On nilpotent and solvable Lie algebras of derivations

JOURNAL OF ALGEBRA

卷:401

Article

Makedonskyi, Ie O.¹  Petravchuk, A. P.¹ 

[1] Kyiv Taras Shevchenko Univ, Dept Algebra & Math Log, Fac Mech & Math, UA-01033 Kiev, Ukraine

关键词: Lie algebra;    Vector field;    Solvable algebra;    Derivation;    Commutative ring;   

DOI  :  10.1016/j.jalgebra.2013.11.021

来源: Elsevier

【 摘 要 】

Let K be a field and A be a commutative associative K-algebra which is an integral domain. The Lie algebra Der K 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 RDer(K) A is a vector space over R. It is proved that if L is a nilpotent subalgebra of RDerK A of rank k over R (i.e. such that dim(R) RL = k), then the derived length of L is ai 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 RDer(K) 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 Der(K) A(x) have derived length at most n (2n respectively). 2013 Elsevier Inc. (C) All rights reserved.

【 授权许可】

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