On nilpotent Lie algebras of derivations of fraction fields

Publication date: 9 February 2017

Abstract: 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) = R D e r_{m a t h b b K} A,

the Lie algebra of

m a t h b b K

-derivations of

R

obtained from

D e r_{m a t h b b K} A

via multiplication by elements of

R .

If

L s u b s e t e q W (A)

is a subalgebra of

W (A)

denote by

r k_{R} L

the dimension of the vector space

R L

over the field

R

and by

F = R^{L}

the field of constants of

L

in

R .

Let

L

be a nilpotent subalgebra

L s u b s e t e q W (A)

with

r k_{R} L l e q 3

. It is proven that the Lie algebra

F L

(as a Lie algebra over the field

F

) is isomorphic to a finite dimensional subalgebra of the triangular Lie subalgebra

u_{3} (F)

of the Lie algebra

D e r F [x_{1}, x_{2}, x_{3}],

where

u_{3} (F) = f (x_{2}, x_{3}) f r a c p a r t i a l p a r t i a l x_{1} + g (x_{3}) f r a c p a r t i a l p a r t i a l x_{2} + c f r a c p a r t i a l p a r t i a l x_{3}

with

f i n F [x_{2}, x_{3}], g i n F [x_{3}]

,

c i n F .

In particular, a characterization of nilpotent Lie algebras of vector fields with polynomial coefficients in three variables is obtained.