Wedderburn-Mal'tsev theorem
Let be a finite-dimensional associative algebra (cf. Associative rings and algebras) over a field
with radical
, and let the quotient algebra
be a separable algebra (for algebras over a field of characteristic zero this is always true). Then
can be decomposed (as a linear space) into a direct sum of the radical
and some semi-simple subalgebra
:
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and if there exists another decomposition , where
is a semi-simple subalgebra, then there exists an automorphism
of the algebra
which maps
onto
(the automorphism
is inner, i.e. there exist elements
such that
and
for all
, where
). 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 be a finite-dimensional Lie algebra over a field of characteristic zero with radical
. Then there exists a semi-simple subalgebra
of
such that
. Such a decomposition is called a Levi decomposition and
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)) |
Wedderburn-Mal'tsev theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wedderburn-Mal%27tsev_theorem&oldid=23127