Lie structures on differential algebras (Q1824679)

Let L be an n-dimensional Lie algebra over a field t of characteristic zero, \(A\supseteq L\) a commutative associative algebra over t with an identity. In this paper, the author extends the Lie structure from L to A by means of derivations \(d_ 1,...,d_ n\) of A satisfying \(d_ id_ j=d_ jd_ i\), \(d_ i(x_ j)=\delta_{ij}\) \((x_ i\) is a basis of L): \([a,b]=\sum [x_ i,x_ j]d_ i(a)d_ j(b)\). After showing a way to give a Lie structure on a localization of A (A is assumed to be an integral domain), the author studies the Lie structures on the formal power series \(t[x_ 1,...,x_ n]\) and on factor algebras of polynomial algebras \(t[x_ 1,...,x_ n,y]/(y^ 2-2y+\beta).\)