Reductive Lie algebra
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In mathematics, a Lie algebra is reductive if its
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
is completely reducible, whence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a
semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...
and an
abelian Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
: \mathfrak = \mathfrak \oplus \mathfrak; there are alternative characterizations, given below.


Examples

The most basic example is the Lie algebra \mathfrak_n of n \times n matrices with the commutator as Lie bracket, or more abstractly as the
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
algebra of an ''n''-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, \mathfrak(V). This is the Lie algebra of the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
GL(''n''), and is reductive as it decomposes as \mathfrak_n = \mathfrak_n \oplus \mathfrak, corresponding to
traceless In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace o ...
matrices and
scalar matrices In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...
. Any
semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...
or
abelian Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
is ''a fortiori'' reductive. Over the real numbers,
compact Lie algebra In the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algebr ...
s are reductive.


Definitions

A Lie algebra \mathfrak over a field of characteristic 0 is called reductive if any of the following equivalent conditions are satisfied: # The
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
(the action by bracketing) of \mathfrak is completely reducible (a direct sum of irreducible representations). # \mathfrak admits a faithful, completely reducible, finite-dimensional representation. # The radical of \mathfrak equals the center: \mathfrak(\mathfrak) = \mathfrak(\mathfrak). #:The radical always contains the center, but need not equal it. # \mathfrak is the direct sum of a semisimple ideal \mathfrak_0 and its center \mathfrak(\mathfrak): \mathfrak = \mathfrak_0 \oplus \mathfrak(\mathfrak). #:Compare to the
Levi decomposition In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by , states that any finite-dimensional real Lie algebra ''g'' is the semidirect product of a solvable ideal and a semi ...
, which decomposes a Lie algebra as its radical (which is solvable, not abelian in general) and a Levi subalgebra (which is semisimple). # \mathfrak is a direct sum of a semisimple Lie algebra \mathfrak and an abelian Lie algebra \mathfrak: \mathfrak = \mathfrak \oplus \mathfrak. # \mathfrak is a direct sum of prime ideals: \mathfrak = \textstyle. Some of these equivalences are easily seen. For example, the center and radical of \mathfrak \oplus \mathfrak is \mathfrak, while if the radical equals the center the Levi decomposition yields a decomposition \mathfrak = \mathfrak_0 \oplus \mathfrak(\mathfrak). Further, simple Lie algebras and the trivial 1-dimensional Lie algebra \mathfrak are prime ideals.


Properties

Reductive Lie algebras are a generalization of semisimple Lie algebras, and share many properties with them: many properties of semisimple Lie algebras depend only on the fact that they are reductive. Notably, the
unitarian trick In mathematics, the unitarian trick is a device in the representation theory of Lie groups, introduced by for the special linear group and by Hermann Weyl for general semisimple groups. It applies to show that the representation theory of some g ...
of Hermann Weyl works for reductive Lie algebras. The associated
reductive Lie group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
s are of significant interest: the
Langlands program In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...
is based on the premise that what is done for one reductive Lie group should be done for all. The intersection of reductive Lie algebras and solvable Lie algebras is exactly abelian Lie algebras (contrast with the intersection of semisimple and solvable Lie algebras being trivial).


External links

*
Lie algebra, reductive
'' A.L. Onishchik, in ''Encyclopaedia of Mathematics,'' , SpringerLink {{Authority control Properties of Lie algebras