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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]].
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| A filtered algebra over the [[field (mathematics)|field]] <math>k</math> is an [[Algebra over a field|algebra]] <math>(A,\cdot)</math> over <math>k</math> which has an increasing sequence <math> \{0\} \subset F_0 \subset F_1 \subset \cdots \subset F_i \subset \cdots \subset A </math> of subspaces of <math>A</math> such that
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| :<math>A=\cup_{i\in \mathbb{N}} F_i</math>
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| and that is compatible with the multiplication in the following sense
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| :<math> \forall m,n \in \mathbb{N},\qquad F_m\cdot F_n\subset F_{n+m}.</math>
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| ==Associated graded algebra==
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| In general there is the following construction that produces a graded algebra out of a filtered algebra.
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| If <math>A</math> is a filtered algebra then the ''associated graded algebra'' <math>\mathcal{G}(A)</math> is defined as follows: {{unordered list
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| |1= As a [[linear space|vector space]]
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| :<math> \mathcal{G}(A)=\bigoplus_{n\in \mathbb{N}}G_n\,, </math>
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| where,
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| :<math> G_0=F_0,</math> and
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| :<math> \forall n>0, \quad G_n=F_n/F_{n-1}\,,</math>
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| |2= the multiplication is defined by
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| :<math> (x+F_{n-1})(y+F_{m-1})=x\cdot y+F_{n+m-1}</math>
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| 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
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| :<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>
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| for all <math>n\geq 0</math> and <math>m\geq 0</math>.)
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| }}
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| 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.
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| 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 [[isomorphism|isomorphic]].
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| ==Examples==
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| Any [[graded algebra]] graded by ℕ, for example <math>A = \oplus_{n\in \mathbb{N}} A_n </math>, has a filtration given by <math> F_n = \oplus_{i=0}^n A_i </math>.
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| An example of a filtered algebra is the [[Clifford algebra]] <math>\mathrm{Cliff}(V,q)</math> of a vector space <math>V</math> endowed with a [[polarization identity|quadratic form]] <math>q.</math> The associated graded algebra is <math>\bigwedge V</math>, the [[exterior algebra]] of <math>V.</math>
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| 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.
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| 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>.
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| Scalar [[differential operator]]s on a manifold <math>M</math> 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 <math>T^*M</math> which are polynomial along the fibers of the projection <math>\pi\colon T^*M\rightarrow M</math>.
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| The [[group ring|group algebra]] of a group with a [[length function]] is a filtered algebra.
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| ==See also==
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| * [[Filtration (mathematics)]]
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| * [[Length function]]
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| ==References==
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| *{{cite book|last=Abe|first=Eiichi|title=Hopf Algebras|year=1980|publisher=Cambridge University Press|location=Cambridge|isbn=0-521-22240-0|url=http://books.google.ch/books?id=D0AIcewz5-8C&pg=PR4&lpg=PR4&dq=isbn+0-521-22240-0&source=bl&ots=hghAhwKyW6&sig=q9kGyfSJ3GeWFgO51uecxVCu7bo&hl=en&sa=X&ei=1AVNT7HjDtP64QTx08TyAg&ved=0CCUQ6AEwAQ#v=onepage&q=isbn%200-521-22240-0&f=false}}
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| {{PlanetMath attribution|id=3938|title=Filtered algebra}}
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| [[Category:Algebras]]
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| [[Category:Homological algebra]]
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