Filtered algebra
An algebra in which there are distinguished subspaces
, indexed by the elements of a totally ordered group
(most often
is the additive group of integers
), such that
for
and
(an increasing filtration). Sometimes one considers the case where
for
(a decreasing filtration), but it reduces to the preceding case by reversing the order in
. With each filtered algebra
one associates the graded algebra
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where (if
, then
), and the product of the elements
and
is defined by the formula
, where
and
are representatives of the cosets
and
, and
is the coset of
generated by
. If some multilinear identity is satisfied in
(for example, commutativity, associativity, or the Jacobi identity), then it is also satisfied in
.
Examples.
1) Let be a Clifford algebra and let
,
, be the collection of elements that can be represented in the form of (non-commutative) polynomials of degree
in the generators. One obtains an increasing
-filtration of
in which
for
. The associated graded algebra is the exterior algebra with the same number of generators.
2) In the universal enveloping algebra of a Lie algebra one can define an increasing -filtration in the same way as in the preceding example. By the Birkhoff–Witt theorem, the associated graded algebra is the polynomial algebra.
Filtered algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Filtered_algebra&oldid=46920