A quadratic Lie algebra is a

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 ...

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) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the specia ...

and sl(n,R).

Definition

A quadratic Lie algebra is a Lie algebra (g, ,. together with a non-degenerate symmetric bilinear form

(.,.)\colon \mathfrak\otimes\mathfrak\to \mathbb

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 In a field of mathematics known as differential geometry, a Courant geometry was originally introduced by Zhang-Ju Liu, Alan Weinstein and Ping Xu in their investigation of doubles of Lie bialgebroids in 1997. Liu, Weinstein and Xu named it after ...

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_jy_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 :

,Y Z,\quad,Z X,\quad,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

, as well as

direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...

s of them. Let thus g be a semi-simple Lie algebra with adjoint representation ''ad'', i.e. :

\mathrm\colon\mathfrak\to\mathrm(\mathfrak):X\mapsto (\mathrm_X\colon Y\mapsto,Y

. Define now the

Killing form In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show ...

k\colon\mathfrak\otimes\mathfrak\to\mathbb: X\otimes Y \mapsto -\mathrm(\mathrm_X\circ\mathrm_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

Lie algebras Theoretical physics {{algebra-stub