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]\displaystyle{ k }[/math] is an algebra [math]\displaystyle{ (A,\cdot) }[/math] over [math]\displaystyle{ k }[/math] that has an increasing sequence [math]\displaystyle{ \{0\} \subseteq F_0 \subseteq F_1 \subseteq \cdots \subseteq F_i \subseteq \cdots \subseteq A }[/math] of subspaces of [math]\displaystyle{ A }[/math] such that
- [math]\displaystyle{ A=\bigcup_{i\in \mathbb{N}} F_{i} }[/math]
and that is compatible with the multiplication in the following sense:
- [math]\displaystyle{ \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]\displaystyle{ A }[/math] is a filtered algebra then the associated graded algebra [math]\displaystyle{ \mathcal{G}(A) }[/math] is defined as follows:
- As a vector space
- [math]\displaystyle{ \mathcal{G}(A)=\bigoplus_{n\in \mathbb{N}}G_n\,, }[/math]
where,
- [math]\displaystyle{ G_0=F_0, }[/math] and
- [math]\displaystyle{ \forall n\gt 0,\ G_n = F_n/F_{n-1}\,, }[/math]
- the multiplication is defined by
- [math]\displaystyle{ (x+F_{n-1})(y+F_{m-1}) = x\cdot y+F_{n+m-1} }[/math]
for all [math]\displaystyle{ x \in F_n }[/math] and [math]\displaystyle{ y \in F_m }[/math]. (More precisely, the multiplication map [math]\displaystyle{ \mathcal{G}(A)\times \mathcal{G}(A) \to \mathcal{G}(A) }[/math] is combined from the maps
- [math]\displaystyle{ (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]
The multiplication is well-defined and endows [math]\displaystyle{ \mathcal{G}(A) }[/math] with the structure of a graded algebra, with gradation [math]\displaystyle{ \{G_n\}_{n \in \mathbb{N}}. }[/math] Furthermore if [math]\displaystyle{ A }[/math] is associative then so is [math]\displaystyle{ \mathcal{G}(A) }[/math]. Also if [math]\displaystyle{ A }[/math] is unital, such that the unit lies in [math]\displaystyle{ F_0 }[/math], then [math]\displaystyle{ \mathcal{G}(A) }[/math] will be unital as well.
As algebras [math]\displaystyle{ A }[/math] and [math]\displaystyle{ \mathcal{G}(A) }[/math] are distinct (with the exception of the trivial case that [math]\displaystyle{ A }[/math] is graded) but as vector spaces they are isomorphic. (One can prove by induction that [math]\displaystyle{ \bigoplus_{i=0}^nG_i }[/math] is isomorphic to [math]\displaystyle{ F_n }[/math] as vector spaces).
Examples
Any graded algebra graded by [math]\displaystyle{ \mathbb{N} }[/math], for example [math]\displaystyle{ A = \bigoplus_{n\in \mathbb{N}} A_n }[/math], has a filtration given by [math]\displaystyle{ F_n = \bigoplus_{i=0}^n A_i }[/math].
An example of a filtered algebra is the Clifford algebra [math]\displaystyle{ \operatorname{Cliff}(V,q) }[/math] of a vector space [math]\displaystyle{ V }[/math] endowed with a quadratic form [math]\displaystyle{ q. }[/math] The associated graded algebra is [math]\displaystyle{ \bigwedge V }[/math], the exterior algebra of [math]\displaystyle{ 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]\displaystyle{ \mathfrak{g} }[/math] is also naturally filtered. The PBW theorem states that the associated graded algebra is simply [math]\displaystyle{ \mathrm{Sym} (\mathfrak{g}) }[/math].
Scalar differential operators on a manifold [math]\displaystyle{ 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]\displaystyle{ T^*M }[/math] which are polynomial along the fibers of the projection [math]\displaystyle{ \pi\colon T^*M\rightarrow M }[/math].
The group algebra of a group with a length function is a filtered algebra.
See also
References
- Abe, Eiichi (1980). Hopf Algebras. Cambridge: Cambridge University Press. ISBN 0-521-22240-0. https://books.google.com/books?id=D0AIcewz5-8C&q=isbn+0-521-22240-0&pg=PR4.
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