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 An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification 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
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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. 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 ${\displaystyle (.,.)\colon {\mathfrak {g}}\otimes {\mathfrak {g}}\to \mathbb {R} }$ 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 Rⁿ with zero-bracket and standard inner product

${\displaystyle ((x_{1},\dots ,x_{n}),(y_{1},\dots ,y_{n})):=\sum _{j}x_{j}y_{j}}$.

Since the bracket is trivial the invariance is trivially fulfilled.

As a more elaborate example consider so(3), i.e. R³ with base X,Y,Z, standard inner product, and Lie bracket

${\displaystyle [X,Y]=Z,\quad [Y,Z]=X,\quad [Z,X]=Y}$.

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.

${\displaystyle \mathrm {ad} \colon {\mathfrak {g}}\to \mathrm {End} ({\mathfrak {g}}):X\mapsto (\mathrm {ad} _{X}\colon Y\mapsto [X,Y])}$.

Define now the Killing form

${\displaystyle k\colon {\mathfrak {g}}\otimes {\mathfrak {g}}\to \mathbb {R} :X\otimes Y\mapsto -\mathrm {tr} (\mathrm {ad} _{X}\circ \mathrm {ad} _{Y})}$.

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.

References

This article incorporates material from Quadratic Lie algebra on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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This page was last edited on 18 May 2024, at 23:40