In algebra, a linear Lie algebra is a subalgebra
of the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
consisting of
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 g ...
s of a
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 ...
''V''. In other words, a linear Lie algebra is the image of a
Lie algebra representation
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is g ...
.
Any Lie algebra is a linear Lie algebra in the sense that there is always a faithful representation of
(in fact, on a finite-dimensional vector space by
Ado's theorem In abstract algebra, Ado's theorem is a theorem characterizing finite-dimensional Lie algebras.
Statement
Ado's theorem states that every finite-dimensional Lie algebra ''L'' over a field ''K'' of characteristic zero can be viewed as a Lie algebr ...
if
is itself finite-dimensional.)
Let ''V'' be a finite-dimensional vector space over a field of characteristic zero and
a subalgebra of
. Then ''V'' is semisimple as a module over
if and only if (i) it is a direct sum of the center and a semisimple ideal and (ii) the elements of the center are
diagonalizable
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
(over some extension field).
Notes
References
* {{cite book , last=Jacobson , first=Nathan , title=Lie algebras , year=1979 , orig-year=1962 , publisher=Dover Publications, Inc. , location=New York , isbn=978-0-486-13679-0 , oclc=867771145 , url=http://www.freading.com/ebooks/details/r:download/ZnJlYWQ6OTc4MDQ4NjEzNjc5MDpl
Lie algebras