Impacts of Some Definitions on Algebra of Differential Operators for Noncommutative Algebras
SAMA Anzoumana1, Bah S.B. Kouame2, Kouakou Konan Mathias3
1Dr. SAMA Anzoumana, Département de Mathématiques, Classes Universitaires Préparatoires aux Grandes Ecoles (CUPGE) Université Polytechnique de San-Pédro, Côte d’Ivoire. 01 BP 1800 San-Pédro 01.
2Sir Bénédic Kouamé Bah, UFR-MI, Laboratoire de Mathématiques et Applications Université Félix Houphouët-Boigny d’Abidjan, Côte d’Ivoire 22 BP 582 Abidjan 22.
3Dr. Konan Mathias Kouakou, UFR-MI, Laboratoire de Mathématiques et Applications Université Félix Houphouët-Boigny d’Abidjan, Côte d’Ivoire 22 BP 582 Abidjan 22.
Manuscript received on 01 October 2024 | First Revised Manuscript received on 24 October 2024 | Second Revised Manuscript received on 21 February 2025 | Manuscript Accepted on 15 April 2025 | Manuscript published on 30 April 2025 | PP: 1-4 | Volume-5 Issue-1, April 2025 | Retrieval Number: 100.1/ijam.B117804021024 | DOI: 10.54105/ijam.B1178.05010425
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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.
Keywords: Differential Operators, Weyl Algebra, Noncommutative Algebra, Derivation on an Algebra.
Scope of the Article: Algebra