Filtered algebra

Remarks

The associated graded algebra is $\operatorname{gr} A = \bigoplus_{i \ge 0} F_i / F_{i-1}$ with $F_{-1} = 0$; clause 3 makes the product $(a + F_{i-1})(b + F_{j-1}) = ab + F_{i+j-1}$ well defined. Example: $A = K[x]$ with $F_i$ the polynomials of degree at most $i$. Clauses 1, 2 and 4 are clear, and clause 3 holds since $\deg(fg) \le \deg f + \deg g$. Here $\operatorname{gr} A \cong K[x]$. Non-example: in $K[x]$ take $F_0 = K + Kx$ and $F_i$ the polynomials of degree at most $i + 1$ for $i \ge 1$. Clauses 1, 2 and 4 hold, but clause 3 fails: $x \cdot x = x^2 \notin F_0$. Let $U(\mathfrak{g})$ be the universal enveloping algebra of a Lie algebra $\mathfrak{g}$ over a field, with $F_i$ spanned by products of at most $i$ elements of $\mathfrak{g}$. Then the Poincaré–Birkhoff–Witt theorem gives $\operatorname{gr} U(\mathfrak{g}) \cong S(\mathfrak{g})$.