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 $\mathcal{A}_1(C)$.
Highlights
In [11], Wood considered the differential operators Dk = i xki +1 ∂ ∂xi, k≥ 1, acting in the usual way on the integral polynomial ring Z[x1, x2, . . .] in a countable number of variables xi of degree 1 in order to give an introductory presentation of the Steenrod algebra from a purely algebraic point of view.The differential operators Dk have been known to topologists for a long time
Interpretations of the Landweber–Novikov algebra in terms of differential operators have been offered in the works of Buhstaber and the others
(1) The free Baxter algebra of weight 1 over ∅ is isomorphic to the C–algebra A1 (C ); (2) The λ–divided power Hopf algebra Aλ [1] is represented by a suitable multiple of generators of A1(C); (3) For a field F of characteristic 0, the F–algebra A1(F) is a polynomial algebra
Summary
In [11], Wood considered the differential operators Wood named this algebra as the divided differential operator algebra D. Representations of several algebras are expressed in the literature of the differential operators algebra A1(C).
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