Английская Википедия: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 <math>k</math> is an algebra <math>(A,\cdot)</math> over <math>k</math> that has an increasing sequence <math> \{0\} \subseteq F_0 \subseteq F_1 \subseteq \cdots \subseteq F_i \subseteq \cdots \subseteq A </math> of subspaces of <math>A</math> such that
- <math>A=\bigcup_{i\in \mathbb{N}} F_{i}</math>
and that is compatible with the multiplication in the following sense:
- <math> \forall m,n \in \mathbb{N},\quad F_m\cdot F_n\subseteq F_{n+m}.</math>
Associated graded algebra
In general there is the following construction that produces a graded algebra out of a filtered algebra.
If <math>A</math> is a filtered algebra then the associated graded algebra <math>\mathcal{G}(A)</math> is defined as follows: Шаблон:Unordered listG_n\,, </math>
where,
- <math>G_0=F_0,</math> and
- <math>\forall n>0,\ G_n = F_n/F_{n-1}\,,</math>
|2= the multiplication is defined by
- <math>(x+F_{n-1})(y+F_{m-1}) = x\cdot y+F_{n+m-1}</math>
for all <math>x \in F_n</math> and <math>y \in F_m</math>. (More precisely, the multiplication map <math> \mathcal{G}(A)\times \mathcal{G}(A) \to \mathcal{G}(A)</math> is combined from the maps
- <math>(F_n / F_{n-1}) \times (F_m / F_{m-1}) \to F_{n+m}/F_{n+m-1}, \ \ \ \ \ \left(x+F_{n-1},y+F_{m-1}\right) \mapsto x\cdot y+F_{n+m-1}</math>
for all <math>n\geq 0</math> and <math>m\geq 0</math>.) }} The multiplication is well-defined and endows <math>\mathcal{G}(A)</math> with the structure of a graded algebra, with gradation <math>\{G_n\}_{n \in \mathbb{N}}.</math> Furthermore if <math>A</math> is associative then so is <math>\mathcal{G}(A)</math>. Also if <math>A</math> is unital, such that the unit lies in <math>F_0</math>, then <math>\mathcal{G}(A)</math> will be unital as well.
As algebras <math>A</math> and <math>\mathcal{G}(A)</math> are distinct (with the exception of the trivial case that <math>A</math> is graded) but as vector spaces they are isomorphic. (One can prove by induction that <math>\bigoplus_{i=0}^nG_i</math> is isomorphic to <math>F_n</math> as vector spaces).
Examples
Any graded algebra graded by <math>\mathbb{N}</math>, for example <math display="inline">A = \bigoplus_{n\in \mathbb{N}} A_n </math>, has a filtration given by <math display="inline"}> F_n = \bigoplus_{i=0}^n A_i </math>.
An example of a filtered algebra is the Clifford algebra <math>\operatorname{Cliff}(V,q)</math> of a vector space <math>V</math> endowed with a quadratic form <math>q.</math> The associated graded algebra is <math>\bigwedge V</math>, the exterior algebra of <math>V.</math>
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 <math>\mathfrak{g}</math> is also naturally filtered. The PBW theorem states that the associated graded algebra is simply <math>\mathrm{Sym} (\mathfrak{g})</math>.
Scalar differential operators on a manifold <math>M</math> 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 <math>T^*M</math> which are polynomial along the fibers of the projection <math>\pi\colon T^*M\rightarrow M</math>.
The group algebra of a group with a length function is a filtered algebra.
See also
References
