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 :

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))