2016/01/17 by Petravchuk, A. P.
#13N15 #FOS: Mathematics #Primary 17B66 #Rings and Algebras (math.RA) #Secondary 17B05
paper · doi:10.48550/arxiv.1601.04313
Let K be an arbitrary field of characteristic zero and A a commutative associative K-algebra which is an integral domain. Denote by R the fraction field of A and by W(A)=RDer\mathbb KA, the Lie algebra of \mathbb K-derivations of R obtained from Der\mathbb KA via multiplication by elements of R. If L⊆ W(A) is a subalgebra of W(A) denote by rkRL the dimension of the vector space RL over the field R and by F=RL the field of constants of L in R. Let L be a nilpotent subalgebra L⊆ W(A) with rkRL≤ 3. It is proven that the Lie algebra FL (as a Lie algebra over the field F) is isomorphic to a finite dimensional subalgebra of the triangular Lie subalgebra u3(F) of the Lie algebra Der F[x1, x2, x3], where u3(F)=\f(x2, x3)\frac∂∂ x1+g(x3)\frac∂∂ x2+c\frac∂∂ x3\ with f∈ F[x2, x3], g∈ F[x3], c∈ F. In particular, a characterization of nilpotent Lie algebras of vector fields with polynomial coefficients in three variables is obtained.