filtered algebra
Definition 1.
A filtered algebra over the field is an algebra over
which is endowed with a filtration
by subspaces
, compatible
with the multiplication
in
the following sense
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 be a filtered algebra then the associated http://planetmath.org/node/3071graded algebra is defined as follows:
- •
-
•
the multiplication is defined by
Theorem 3.
The multiplication is well defined and endows with the of a graded algebra, with gradation . Furthermore if is associative then so is .
An example of a filtered algebra is the Clifford algebra
of a vector space endowed with a quadratic form
. The associated
graded algebra is , the exterior algebra
of .
As algebras and are distinct (with the exception of the trivial case that is graded) but as vector spaces they are isomorphic.
Theorem 4.
The underlying vector spaces of and are isomorphic.
Title | filtered algebra |
---|---|
Canonical name | FilteredAlgebra |
Date of creation | 2013-03-22 13:23:55 |
Last modified on | 2013-03-22 13:23:55 |
Owner | Dr_Absentius (537) |
Last modified by | Dr_Absentius (537) |
Numerical id | 11 |
Author | Dr_Absentius (537) |
Entry type | Definition |
Classification | msc 08A99 |