathbb{Q}_p$, and $\mathbb{F}_2$ the 2-element filed. In this article, first, the $C$-algebra $\mathcal{A}_1(C)$ of differential operators is constructed by the divided differential operators $(d^1)^{\vee k}/k!$ as its generators, where $\vee$ stands for the wedge product. Then, the free Baxter algebra of weight $1$ over $\varnothing$, the $\lambda$--divided power Hopf algebra $\mathcal{A}_\lambda$, the algebra $C(\mathbb{Z}_p,K)$ of continuous functions from $\mathbb{Z}_p$ to $K$, and the algebra of all $\mathbb{F}_2$--valued continuous functions on the ternary Cantor set are represented in terms of the differential operators algebra $\mathcal{A}_1(C)$.

Keywords

Differential operator, integral Steenrod operator, $\lambda$-divided power Hopf algebra, Baxter algebra

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