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An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f0401401.png" /> in which there are distinguished subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f0401402.png" />, indexed by the elements of a totally ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f0401403.png" /> (most often <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f0401404.png" /> is the additive group of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f0401405.png" />), such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f0401406.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f0401407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f0401408.png" /> (an increasing filtration). Sometimes one considers the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f0401409.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014010.png" /> (a decreasing filtration), but it reduces to the preceding case by reversing the order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014011.png" />. With each filtered algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014012.png" /> one associates the [[Graded algebra|graded algebra]]
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014013.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014014.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014016.png" />), and the product of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014018.png" /> is defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014021.png" /> are representatives of the cosets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014023.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014024.png" /> is the coset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014025.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014026.png" />. If some multilinear identity is satisfied in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014027.png" /> (for example, commutativity, associativity, or the Jacobi identity), then it is also satisfied in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014028.png" />.
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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  $
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for  $  \alpha < \beta $
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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]]
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 +
$$
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\mathop{\rm gr}  S  = \
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\oplus _  \alpha  \overline{S}\; _  \alpha  ,
 +
$$
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 +
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  $
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and $  \overline{y}\; \in \overline{S}\; _  \beta  $
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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 $
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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 $
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in which  $  S _ {n} = 0 $
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for  $  n < 0 $.
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The associated graded algebra is the [[Exterior algebra|exterior algebra]] with the same number of generators.
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014029.png" /> be a [[Clifford algebra|Clifford algebra]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014031.png" />, be the collection of elements that can be represented in the form of (non-commutative) polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014032.png" /> in the generators. One obtains an increasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014033.png" />-filtration of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014034.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014035.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014036.png" />. The associated graded algebra is the [[Exterior algebra|exterior algebra]] with the same number of generators.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040140/f04014037.png" />-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.
 

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.

How to Cite This Entry:
Filtered algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Filtered_algebra&oldid=46920
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article