Filtered algebra

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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

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 .


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.

How to Cite This Entry:
Filtered algebra. E.B. Vinberg (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098