Difference between revisions of "Filtered algebra"
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| − | where < | + | 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|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.=== | ===Examples.=== | ||
| + | 1) Let $ S $ | ||
| + | be a [[Clifford algebra|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|exterior algebra]] with the same number of generators. | ||
| − | + | 2) In the [[Universal enveloping algebra|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|Birkhoff–Witt theorem]], the associated graded algebra is the polynomial algebra. | |
| − | 2) In the [[Universal enveloping algebra|universal enveloping algebra]] of a Lie algebra one can define an increasing | ||
Latest revision as of 19:39, 5 June 2020
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.
Filtered algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Filtered_algebra&oldid=12513