Definition:Filtered Algebra

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Definition

A filtered algebra is a generalization of the notion of a graded algebra.


A filtered algebra over the field $k$ is an algebra $\struct {A_k, \oplus}$ over $k$ which has an increasing sequence $\set 0 \subset F_0 \subset F_1 \subset \cdots \subset F_i \subset \cdots \subset A$ of substructures of $A$ such that:

$\ds A = \bigcup_{i \mathop \in \N} F_i$

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

$\forall m, n \in \N: F_m \cdot F_n \subset F_{n + m}$


Also see

  • Results about filtered algebras can be found here.


Sources