Filtered algebra

In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory.

A filtered algebra over the field ${\displaystyle k}${\displaystyle k} is an algebra ${\displaystyle (A,\cdot )}${\displaystyle (A,\cdot )} over ${\displaystyle k}${\displaystyle k} that has an increasing sequence ${\displaystyle \{0\}\subseteq F_{0}\subseteq F_{1}\subseteq \cdots \subseteq F_{i}\subseteq \cdots \subseteq A}${\displaystyle \{0\}\subseteq F_{0}\subseteq F_{1}\subseteq \cdots \subseteq F_{i}\subseteq \cdots \subseteq A} of subspaces of ${\displaystyle A}${\displaystyle A} such that

${\displaystyle A=\bigcup _{i\in \mathbb {N} }F_{i}}${\displaystyle A=\bigcup _{i\in \mathbb {N} }F_{i}}

and that is compatible with the multiplication in the following sense:

${\displaystyle \forall m,n\in \mathbb {N} ,\quad F_{m}\cdot F_{n}\subseteq F_{n+m}.}${\displaystyle \forall m,n\in \mathbb {N} ,\quad F_{m}\cdot F_{n}\subseteq F_{n+m}.}

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

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

The multiplication is well-defined and endows ${\displaystyle {\mathcal {G}}(A)}${\displaystyle {\mathcal {G}}(A)} with the structure of a graded algebra, with gradation ${\displaystyle \{G_{n}\}_{n\in \mathbb {N} }.}${\displaystyle \{G_{n}\}_{n\in \mathbb {N} }.} Furthermore if ${\displaystyle A}${\displaystyle A} is associative then so is ${\displaystyle {\mathcal {G}}(A)}${\displaystyle {\mathcal {G}}(A)}. Also, if ${\displaystyle A}${\displaystyle A} is unital, such that the unit lies in ${\displaystyle F_{0}}${\displaystyle F_{0}}, then ${\displaystyle {\mathcal {G}}(A)}${\displaystyle {\mathcal {G}}(A)} will be unital as well.

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

Any graded algebra graded by ${\displaystyle \mathbb {N} }${\displaystyle \mathbb {N} }, for example ${\textstyle A=\bigoplus _{n\in \mathbb {N} }A_{n}}${\textstyle A=\bigoplus _{n\in \mathbb {N} }A_{n}}, has a filtration given by ${\textstyle F_{n}=\bigoplus _{i=0}^{n}A_{i}}${\textstyle F_{n}=\bigoplus _{i=0}^{n}A_{i}}.

An example of a filtered algebra is the Clifford algebra ${\displaystyle \operatorname {Cliff} (V,q)}${\displaystyle \operatorname {Cliff} (V,q)} of a vector space ${\displaystyle V}${\displaystyle V} endowed with a quadratic form ${\displaystyle q.}${\displaystyle q.} The associated graded algebra is ${\displaystyle \bigwedge V}${\displaystyle \bigwedge V}, the exterior algebra of ${\displaystyle V.}${\displaystyle 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 ${\displaystyle {\mathfrak {g}}}${\displaystyle {\mathfrak {g}}} is also naturally filtered. The PBW theorem states that the associated graded algebra is simply ${\displaystyle \mathrm {Sym} ({\mathfrak {g}})}${\displaystyle \mathrm {Sym} ({\mathfrak {g}})}.

Scalar differential operators on a manifold ${\displaystyle M}${\displaystyle 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 ${\displaystyle T^{*}M}${\displaystyle T^{*}M} which are polynomial along the fibers of the projection ${\displaystyle \pi \colon T^{*}M\rightarrow M}${\displaystyle \pi \colon T^{*}M\rightarrow M}.

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

• Filtration (mathematics)
• Length function

• Abe, Eiichi (1980). Hopf Algebras. Cambridge: Cambridge University Press. ISBN 0-521-22240-0.

This article incorporates material from Filtered algebra on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

• Algebras
• Homological algebra

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