On nilpotent and solvable Lie algebras of derivations

Ie O. Makedonskyi, A. P. Petravchuk

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


Let K be a field and A be a commutative associative K-algebra which is an integral domain. The Lie algebra DerKA of all K-derivations of A is an A-module in a natural way, and if R is the quotient field of A then RDerKA is a vector space over R. It is proved that if L is a nilpotent subalgebra of RDerKA of rank k over R (i.e. such that dimRR L = k), then the derived length of L is at most k and L is finite dimensional over its field of constants. In case of solvable Lie algebras over a field of characteristic zero their derived length does not exceed 2k. Nilpotent and solvable Lie algebras of rank 1 and 2 (over R) from the Lie algebra RDerKA are characterized. As a consequence we obtain the same estimations for nilpotent and solvable Lie algebras of vector fields with polynomial, rational, or formal coefficients. Analogously, if X is an irreducible affine variety of dimension n over an algebraically closed field K of characteristic zero and A X is its coordinate ring, then all nilpotent (solvable) subalgebras of DerKA X have derived length at most n (2n respectively).

Original languageEnglish
Pages (from-to)245-257
Number of pages13
JournalJournal of Algebra
Publication statusPublished - 1 Mar 2014
Externally publishedYes


  • Commutative ring
  • Derivation
  • Lie algebra
  • Primary
  • Secondary
  • Solvable algebra
  • Vector field


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