# Quadratic Lie algebra

Lie groups and Lie algebras :
Classical groups
General linear GL(n)
Special linear SL(n)
Orthogonal O(n)
Special orthogonal SO(n)
Unitary U(n)
Special unitary SU(n)
Symplectic Sp(n) :
Simple Lie groups
Classical
A$_n$
B$_n$
C$_n$
D$_n$
Exceptional
G$_2$
F$_4$
E$_6$
E$_7$
E$_8$ :
Other Lie groups
Circle
Lorentz
Poincaré
Conformal group
Diffeomorphism
Loop
Euclidean :
Lie algebras
Lie group–Lie algebra correspondence
Exponential map
Adjoint representation
Killing formIndex
Simple Lie algebra
Loop algebra
Affine Lie algebra :
Semisimple Lie algebra
Dynkin diagrams
Cartan subalgebra
Root systemWeyl group
Real formComplexification
Split Lie algebra
Compact Lie algebra :
Representation theory
Lie group representation
Lie algebra representation
Representation theory of semisimple Lie algebras
Representations of classical Lie groups
Theorem of the highest weight
Borel–Weil–Bott theorem :
Lie groups in physics
Particle physics and representation theory
Lorentz group representations
Poincaré group representations
Galilean group representations :
Scientists
Sophus Lie
Henri Poincaré
Wilhelm Killing
Élie Cartan
Hermann Weyl
Claude Chevalley
Harish-Chandra
Armand Borel :
Glossary
Table of Lie groups :
vte :
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. Examples of such are semisimple Lie algebras, such as su(n) and sl(n,R).

## Definition

A quadratic Lie algebra is a Lie algebra (g,[.,.]) together with a non-degenerate symmetric bilinear form $(.,.)\colon {\mathfrak {g}}\otimes {\mathfrak {g}}\to \mathbb {R}$ 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$^n$ with zero-bracket and standard inner product
    $((x_{1},\dots ,x_{n}),(y_{1},\dots ,y_{n})):=\sum _{j}x_{j}y_{j}$.
Since the bracket is trivial the invariance is trivially fulfilled.
As a more elaborate example consider so(3), i.e. R$^3$ with base X,Y,Z, standard inner product, and Lie bracket
    $[X,Y]=Z,\quad [Y,Z]=X,\quad [Z,X]=Y$.
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 sl(n,R) and su(n), as well as direct sums of them. Let thus g be a semi-simple Lie algebra with adjoint representation ad, i.e.
    $\mathrm {ad} \colon {\mathfrak {g}}\to \mathrm {End} ({\mathfrak {g}}):X\mapsto (\mathrm {ad} _{X}\colon Y\mapsto [X,Y])$.
Define now the Killing form
    $k\colon {\mathfrak {g}}\otimes {\mathfrak {g}}\to \mathbb {R} :X\otimes Y\mapsto -\mathrm {tr} (\mathrm {ad} _{X}\circ \mathrm {ad} _{Y})$.
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.