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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w0980601.png" /> be a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w0980602.png" />. Consider the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w0980603.png" />-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/w/w098/w098060/w0980604.png" /></td> </tr></table>
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{{TEX|done}}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w0980605.png" /> be the algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w0980606.png" />-derivations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w0980607.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w0980608.png" /> is known as the Witt algebra. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w0980609.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806010.png" />) are known as the split Jacobson–Witt algebras. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806011.png" /> is a simple [[Lie algebra|Lie algebra]], except when it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806012.png" />-dimensional. The dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806013.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806014.png" />.
+
Let $  k $
 +
be a field of characteristic  $  p \neq 0 $.  
 +
Consider the  $  k $-
 +
algebra
  
More generally one considers the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806015.png" />-algebras
+
$$
 +
A _ {n}  = k [ X _ {1} \dots X _ {n} ] /
 +
( X _ {1}  ^ {p} \dots X _ {n}  ^ {p} ) .
 +
$$
  
<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/w/w098/w098060/w09806016.png" /></td> </tr></table>
+
Let  $  V _ {n} $
 +
be the algebra of  $  k $-
 +
derivations of  $  A _ {n} $.  
 +
The algebra  $  V _ {1} $
 +
is known as the Witt algebra. The  $  V _ {n} $(
 +
$  n \geq  2 $)
 +
are known as the split Jacobson–Witt algebras. The algebra  $  V _ {n} $
 +
is a simple [[Lie algebra|Lie algebra]], except when it is  $  2 $-
 +
dimensional. The dimension of  $  V _ {n} $
 +
is  $  np  ^ {n} $.
  
and their algebras of derivations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806017.png" />, the Jacobson–Witt algebras. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806019.png" /> are (obviously) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806020.png" />-forms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806023.png" /> (cf. [[Form of an (algebraic) structure|Form of an (algebraic) structure]]). Many simple Lie algebras in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806024.png" /> arise as subalgebras of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806025.png" />.
+
More generally one considers the $  k $-
 +
algebras
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806026.png" /> be an additive group of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806027.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806028.png" /> such that the only element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806029.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806030.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806031.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806032.png" /> is the zero element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806033.png" />. For instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806034.png" /> can be the set of all functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806035.png" /> to some additive subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806036.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806037.png" /> is finite, it is of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806038.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806039.png" />. Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806040.png" /> be a [[Vector space|vector space]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806041.png" /> with basis elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806044.png" />, and define a bilinear product on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806045.png" /> by
+
$$
 +
A _ {n} ( \xi )  = k [ X _ {1} \dots X _ {n} ] /
 +
( X _ {1}  ^ {p} - \xi _ {1} \dots X _ {n}  ^ {p} - \xi _ {n} ) ,
 +
$$
  
<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/w/w098/w098060/w09806046.png" /></td> </tr></table>
+
and their algebras of derivations  $  V _ {n} ( \xi ) $,
 +
the Jacobson–Witt algebras. The  $  A _ {n} ( \xi ) $
 +
and  $  V _ {n} ( \xi ) $
 +
are (obviously)  $  k  ^  \prime  / k $-
 +
forms of  $  A _ {n} $
 +
and  $  V _ {n} $,
 +
where  $  k  ^  \prime  = k ( \xi _ {1}  ^ {1/p} \dots \xi _ {n}  ^ {1/p} ) $(
 +
cf. [[Form of an (algebraic) structure|Form of an (algebraic) structure]]). Many simple Lie algebras in characteristic  $  p $
 +
arise as subalgebras of the  $  V _ {n} $.
  
There results a Lie algebra, called a generalized Witt algebra. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806047.png" /> is finite of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806048.png" />, the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806049.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806050.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806051.png" /> is a simple Lie algebra if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806052.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806053.png" />.
+
Let  $  G $
 +
be an additive group of functions on  $  \{ 1 \dots m \} $
 +
into  $  k $
 +
such that the only element  $  f $
 +
of  $  G $
 +
such that  $  \sum f( i) g( i) = 0 $
 +
for all  $  g \in G $
 +
is the zero element  $  f = 0 $.  
 +
For instance,  $  G $
 +
can be the set of all functions from  $  \{ 1 \dots m \} $
 +
to some additive subgroup of  $  k $.  
 +
If  $  G $
 +
is finite, it is of order $  p  ^ {n} $
 +
for some  $  n $.  
 +
Now, let  $  V $
 +
be a [[Vector space|vector space]] over  $  k $
 +
with basis elements  $  e _ {g}  ^ {i} $,  
 +
$  i = 1 \dots m $,
 +
$  g \in G $,  
 +
and define a bilinear product on  $  V $
 +
by
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806054.png" /> is of characteristic zero, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806056.png" /> is the additive subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806057.png" />, the same construction results in the [[Virasoro algebra|Virasoro algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806058.png" />.
+
$$
 +
[ e _ {g}  ^ {i} , e _ {h}  ^ {j} ] = \
 +
h( i) e _ {g+ h }  ^ {j} -
 +
g( j) e _ {g+ h }  ^ {i} .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806059.png" /> is of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806061.png" /> is the group of all functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806062.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806063.png" />, one recovers the Jacobson–Witt algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806064.png" />.
+
There results a Lie algebra, called a generalized Witt algebra. If $  G $
 +
is finite of order  $  p  ^ {n} $,
 +
the dimension of $  V $
 +
is  $  m p  ^ {n} $,
 +
and $  V $
 +
is a simple Lie algebra if  $  m > 1 $
 +
or  $  p > 2 $.
  
There are no isomorphisms between the Jacobson–Witt algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806065.png" /> and the classical Lie algebras in positive characteristic when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806066.png" />. Several more classes of simple Lie algebras different from the classical ones and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806067.png" /> are known, [[#References|[a1]]].
+
If  $  k $
 +
is of characteristic zero,  $  m = 1 $
 +
and  $  G $
 +
is the additive subgroup  $  \mathbf Z \subset  k $,
 +
the same construction results in the [[Virasoro algebra|Virasoro algebra]]  $  [ e _ {g} , e _ {h} ] = ( h- g) e _ {g+} h $.
 +
 
 +
If  $  k $
 +
is of characteristic  $  p $
 +
and  $  G $
 +
is the group of all functions on  $  \{ 1 \dots n \} $
 +
with values in  $  \mathbf Z / ( p) \subset  k $,
 +
one recovers the Jacobson–Witt algebras  $  V _ {n} $.
 +
 
 +
There are no isomorphisms between the Jacobson–Witt algebras $  V _ {n} $
 +
and the classical Lie algebras in positive characteristic when $  \mathop{\rm char} ( k) \neq 2, 3 $.  
 +
Several more classes of simple Lie algebras different from the classical ones and the $  V _ {n} $
 +
are known, [[#References|[a1]]].
  
 
The Witt algebra(s) described here should of course not be confused with the [[Witt ring|Witt ring]] of quadratic forms over a field, nor with the various rings of Witt vectors, cf. [[Witt vector|Witt vector]].
 
The Witt algebra(s) described here should of course not be confused with the [[Witt ring|Witt ring]] of quadratic forms over a field, nor with the various rings of Witt vectors, cf. [[Witt vector|Witt vector]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.B. Seligman,   "Modular Lie algebras" , Springer (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Jacobson,   "Classes of restricted Lie algebras of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806068.png" />, II" ''Duke Math. J.'' , '''10''' (1943) pp. 107–121</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R. Ree,   "On generalised Witt algebras" ''Trans. Amer. Math. Soc.'' , '''83''' (1956) pp. 510–546</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.B. Seligman, "Modular Lie algebras" , Springer (1967) {{MR|0245627}} {{ZBL|0189.03201}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Jacobson, "Classes of restricted Lie algebras of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098060/w09806068.png" />, II" ''Duke Math. J.'' , '''10''' (1943) pp. 107–121</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R. Ree, "On generalised Witt algebras" ''Trans. Amer. Math. Soc.'' , '''83''' (1956) pp. 510–546</TD></TR></table>

Latest revision as of 08:29, 6 June 2020


Let $ k $ be a field of characteristic $ p \neq 0 $. Consider the $ k $- algebra

$$ A _ {n} = k [ X _ {1} \dots X _ {n} ] / ( X _ {1} ^ {p} \dots X _ {n} ^ {p} ) . $$

Let $ V _ {n} $ be the algebra of $ k $- derivations of $ A _ {n} $. The algebra $ V _ {1} $ is known as the Witt algebra. The $ V _ {n} $( $ n \geq 2 $) are known as the split Jacobson–Witt algebras. The algebra $ V _ {n} $ is a simple Lie algebra, except when it is $ 2 $- dimensional. The dimension of $ V _ {n} $ is $ np ^ {n} $.

More generally one considers the $ k $- algebras

$$ A _ {n} ( \xi ) = k [ X _ {1} \dots X _ {n} ] / ( X _ {1} ^ {p} - \xi _ {1} \dots X _ {n} ^ {p} - \xi _ {n} ) , $$

and their algebras of derivations $ V _ {n} ( \xi ) $, the Jacobson–Witt algebras. The $ A _ {n} ( \xi ) $ and $ V _ {n} ( \xi ) $ are (obviously) $ k ^ \prime / k $- forms of $ A _ {n} $ and $ V _ {n} $, where $ k ^ \prime = k ( \xi _ {1} ^ {1/p} \dots \xi _ {n} ^ {1/p} ) $( cf. Form of an (algebraic) structure). Many simple Lie algebras in characteristic $ p $ arise as subalgebras of the $ V _ {n} $.

Let $ G $ be an additive group of functions on $ \{ 1 \dots m \} $ into $ k $ such that the only element $ f $ of $ G $ such that $ \sum f( i) g( i) = 0 $ for all $ g \in G $ is the zero element $ f = 0 $. For instance, $ G $ can be the set of all functions from $ \{ 1 \dots m \} $ to some additive subgroup of $ k $. If $ G $ is finite, it is of order $ p ^ {n} $ for some $ n $. Now, let $ V $ be a vector space over $ k $ with basis elements $ e _ {g} ^ {i} $, $ i = 1 \dots m $, $ g \in G $, and define a bilinear product on $ V $ by

$$ [ e _ {g} ^ {i} , e _ {h} ^ {j} ] = \ h( i) e _ {g+ h } ^ {j} - g( j) e _ {g+ h } ^ {i} . $$

There results a Lie algebra, called a generalized Witt algebra. If $ G $ is finite of order $ p ^ {n} $, the dimension of $ V $ is $ m p ^ {n} $, and $ V $ is a simple Lie algebra if $ m > 1 $ or $ p > 2 $.

If $ k $ is of characteristic zero, $ m = 1 $ and $ G $ is the additive subgroup $ \mathbf Z \subset k $, the same construction results in the Virasoro algebra $ [ e _ {g} , e _ {h} ] = ( h- g) e _ {g+} h $.

If $ k $ is of characteristic $ p $ and $ G $ is the group of all functions on $ \{ 1 \dots n \} $ with values in $ \mathbf Z / ( p) \subset k $, one recovers the Jacobson–Witt algebras $ V _ {n} $.

There are no isomorphisms between the Jacobson–Witt algebras $ V _ {n} $ and the classical Lie algebras in positive characteristic when $ \mathop{\rm char} ( k) \neq 2, 3 $. Several more classes of simple Lie algebras different from the classical ones and the $ V _ {n} $ are known, [a1].

The Witt algebra(s) described here should of course not be confused with the Witt ring of quadratic forms over a field, nor with the various rings of Witt vectors, cf. Witt vector.

References

[a1] G.B. Seligman, "Modular Lie algebras" , Springer (1967) MR0245627 Zbl 0189.03201
[a2] N. Jacobson, "Classes of restricted Lie algebras of characteristic , II" Duke Math. J. , 10 (1943) pp. 107–121
[a3] R. Ree, "On generalised Witt algebras" Trans. Amer. Math. Soc. , 83 (1956) pp. 510–546
How to Cite This Entry:
Witt algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Witt_algebra&oldid=17520