A filtered algebra over the field $k$ is an algebra $(A,\cdot)$ over $k$ which is endowed with a filtration $\mathcal{F}=\{F_{i}\}_{{i\in\mathbb{N}}}$ by subspaces, compatible with the multiplication in the following sense

Let $(A,\cdot,\mathcal{F})$ be a filtered algebra then the associated graded algebra $\mathcal{G}(A)$ is defined as follows:

• As a vector space

  $$\mathcal{G}(A)=\bigoplus_{{n\in\mathbb{N}}}G_{n}\,,$$

  where,

  $$G_{0}=F_{0},\quad\text{and }\forall n>0,\quad G_{n}=F_{n}/F_{{n-1}}\,,$$

• the multiplication is defined by

  $$(x+F_{{n}})(y+F_{{m}})=x\cdot y+F_{{n+m}}$$