SOME ALGEBRAS IN TERMS OF DIFFERENTIAL OPERATORS
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,
___
- G. E. Andrews, L. Guo, W. Keigher and K. Ono, Baxter algebras and Hopf algebras, Trans. Amer. Math. Soc., 355(11) (2003), 4639-4656.
- P. Berthelot and A. Ogus, Notes on Crystalline Cohomology, Princeton Uni- versity Press, 1978.
- V. M. Buchstaber, Semigroups of maps into groups, operator doubles, and complex cobordisms, Amer. Math. Soc. Trans., Series 2, 170 (1995), 9-35.
- V. M. Buchstaber and A. N. Kholodov, Groups of formal diffeomorphisms of the superline, generating functions for sequences of polynomials, and functional equations, Izv. Akad. Nauk SSSR Ser. Mat., 53(5) (1989), 944-970.
- V. M. Buchstaber and A. V. Shokurov, The Landweber-Novikov algebra and formal vector fields on the line, Funktsional. Anal. i Prilozhen, 12(3) (1978), 1-11.
- L. Guo, Baxter algebras and the umbral calculus, Adv. in Appl. Math., 27 (2001), 405-426.
- W. Keigher, On the ring of Hurwitz series, Comm. Algebra, 25 (1997), 1845- 1859.
- S. Roman and G. C. Rota, The umbral calculus, Adv. Math., 27(2) (1978), 95-188.
- W. H. Schikhof, Ultrametric Calculus, Cambridge University Press, 1984.
- G. Walker and R. M. W. Wood, Polynomials and the mod-2 Steenrod Algebra, Volume 1, The Peterson hit problem, Cambridge University Press, 2018.
- R. M. W.Wood, Differential operations and the Steenrod algebra, Proc. London Math. Soc., 75(3) (1997), 194-220.
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI