View source for Wedderburn-Mal'tsev theorem

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" />:

<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>

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.

====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>



====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.

====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>