Difference between revisions of "Wedderburn-Mal'tsev theorem"
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| + | $#C+1 = 26 : ~/encyclopedia/old_files/data/W097/W.0907350 Wedderburn\ANDMal\AAptsev theorem | ||
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| − | and if there exists another decomposition | + | 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 | + | 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=23127
Wedderburn-Mal'tsev theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wedderburn-Mal%27tsev_theorem&oldid=23127
This article was adapted from an original article by L.A. Bokut' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article