Motivated by the recent developments of the theory of Cherednik algebras in positive characteristic, we study rational Cherednik algebras with divided powers. In our research we have started with the simplest case, the rational Cherednik algebra of type A₁. We investigate its maximal divided power extensions over R[c] and R for arbitrary principal ideal domains R of characteristic zero. In these cases, we prove that the maximal divided power extensions are free modules over the base rings, and construct an explicit basis in the case of R[c]. In addition, we provide an abstract construction of the rational Cherednik algebra of type A₁ over an arbitrary ring, and prove that this generalization expands the rational Cherednik algebra to include all of the divided powers.

This project was done under the MIT PRIMES-USA program, which the authors would like to thank for this opportunity. The authors would also like to thank Professor Pavel Etingof for suggesting this project, and for useful discussions on the research.