In the Poincare-Birkhoff-Witt theorem, given a Lie algebra $L$, we consider symmetric algebra on $L$ given by $\mathfrak{S}=T(L)/I$ where $I$ is the ideal $\langle x\otimes y-y\otimes x:x,y,\in L\rangle$. The ideal in this definition is generated by homogenous elements, so the grading on tensor algebra $T(L)$ induces a natural grading on $\mathfrak{S}$: $$\mathfrak{S}=S_0\oplus S_1\oplus S_2\oplus \cdots$$ where $_i$ is the vector subspace of homogeneous polynomials of degree $i$. Associated to this grading is a natural filtration $$S_0 \subset (S_0\oplus S_2) \subset (S_0\oplus S_1\oplus S_2)\subset \cdots$$ Then in the PBW-theorem, the representation of $L$ on $\mathfrak{S}$ is defined inductively by defining representation each member of the filtration and extending linearly to next member of the filtration. My question is a basic one related to such kind of representation:
Are there other examples of representations of a group or algebra, which are defined inductively considering filtration of the representation space, rather than directly defining action on arbitrary element of the (vector) space (of representation) (i.e. defining them directly is tedious, but defining inductively is easier).