SOME ALGEBRAS IN TERMS OF DIFFERENTIAL OPERATORS

Let C be a commutative ring and C[x1, x2, . . .] the polynomial ring in a countable number of variables xi of degree 1. Suppose that the differential operator d 1 = P i xi∂i acts on C[x1, x2, . . .]. Let Zp be the p–adic integers, K the extension field of the p–adic numbers Qp, and F2 the 2-element filed. In this article, first, the C-algebra A1(C) of differential operators is constructed by the divided differential operators (d 1 ) ∨k/k! as its generators, where ∨ stands for the wedge product. Then, the free Baxter algebra of weight 1 over ∅, the λ–divided power Hopf algebra Aλ, the algebra C(Zp, K) of continuous functions from Zp to K, and the algebra of all F2–valued continuous functions on the ternary Cantor set are represented in terms of the differential operators algebra A1(C).