Filtered algebra

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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 {\displaystyle k} {\displaystyle k} is an algebra ( A , ⋅ ) {\displaystyle (A,\cdot )} {\displaystyle (A,\cdot )} over k {\displaystyle k} {\displaystyle k} that has an increasing sequence { 0 } ⊆ F 0 ⊆ F 1 ⊆ ⋯ ⊆ F i ⊆ ⋯ ⊆ A {\displaystyle \{0\}\subseteq F_{0}\subseteq F_{1}\subseteq \cdots \subseteq F_{i}\subseteq \cdots \subseteq A} {\displaystyle \{0\}\subseteq F_{0}\subseteq F_{1}\subseteq \cdots \subseteq F_{i}\subseteq \cdots \subseteq A} of subspaces of A {\displaystyle A} {\displaystyle A} such that

A = ⋃ i ∈ N F i {\displaystyle A=\bigcup _{i\in \mathbb {N} }F_{i}} {\displaystyle A=\bigcup _{i\in \mathbb {N} }F_{i}}

and that is compatible with the multiplication in the following sense:

∀ m , n ∈ N , F m ⋅ F n ⊆ F n + m . {\displaystyle \forall m,n\in \mathbb {N} ,\quad F_{m}\cdot F_{n}\subseteq F_{n+m}.} {\displaystyle \forall m,n\in \mathbb {N} ,\quad F_{m}\cdot F_{n}\subseteq F_{n+m}.}

Associated graded algebra

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

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

  • As a vector space
    G ( A ) = ⨁ n ∈ N G n , {\displaystyle {\mathcal {G}}(A)=\bigoplus _{n\in \mathbb {N} }G_{n}\,,} {\displaystyle {\mathcal {G}}(A)=\bigoplus _{n\in \mathbb {N} }G_{n}\,,}

    where,

    G 0 = F 0 , {\displaystyle G_{0}=F_{0},} {\displaystyle G_{0}=F_{0},} and
    ∀ n > 0 ,   G n = F n / F n − 1 , {\displaystyle \forall n>0,\ G_{n}=F_{n}/F_{n-1}\,,} {\displaystyle \forall n>0,\ G_{n}=F_{n}/F_{n-1}\,,}
  • the multiplication is defined by
    ( x + F n − 1 ) ( y + F m − 1 ) = x ⋅ y + F n + m − 1 {\displaystyle (x+F_{n-1})(y+F_{m-1})=x\cdot y+F_{n+m-1}} {\displaystyle (x+F_{n-1})(y+F_{m-1})=x\cdot y+F_{n+m-1}}

    for all x ∈ F n {\displaystyle x\in F_{n}} {\displaystyle x\in F_{n}} and y ∈ F m {\displaystyle y\in F_{m}} {\displaystyle y\in F_{m}}. (More precisely, the multiplication map G ( A ) × G ( A ) → G ( A ) {\displaystyle {\mathcal {G}}(A)\times {\mathcal {G}}(A)\to {\mathcal {G}}(A)} {\displaystyle {\mathcal {G}}(A)\times {\mathcal {G}}(A)\to {\mathcal {G}}(A)} is combined from the maps

    ( F n / F n − 1 ) × ( F m / F m − 1 ) → F n + m / F n + m − 1 ,           ( x + F n − 1 , y + F m − 1 ) ↦ x ⋅ y + F n + m − 1 {\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}} {\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}}
    for all n ≥ 0 {\displaystyle n\geq 0} {\displaystyle n\geq 0} and m ≥ 0 {\displaystyle m\geq 0} {\displaystyle m\geq 0}.)

The multiplication is well-defined and endows G ( A ) {\displaystyle {\mathcal {G}}(A)} {\displaystyle {\mathcal {G}}(A)} with the structure of a graded algebra, with gradation { G n } n ∈ N . {\displaystyle \{G_{n}\}_{n\in \mathbb {N} }.} {\displaystyle \{G_{n}\}_{n\in \mathbb {N} }.} Furthermore if A {\displaystyle A} {\displaystyle A} is associative then so is G ( A ) {\displaystyle {\mathcal {G}}(A)} {\displaystyle {\mathcal {G}}(A)}. Also, if A {\displaystyle A} {\displaystyle A} is unital, such that the unit lies in F 0 {\displaystyle F_{0}} {\displaystyle F_{0}}, then G ( A ) {\displaystyle {\mathcal {G}}(A)} {\displaystyle {\mathcal {G}}(A)} will be unital as well.

As algebras A {\displaystyle A} {\displaystyle A} and G ( A ) {\displaystyle {\mathcal {G}}(A)} {\displaystyle {\mathcal {G}}(A)} are distinct (with the exception of the trivial case that A {\displaystyle A} {\displaystyle A} is graded) but as vector spaces they are isomorphic. (One can prove by induction that ⨁ i = 0 n G i {\displaystyle \bigoplus _{i=0}^{n}G_{i}} {\displaystyle \bigoplus _{i=0}^{n}G_{i}} is isomorphic to F n {\displaystyle F_{n}} {\displaystyle F_{n}} as vector spaces).

Examples

Any graded algebra graded by N {\displaystyle \mathbb {N} } {\displaystyle \mathbb {N} }, for example A = ⨁ n ∈ N A n {\textstyle A=\bigoplus _{n\in \mathbb {N} }A_{n}} {\textstyle A=\bigoplus _{n\in \mathbb {N} }A_{n}}, has a filtration given by F n = ⨁ i = 0 n A i {\textstyle F_{n}=\bigoplus _{i=0}^{n}A_{i}} {\textstyle F_{n}=\bigoplus _{i=0}^{n}A_{i}}.

An example of a filtered algebra is the Clifford algebra Cliff ⁡ ( V , q ) {\displaystyle \operatorname {Cliff} (V,q)} {\displaystyle \operatorname {Cliff} (V,q)} of a vector space V {\displaystyle V} {\displaystyle V} endowed with a quadratic form q . {\displaystyle q.} {\displaystyle q.} The associated graded algebra is ⋀ V {\displaystyle \bigwedge V} {\displaystyle \bigwedge V}, the exterior algebra of V . {\displaystyle V.} {\displaystyle 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 g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}} is also naturally filtered. The PBW theorem states that the associated graded algebra is simply S y m ( g ) {\displaystyle \mathrm {Sym} ({\mathfrak {g}})} {\displaystyle \mathrm {Sym} ({\mathfrak {g}})}.

Scalar differential operators on a manifold M {\displaystyle M} {\displaystyle M} 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 T ∗ M {\displaystyle T^{*}M} {\displaystyle T^{*}M} which are polynomial along the fibers of the projection π : T ∗ M → M {\displaystyle \pi \colon T^{*}M\rightarrow M} {\displaystyle \pi \colon T^{*}M\rightarrow M}.

The group algebra of a group with a length function is a filtered algebra.

See also

References

This article incorporates material from Filtered algebra on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.