Reductive Lie algebra
Encyclopedia
In mathematics
, a Lie algebra
is reductive if its adjoint representation
is completely reducible, whence the name. More concretely, a Lie algebra is reductive if is a direct sum of a semisimple Lie algebra and an abelian Lie algebra:
there are alternative characterizations, given below.
of 
matrices with the commutator as Lie bracket, or more abstractly as the endomorphism algebra of an n-dimensional vector space
,
This is the Lie algebra of the general linear group
GL(n), and is reductive as it decomposes as
corresponding to traceless matrices and scalar matrices.
Any semisimple Lie algebra or abelian Lie algebra is a fortiori reductive.
Over the real numbers, compact Lie algebra
s are reductive.
over a field of characteristic 0 is called reductive if any of the following equivalent conditions are satisfied:
Some of these equivalences are easily seen. For example, the center and radical of
is 
while if the radical equals the center the Levi decomposition yields a decomposition 
Further, simple Lie algebras and the trivial 1-dimensional Lie algebra 
are prime ideals.
of Hermann Weyl
works for reductive Lie algebras.
The associated reductive Lie groups are of significant interest: the Langlands program
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).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
is reductive if its adjoint representation
Adjoint endomorphism
In mathematics, the adjoint endomorphism or adjoint action is an endomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras and Lie groups....
is completely reducible, whence the name. More concretely, a Lie algebra is reductive if is a direct sum of a semisimple Lie algebra and an abelian Lie algebra:
there are alternative characterizations, given below.Examples
The most basic example is the Lie algebra of matrices with the commutator as Lie bracket, or more abstractly as the endomorphism algebra of an n-dimensional vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
,
This is the Lie algebra of the general linear groupGeneral linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
GL(n), and is reductive as it decomposes as
corresponding to traceless matrices and scalar matrices.Any semisimple Lie algebra or abelian Lie algebra is a fortiori reductive.
Over the real numbers, compact Lie algebra
Compact Lie algebra
In the mathematical field of Lie theory, a Lie algebra is compact if it is the Lie algebra of a compact Lie group. Intrinsically, a compact Lie algebra is a real Lie algebra whose Killing form is negative definite, though this definition does not quite agree with the previous...
s are reductive.
Definitions
A Lie algebra over a field of characteristic 0 is called reductive if any of the following equivalent conditions are satisfied:
- The adjoint representationAdjoint representationIn mathematics, the adjoint representation of a Lie group G is the natural representation of G on its own Lie algebra...
(the action by bracketing) of
is completely reducible (a direct sum of irreducible representations). -
admits a faithful, completely reducible, finite-dimensional representation. - The radical of
equal the center: 
- The radical always contains the center, but need not equal it.
-
is the direct sum of its derived ideal 
and its center 
-
is the direct sum of a semisimple ideal 
and its center 
- Compare to the Levi decompositionLevi decompositionIn Lie theory and representation theory, the Levi decomposition, conjectured by Killing and Cartan and proved by , states that any finite dimensional real Lie algebra g is the semidirect product of a solvable ideal and a semisimple subalgebra....
, which decomposes a Lie algebra as its radical (which is solvable, not abelian in general) and a Levi subalgebra (which is semisimple).
- Compare to the Levi decomposition
-
is a direct sum of a semisimple Lie algebra 
and an abelian Lie algebra 
-
is a direct sum of prime ideals: 
Some of these equivalences are easily seen. For example, the center and radical of
is while if the radical equals the center the Levi decomposition yields a decomposition Further, simple Lie algebras and the trivial 1-dimensional Lie algebra 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 trickUnitarian 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 group G is in a qualitative way controlled by that...
of Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
works for reductive Lie algebras.
The associated reductive Lie groups are of significant interest: the Langlands program
Langlands program
The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
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, ISBN 1-40200609-8, SpringerLink