Filtered algebra

An algebra $ S $ in which there are distinguished subspaces $ S _ \alpha $, indexed by the elements of a totally ordered group $ A $( most often $ A $ is the additive group of integers $ \mathbf Z $), such that $ S _ \alpha \subseteq S _ \beta $ for $ \alpha < \beta $ and $ S _ \alpha S _ \beta \subseteq S _ {\alpha + \beta } $( an increasing filtration). Sometimes one considers the case where $ S _ \alpha \supseteq S _ \beta $ for $ \alpha < \beta $( a decreasing filtration), but it reduces to the preceding case by reversing the order in $ A $. With each filtered algebra $ S $ one associates the graded algebra

$$ \mathop{\rm gr} S = \ \oplus _ \alpha \overline{S}\; _ \alpha , $$

where $ \overline{S}\; _ \alpha = S _ \alpha / \sum _ {\beta < \alpha } S _ \beta $( if $ A = \mathbf Z $, then $ \overline{S}\; _ \alpha = S _ \alpha /S _ {\alpha - 1 } $), and the product of the elements $ \overline{x}\; \in \overline{S}\; _ \alpha $ and $ \overline{y}\; \in \overline{S}\; _ \beta $ is defined by the formula $ \overline{x}\; \overline{y}\; = \overline{xy}\; $, where $ x $ and $ y $ are representatives of the cosets $ \overline{x}\; $ and $ \overline{y}\; $, and $ \overline{xy}\; $ is the coset of $ \sum _ {\gamma < \alpha + \beta } S _ \gamma $ generated by $ xy \in S _ {\alpha + \beta } $. If some multilinear identity is satisfied in $ S $( for example, commutativity, associativity, or the Jacobi identity), then it is also satisfied in $ \mathop{\rm gr} S $.

Examples.

1) Let $ S $ be a Clifford algebra and let $ S _ {n} $, $ n \in \mathbf Z $, be the collection of elements that can be represented in the form of (non-commutative) polynomials of degree $ \leq n $ in the generators. One obtains an increasing $ \mathbf Z $- filtration of $ S $ in which $ S _ {n} = 0 $ for $ n < 0 $. 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 $ \mathbf Z $- filtration in the same way as in the preceding example. By the Birkhoff–Witt theorem, the associated graded algebra is the polynomial algebra.