Solution
= Solution
The relation $[h,e]=2e$ forces $ew_i$ to have weight $a+2(i+1)$, so
$$
ew_i=c_iw_{i+1}
$$
for scalars $c_i$. The relation $[e,f]=h$ becomes
$$
(c_{i-1}-c_i)w_i=(a+2i)w_i.
$$
With $c_0=b$, this recurrence has the unique solution
$$
\boxed{c_i=b-ia-i(i+1)}
$$
for every $i\in\mathbb Z$. Direct substitution also verifies $[h,f]=-2f$ and $[h,e]=2e$, so these formulas define the unique required <sl2 Lie algebra> action. They form an <Intermediate-series sl2 module>.