Extensions of Poisson algebras by derivations (Q752161)

Let F be a Poisson algebra with bracket [, ] \(([ab,c]=a[b,c]+b[a,c]\) for a,b\(\in F)\), and D a derivation of the associative algebra F. Define a D- extension \((F,<, >)\) of F whose bracket \(<, >\) on F is given by \(<a,b>=[a,b]+D(a)b-aD(b)\) for a,b\(\in F\). The authors first give a result on the alternating Schouten product and then use it to prove tat \((F,<, >)\) is a Lie algebra if and only if for any a,b,c\(\in F\) the equations \(D([a,b])=[D(a),b]+[a,D(b)]\) and \([a,b]D(c)+[b,c]D(a)+[c,a]D(b)=0\) hold.