Derivations of Lie Algebras of Vector Fields in Infinitely Many Variables

2025/07/06 by Bezushchak, Oksana, Kashuba, Iryna
#17B40 #17B56 #17B66 #FOS: Mathematics #Primary 16W25 #Rings and Algebras (math.RA) #Secondary 17A36

paper · doi:10.48550/arxiv.2507.04541

Abstract

Let W_X(\mathbbF) be the Lie algebra of all derivations of the polynomial algebra \mathbbF[X] in infinitely many variables. We describe all derivations of W_X(\mathbbF) over a field of characteristic zero and prove that all such derivations are inner. We also consider the subalgebras W_∞(\mathbbF) and W_fin(\mathbbF) of the algebra W_X(\mathbbF) and describe all of their derivations.

Citations

• On Lie isomorphisms of rings
• Automorphisms and derivations of affine commutative and PI-algebras
• Derivations and automorphisms of locally matrix algebras
• On primary decompositions of unital locally matrix algebras
• Automorphisms of the Lie algebra of vector fields on affine n-space
• On nilpotent and solvable Lie algebras of derivations
• The group of automorphisms of the Lie algebra of derivations of a field of rational functions
• The group of automorphisms of the Lie algebra of derivations of a polynomial algebra
• The groups of automorphisms of the Lie algebras of unitriangular polynomial derivations

Related

• Almost inner derivations of Lie algebras II
• On Lie and associative algebras containing inner derivations
• Lie algebras whose derivation algebras are simple
• The structure of Lie algebras with a derivation satisfying a polynomial identity
• Wildness of the problem of classifying nilpotent Lie algebras of vector fields in four variables