Filtered algebra - Citizendia

In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure Examples appear in many branches of mathematics, especially in homological algebra and representation theory. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of

A filtered algebra over the field $k$ is an algebra $(A,\cdot)$ over $k$ which is endowed with a filtration

$\mathcal{F}=\{F_i\}_{i\in \mathbb{N}}$

compatible with the multiplication in the following sense

$\forall m,n \in \mathbb{N},\qquad F_m\cdot F_n\subset F_{n+m}.$

A special case of filtered algebra is a graded algebra. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with In Mathematics, a filtration is an Indexed set Si of Subobjects of a given Algebraic structure S, with the index

Associated graded

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

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

As a vector space

$\mathcal{G}(A)=\bigoplus_{n\in \mathbb{N}}G_n\,,$

where,

$G 0 = F 0,$ and

$\forall n>0, \quad G_n=F_n/F_{n-1}\,,$
the multiplication is defined by

$(x+F_{n})(y+F_{m})=x\cdot y+F_{n+m}$

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 $\mathcal{G}(A).$

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. In Mathematics a linear space can mean one of two things In Linear algebra or Mathematical analysis, a Vector space In Mathematics, associativity is a property that a Binary operation can have In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

Examples

An example of a filtered algebra is the Clifford algebra $Cliff(V, q)$ of a vector space $V$ endowed with a quadratic form $q . In Mathematical physics, a geometric algebra is a Multilinear algebra described technically as a Clifford algebra over a real vector space equipped In Mathematics, and more specifically in the theory of Normed spaces and Pre-Hilbert spaces in Functional analysis, a Vector space over the real$ The associated graded algebra is $\bigwedge V$ , 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. In Mathematics, the symmetric algebra S ( V) (also denoted Sym ( V) on a Vector space V over a field In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added

The universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is also naturally filtered. In Mathematics, for any Lie algebra L one can construct its universal enveloping algebra U ( L) In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie The PBW theorem states that the associated graded algebra is simply $\mathrm{Sym} (\mathfrak{g})$ . In the theory of Lie algebras the Poincaré–Birkhoff–Witt theorem (stated by Henri Poincaré (1900 and proved by Garrett Birkhoff (1937 and

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 $\pi:T^*M\rightarrow M$ .

This article incorporates material from Filtered algebra on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative online Mathematics Encyclopedia.

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