﻿

# 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 $k$ is an algebra $\left(A,cdot\right)$ over $k$ which has an increasing sequence $\left\{0\right\} subset F_0 subset F_1 subset cdots subset F_i subset cdots subset A$ of subspaces of $A$ such that

:$A=cup_\left\{iin mathbb\left\{N F_i$

and that is compatible with the multiplication in the following sense

:$forall m,n in mathbb\left\{N\right\},qquad F_mcdot F_nsubset F_\left\{n+m\right\}.$

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

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

• As a vector space

:

where,

:$G_0=F_0,$ and

:$forall n>0, quad G_n=F_n/F_\left\{n-1\right\},,$

• the multiplication is defined by

:$\left(x+F_\left\{n\right\}\right)\left(y+F_\left\{m\right\}\right)=xcdot y+F_\left\{n+m+1\right\}$

The multiplication is well defined and endows $mathcal\left\{G\right\}\left(A\right)$ with the structure of a graded algebra, with gradation $\left\{G_n\right\}_\left\{n in mathbb\left\{N.$ Furthermore if $A$ is associative then so is $mathcal\left\{G\right\}\left(A\right).$. Also if $A$ is unital, such that the unit lies in $F_0$, then $mathcal\left\{G\right\}\left(A\right).$ will be unital as well.

As algebras $A$ and $mathcal\left\{G\right\}\left(A\right)$ are distinct (with the exception of the trivial case that $A$ is graded) but as vector spaces they are isomorphic.

Examples

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

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 is the commutative algebra of smooth functions on the cotangent bundle $T^*M$ which are polynomial along the fibers of the projection $pi:T^*M ightarrow M$.

----

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Clifford algebra — In mathematics, Clifford algebras are a type of associative algebra. They can be thought of as one of the possible generalizations of the complex numbers and quaternions. The theory of Clifford algebras is intimately connected with the… …   Wikipedia

• Symmetric algebra — In mathematics, the symmetric algebra S ( V ) (also denoted Sym ( V )) on a vector space V over a field K is the free commutative unital associative K algebra containing V .It corresponds to polynomials with indeterminates in V , without choosing …   Wikipedia

• Graded algebra — In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a gradation (or grading ). Graded rings A graded ring A is a ring that has a direct sum… …   Wikipedia

• Poincaré–Birkhoff–Witt theorem — In the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (Poincaré (1900), G. D. Birkhoff (1937), Witt (1937); frequently contracted to PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie… …   Wikipedia

• Filtration (mathematics) — In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with the index i running over some index set I that is a totally ordered set, subject to the condition that if i ≤ j in I then Si ⊆ Sj. The concept… …   Wikipedia

• List of mathematics articles (F) — NOTOC F F₄ F algebra F coalgebra F distribution F divergence Fσ set F space F test F theory F. and M. Riesz theorem F1 Score Faà di Bruno s formula Face (geometry) Face configuration Face diagonal Facet (mathematics) Facetting… …   Wikipedia

• Group ring — This page discusses the algebraic group ring of a discrete group; for the case of a topological group see group algebra, and for a general group see Group Hopf algebra. In algebra, a group ring is a free module and at the same time a ring,… …   Wikipedia

• Boolean algebras canonically defined — Boolean algebras have been formally defined variously as a kind of lattice and as a kind of ring. This article presents them more neutrally but equally formally as simply the models of the equational theory of two values, and observes the… …   Wikipedia

• Spectral sequence — In the area of mathematics known as homological algebra, especially in algebraic topology and group cohomology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a… …   Wikipedia