View source for Quadratic Lie algebra

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A '''quadratic Lie algebra''' is a [[Lie algebra]] together with a compatible [[symmetric bilinear form]].  Compatibility means that it is invariant under the [[adjoint representation of a Lie algebra|adjoint representation]].  Examples of such are [[semisimple Lie algebra]]s, such as [[SU(N)#Lie algebra|su(''n'')]] and [[special linear group#Lie subgroup|sl(''n'','''R''')]].

== Definition ==
A quadratic Lie algebra is a Lie algebra ('''g''',[.,.]) together with a non-degenerate symmetric bilinear form <math>(.,.)\colon \mathfrak{g}\otimes\mathfrak{g}\to \mathbb{R}</math> that is invariant under the adjoint action, i.e.
:([''X'',''Y''],''Z'')+(''Y'',[''X'',''Z''])=0
where ''X,Y,Z'' are elements of the Lie algebra '''g'''.
A localization/ generalization is the concept of [[Courant algebroid]] where the vector space '''g''' is replaced by (sections of) a [[vector bundle]].

== Examples ==
As a first example, consider ''R''<sup>n</sup> with zero-bracket and standard inner product
:<math>((x_1,\dots,x_n),(y_1,\dots,y_n)):= \sum_j x_jy_j</math>.

Since the bracket is trivial the invariance is trivially fulfilled.

As a more elaborate example consider [[SO(3)#Lie algebra|so(3)]], i.e. ''R''<sup>3</sup> with base ''X,Y,Z'', standard inner product, and Lie bracket
:<math>[X,Y]=Z,\quad [Y,Z]=X,\quad [Z,X]=Y</math>.
Straightforward computation shows that the inner product is indeed preserved.  A generalization is the following.

=== Semisimple Lie algebras ===
A big group of examples fits into the category of semisimple Lie algebras, i.e. Lie algebras whose adjoint representation is faithful.  Examples are [[special linear group#Lie subgroup|sl(n,R)]] and [[SU(N)#Lie algebra|su(n)]], as well as [[direct sum of Lie algebras|direct sum]]s of them.  Let thus '''g''' be a semi-simple Lie algebra with adjoint representation ''ad'', i.e.
:<math>\mathrm{ad}\colon\mathfrak{g}\to\mathrm{End}(\mathfrak{g}):X\mapsto (\mathrm{ad}_X\colon Y\mapsto [X,Y])</math>.
Define now the [[Killing form]]
:<math>k\colon\mathfrak{g}\otimes\mathfrak{g}\to\mathbb{R}: X\otimes Y \mapsto -\mathrm{tr}(\mathrm{ad}_X\circ\mathrm{ad}_Y)</math>.
Due to the [[Cartan criterion]], the Killing form is non-degenerate if and only if the Lie algebra is semisimple.

If '''g''' is in addition a [[simple Lie algebra]], then the Killing form is up to rescaling the only invariant symmetric bilinear form.

== References ==
{{Reflist}}
{{PlanetMath attribution|id=7379|title=Quadratic Lie algebra}}

[[Category:Lie algebras]]
[[Category:Theoretical physics]]


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