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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w0973501.png" /> be a finite-dimensional associative algebra (cf. [[Associative rings and algebras|Associative rings and algebras]]) over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w0973502.png" /> with radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w0973503.png" />, and let the quotient algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w0973504.png" /> be a [[Separable algebra|separable algebra]] (for algebras over a field of characteristic zero this is always true). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w0973505.png" /> can be decomposed (as a linear space) into a direct sum of the radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w0973506.png" /> and some semi-simple subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w0973507.png" />:
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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/w097/w097350/w0973508.png" /></td> </tr></table>
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and if there exists another decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w0973509.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735010.png" /> is a semi-simple subalgebra, then there exists an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735011.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735012.png" /> which maps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735013.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735014.png" /> (the automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735015.png" /> is inner, i.e. there exist elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735020.png" />). The existence of this decomposition was shown by J.H.M. Wedderburn [[#References|[1]]] and the uniqueness, up to an automorphism of the semi-simple term, was proved by A.I. Mal'tsev [[#References|[2]]]. This theorem, together with Wedderburn's theorem (cf. [[Associative rings and algebras|Associative rings and algebras]]) on the structure of semi-simple algebras constitutes the central part of the classical theory of finite-dimensional algebras.
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Let  $  A $
 +
be a finite-dimensional associative algebra (cf. [[Associative rings and algebras|Associative rings and algebras]]) over a field  $  F $
 +
with radical  $  N $,
 +
and let the quotient algebra  $  A/N $
 +
be a [[Separable algebra|separable algebra]] (for algebras over a field of characteristic zero this is always true). Then  $  A $
 +
can be decomposed (as a linear space) into a direct sum of the radical  $  N $
 +
and some semi-simple subalgebra  $  S $:
 +
 
 +
$$
 +
A  =  N \oplus S,
 +
$$
 +
 
 +
and if there exists another decomposition $  A = N \oplus {S _ {1} } $,  
 +
where $  S _ {1} $
 +
is a semi-simple subalgebra, then there exists an automorphism $  \phi $
 +
of the algebra $  A $
 +
which maps $  S $
 +
onto $  S _ {1} $(
 +
the automorphism $  \phi $
 +
is inner, i.e. there exist elements $  a, a  ^  \prime  \in A $
 +
such that $  a \cdot a  ^  \prime  = a  ^  \prime  \cdot a = 0 $
 +
and $  x \phi = a \cdot x \cdot a  ^  \prime  $
 +
for all $  x \in A $,  
 +
where $  x \cdot y = x + y + xy $).  
 +
The existence of this decomposition was shown by J.H.M. Wedderburn [[#References|[1]]] and the uniqueness, up to an automorphism of the semi-simple term, was proved by A.I. Mal'tsev [[#References|[2]]]. This theorem, together with Wedderburn's theorem (cf. [[Associative rings and algebras|Associative rings and algebras]]) on the structure of semi-simple algebras constitutes the central part of the classical theory of finite-dimensional algebras.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.H.M. Wedderburn,  "On hypercomplex numbers"  ''Proc. London Math. Soc. (2)'' , '''6'''  (1908)  pp. 77–118</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Mal'tsev,  "On the representation of an algebra as a direct sum of the radical and a semi-simple subalgebra"  ''Dokl. Akad. Nauk SSSR'' , '''36''' :  1  (1942)  pp. 42–45  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Albert,  "Structure of algebras" , Amer. Math. Soc.  (1939)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.H.M. Wedderburn,  "On hypercomplex numbers"  ''Proc. London Math. Soc. (2)'' , '''6'''  (1908)  pp. 77–118</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.I. Mal'tsev,  "On the representation of an algebra as a direct sum of the radical and a semi-simple subalgebra"  ''Dokl. Akad. Nauk SSSR'' , '''36''' :  1  (1942)  pp. 42–45  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.A. Albert,  "Structure of algebras" , Amer. Math. Soc.  (1939)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A similar theorem holds for Lie algebras. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735021.png" /> be a finite-dimensional Lie algebra over a field of characteristic zero with radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735022.png" />. Then there exists a semi-simple subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735023.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735025.png" />. Such a decomposition is called a Levi decomposition and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097350/w09735026.png" /> is called a Levi factor or Levi subalgebra. It is unique up to inner automorphisms.
+
A similar theorem holds for Lie algebras. Let $  \mathfrak g $
 +
be a finite-dimensional Lie algebra over a field of characteristic zero with radical $  \mathfrak r $.  
 +
Then there exists a semi-simple subalgebra $  \mathfrak h $
 +
of $  \mathfrak g $
 +
such that $  \mathfrak g = \mathfrak h \oplus \mathfrak r $.  
 +
Such a decomposition is called a Levi decomposition and $  \mathfrak h $
 +
is called a Levi factor or Levi subalgebra. It is unique up to inner automorphisms.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,  "Lie algebras" , Dover, reprint  (1962)  pp. 91ff  ((also: Dover, reprint, 1979))</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,  "Lie algebras" , Dover, reprint  (1962)  pp. 91ff  ((also: Dover, reprint, 1979))</TD></TR></table>

Latest revision as of 08:28, 6 June 2020


Let $ A $ be a finite-dimensional associative algebra (cf. Associative rings and algebras) over a field $ F $ with radical $ N $, and let the quotient algebra $ A/N $ be a separable algebra (for algebras over a field of characteristic zero this is always true). Then $ A $ can be decomposed (as a linear space) into a direct sum of the radical $ N $ and some semi-simple subalgebra $ S $:

$$ A = N \oplus S, $$

and if there exists another decomposition $ A = N \oplus {S _ {1} } $, where $ S _ {1} $ is a semi-simple subalgebra, then there exists an automorphism $ \phi $ of the algebra $ A $ which maps $ S $ onto $ S _ {1} $( the automorphism $ \phi $ is inner, i.e. there exist elements $ a, a ^ \prime \in A $ such that $ a \cdot a ^ \prime = a ^ \prime \cdot a = 0 $ and $ x \phi = a \cdot x \cdot a ^ \prime $ for all $ x \in A $, where $ x \cdot y = x + y + xy $). The existence of this decomposition was shown by J.H.M. Wedderburn [1] and the uniqueness, up to an automorphism of the semi-simple term, was proved by A.I. Mal'tsev [2]. This theorem, together with Wedderburn's theorem (cf. Associative rings and algebras) on the structure of semi-simple algebras constitutes the central part of the classical theory of finite-dimensional algebras.

References

[1] J.H.M. Wedderburn, "On hypercomplex numbers" Proc. London Math. Soc. (2) , 6 (1908) pp. 77–118
[2] A.I. Mal'tsev, "On the representation of an algebra as a direct sum of the radical and a semi-simple subalgebra" Dokl. Akad. Nauk SSSR , 36 : 1 (1942) pp. 42–45 (In Russian)
[3] A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939)
[4] C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962)

Comments

A similar theorem holds for Lie algebras. Let $ \mathfrak g $ be a finite-dimensional Lie algebra over a field of characteristic zero with radical $ \mathfrak r $. Then there exists a semi-simple subalgebra $ \mathfrak h $ of $ \mathfrak g $ such that $ \mathfrak g = \mathfrak h \oplus \mathfrak r $. Such a decomposition is called a Levi decomposition and $ \mathfrak h $ is called a Levi factor or Levi subalgebra. It is unique up to inner automorphisms.

References

[a1] N. Jacobson, "Lie algebras" , Dover, reprint (1962) pp. 91ff ((also: Dover, reprint, 1979))
How to Cite This Entry:
Wedderburn-Mal'tsev theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wedderburn-Mal%27tsev_theorem&oldid=49186
This article was adapted from an original article by L.A. Bokut' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article