Impacts of Some Definitions on Algebra of Differential Operators for Noncommutative Algebras
Year: 2025
Type: article
Abstract: 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.
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W4409797898