## Abstract

Let K be a field and A be a commutative associative K-algebra which is an integral domain. The Lie algebra _{DerK}A of all K-derivations of A is an A-module in a natural way, and if R is the quotient field of A then R_{DerK}A is a vector space over R. It is proved that if L is a nilpotent subalgebra of R_{DerK}A of rank k over R (i.e. such that _{dimR}R 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 R_{DerK}A 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 language | English |
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Pages (from-to) | 245-257 |

Number of pages | 13 |

Journal | Journal of Algebra |

Volume | 401 |

DOIs | |

Publication status | Published - 1 Mar 2014 |

Externally published | Yes |

## Keywords

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