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filtered algebra
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(Definition)
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Definition 1 A filtered algebra over the field $k$ is an algebra $(A,\cdot)$ over $k$ which is endowed with a filtration $\mathcal{F}=\{F_i\}_{i\in \mathbb{N}}$ by subspaces, 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 general there is the following construction that produces a graded algebra out of a filtered algebra.
Definition 2 Let $(A,\cdot,\mathcal{F})$ be 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, $$G_0=F_0,\quad \text{and } \forall n>0, \quad G_n=F_n/F_{n-1}\,,$$
- the multiplication is defined by $$(x+F_{n})(y+F_{m})=x\cdot y+F_{n+m}$$
Theorem 3 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 $\mathcal{G}(A)$ .
An example of a filtered algebra is the Clifford algebra $\mathrm{Cliff}(V,q)$ of a vector space $V$ endowed with a quadratic form $q$ . The associated graded algebra is $\bigwedge V$ , the exterior algebra of $V$ .
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.
Theorem 4 The underlying vector spaces of $A$ and $\mathcal{G}(A)$ are isomorphic.
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"filtered algebra" is owned by Dr_Absentius.
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Cross-references: isomorphic, exterior algebra, quadratic form, Clifford algebra, associative, well defined, vector space, graded algebra, multiplication, compatible, subspaces, filtration, algebra, field
There are 2 references to this entry.
This is version 8 of filtered algebra, born on 2003-01-28, modified 2007-02-04.
Object id is 3938, canonical name is FilteredAlgebra.
Accessed 3206 times total.
Classification:
| AMS MSC: | 08A99 (General algebraic systems :: Algebraic structures :: Miscellaneous) |
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Pending Errata and Addenda
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