2023/02/05 by L. Bedratyuk, Bedratyuk, L., Y. Chapovskyi +3
Mathematics · #13N15 #17B05 #17B66 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2302.02441
openalex publication_date 2023/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let K be an algebraically closed field of characteristic zero, A = K[x1,…,xn] the polynomial ring, R = K(x1,…,xn) the field of rational functions, and let Wn(K) = \DerKA be the Lie algebra of all K-derivations on A. If D ∈ Wn(K), D\not =0 is linear (i.e. of the form D = ∑i,j=1n aijxj (∂)/(∂ xi)) we give a description of the centralizer of D in Wn(K) and point out an algorithm for finding generators of CWn(K)(D) as a module over the ring of constants in case when D is the basic Weitzenboeck derivation. In more general case when the ring A is a finitely generated domain over K and D is a locally nilpotent derivation on A, we prove that the centralizer C_\rm DerA(D) is a "large" subalgebra in \rm DerK A, namely \rkA C\Der A(D) := dimR RC\Der A(D) equals \rm tr.°KR, where R is the field of fraction of the ring A.