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 (A,cdot) over k which has an increasing sequence {0} subset F_0 subset F_1 subset cdots subset F_i subset cdots subset A of subspaces of A such that

:A=cup_{iin mathbb{N F_i

and that is compatible with the multiplication in the following sense

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

Associated graded

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{G}(A) is defined as follows:

  • As a vector space

    : mathcal{G}(A)=igoplus_{nin mathbb{NG_n,,

    where,

    : G_0=F_0, and

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

  • the multiplication is defined by

    : (x+F_{n})(y+F_{m})=xcdot y+F_{n+m+1}

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

As algebras A and mathcal{G}(A) 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{Cliff}(V,q) of a vector space V endowed with a quadratic form q. The associated graded algebra is igwedge 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 mathfrak{g} is also naturally filtered. The PBW theorem states that the associated graded algebra is simply mathrm{Sym} (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 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.[1][2] 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

  • Non-associative algebra — This article is about a particular non associative structure known as a non associative algebra. See also the article about non associativity in general. A non associative algebra[1] (or distributive algebra) over a field (or a ring) K is a K… …   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


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.