Impacts of Some Definitions on Algebra of Differential Operators for Noncommutative Algebras

Rings of differential operators are one of the most important noncommutative (associative) algebras. They play an important role in the representation theory of Lie algebras and the algebraic analysis of systems of partial differential equations. However, If A is a commutative and unitary algebra on a field k, Grothendieck defined the ring of differential operators on the algebra A, denoted by D(A), as follows: D(A):=∪Dⁿ(A), where D⁻¹(A)=0 and for n∈ℕ, (1) Dⁿ(A):={u∈ Endk(A):[u, a]=ua-au ∈ Dⁿ⁻¹(A),∀ a∈ A}. In this paper, we show that with this definition, the algebra of differential operators is no longer rich when it is a noncommutative algebra.