Filtered Algebra - Associated Graded Algebra

In general there is the following construction that produces a graded algebra out of a filtered algebra.

If is a filtered algebra then the associated graded algebra is defined as follows:

• As a vector space
  where,

  and

• the multiplication is defined by

  for all and . (More precisely, the multiplication map is combined from the maps

  for all and .)

The multiplication is well defined and endows with the structure of a graded algebra, with gradation Furthermore if is associative then so is . Also if is unital, such that the unit lies in, then will be unital as well.

As algebras and are distinct (with the exception of the trivial case that is graded) but as vector spaces they are isomorphic.