In
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, 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 operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
''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 i ...
.
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
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
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 algeb ...
if
is itself finite-dimensional.)
Let ''V'' be a finite-dimensional vector space over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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. That is, if there exists an invertible matrix P and a diagonal matrix D such that . This is equivalent to (Such D are not ...
(over some
extension field
In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...
).
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