A quadratic Lie algebra is a

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 ...

together with a compatible

symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a biline ...

. Compatibility means that it is invariant under the

adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is \m ...

. Examples of such are

semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of Simple Lie algebra, simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals ...

s, 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\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 differential geometry, a field of mathematics, a Courant algebroid is a vector bundle together with an inner product and a compatible bracket more general than that of a Lie algebroid. It is named after Theodore James Courant, Theodore Courant, ...

where the vector space g is replaced by (sections of) a

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...

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) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is a ...

, 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

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

, 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. As an example, the direct sum of two abelian groups A and B is anothe ...

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) sho ...

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 In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan. A direct sum of ...

, then the Killing form is up to rescaling the only invariant symmetric bilinear form.

References

Lie algebras Theoretical physics {{algebra-stub