Filtered algebra

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

If $A$ is a filtered algebra, then the associated graded algebra $\backslash mathcal\{G\}(A)$ is defined as follows: {{unordered list

|1= As a vector space

:$\backslash mathcal\{G\}(A)=\backslash bigoplus\_\{n\backslash in\; \backslash mathbb\{N\}\}G\_n\backslash ,,$

where,

:$G\_0=F\_0,$ and

:$\backslash forall\; n>0,\backslash \; G\_n\; =\; F\_n/F\_\{n-1\}\backslash ,,$

|2= the multiplication is defined by

:$(x+F\_\{n-1\})(y+F\_\{m-1\})\; =\; x\backslash cdot\; y+F\_\{n+m-1\}$

for all $x\; \backslash in\; F\_n$ and $y\; \backslash in\; F\_m$. (More precisely, the multiplication map $\backslash mathcal\{G\}(A)\backslash times\; \backslash mathcal\{G\}(A)\; \backslash to\; \backslash mathcal\{G\}(A)$ is combined from the maps

:$(F\_n\; /\; F\_\{n-1\})\; \backslash times\; (F\_m\; /\; F\_\{m-1\})\; \backslash to\; F\_\{n+m\}/F\_\{n+m-1\},\; \backslash \; \backslash \; \backslash \; \backslash \; \backslash \; \backslash left(x+F\_\{n-1\},y+F\_\{m-1\}\backslash right)\; \backslash mapsto\; x\backslash cdot\; y+F\_\{n+m-1\}$

for all $n\backslash geq\; 0$ and $m\backslash geq\; 0$.)

}}

The multiplication is well-defined and endows $\backslash mathcal\{G\}(A)$ with the structure of a graded algebra, with gradation $\backslash \{G\_n\backslash \}\_\{n\; \backslash in\; \backslash mathbb\{N\}\}.$ Furthermore if $A$ is associative then so is $\backslash mathcal\{G\}(A)$. Also, if $A$ is unital, such that the unit lies in $F\_0$, then $\backslash mathcal\{G\}(A)$ will be unital as well.

As algebras $A$ and $\backslash mathcal\{G\}(A)$ are distinct (with the exception of the trivial case that $A$ is graded) but as vector spaces they are isomorphic. (One can prove by induction that $\backslash bigoplus\_\{i=0\}^nG\_i$ is isomorphic to $F\_n$ as vector spaces).

Any graded algebra graded by $\backslash mathbb\{N\}$, for example $A\; =\; \backslash bigoplus\_\{n\backslash in\; \backslash mathbb\{N\}\}\; A\_n$, has a filtration given by .

An example of a filtered algebra is the Clifford algebra $\backslash operatorname\{Cliff\}(V,q)$ of a vector space $V$ endowed with a quadratic form $q.$ The associated graded algebra is $\backslash bigwedge\; V$, the exterior algebra of $V.$

The symmetric algebra on the dual of an affine space is a filtered algebra of polynomials; on a vector space, one instead obtains a graded algebra.

The universal enveloping algebra of a Lie algebra $\backslash mathfrak\{g\}$ is also naturally filtered. The PBW theorem states that the associated graded algebra is simply $\backslash mathrm\{Sym\}\; (\backslash mathfrak\{g\})$.

Scalar differential operators on a manifold $M$ form a filtered algebra where the filtration is given by the degree of differential operators. The associated graded algebra is the commutative algebra of smooth functions on the cotangent bundle $T^*M$ which are polynomial along the fibers of the projection $\backslash pi\backslash colon\; T^*M\backslash rightarrow\; M$.

The group algebra of a group with a length function is a filtered algebra.

• Filtration (mathematics)
• Length function

• {{cite book|last=Abe|first=Eiichi|title=Hopf Algebras|year=1980|publisher=Cambridge University Press|location=Cambridge|isbn=0-521-22240-0|url=https://books.google.com/books?id=D0AIcewz5-8C&q=isbn+0-521-22240-0&pg=PR4}}

{{PlanetMath attribution|id=3938|title=Filtered algebra}}

Category:Algebras

Category:Homological algebra