Lie algebra

A Lie algebra is an easy example of an algebraic structure that is not associative. Lie algebras describe infinitesimal symmetries or transformations. In short a Lie algebra is a vector space together with a skew-symmetric bilinear operation denoted as bracket that is subject to the Jacobi identity [X,[Y,Z]] +[Y,[Z,X]] +[Z,[X,Y]] = 0 where X, Y, and Z run over all elements of the Lie algebra.

In particular to every Lie group there is associated a Lie algebra that covers the infinitesimal structure of that group.

Examples

The simplest example is the three dimensional space R³ together with the vector product. In the standard base i,j,k this is defined as ixj=k, jxk=i, kxi=j, and extended skew-symmetric and linear. This Lie algebra is also denoted so(3) or su(2) as it is the Lie algebra associated to either of the Lie groups SO(3) or SU(2).

Other examples are:

• kⁿ where k is a field with 0-bracket [X,Y]=0 called abelian Lie algebras.
• Mat_n(k) the nxn matrices over a field k together with the commutator of matrix multiplication, i.e. [A,B]=AoB -BoA. A straightforward computation shows that the Jacobi identity holds.
• subalgebras, such as: so(n), also denoted o(n), the real nxn matrices that are skew-symmetric,
• sl_n(k) the nxn matrices that are traceless, i.e. the sum of the diagonal elements is 0.
• u(n) the complex nxn matrices that are skew-hermitean. Note that these form only a real Lie algebra, as multiplication with a general complex number does not preserve skew-hermiticity.
• su(n) the skew-hermitean matrices with trace 0.
• sp_n(k) the 2nx2n matrices that preserve the standard symplectic form ω – that is the 2nx2n matrix J that has 2x2 block structure and -Id in the lower left block, Id (the identity matrix) in the upper right block, and 0 in the rest. Infinitesimal symmetry of J means

:AToJ +JoA = 0
where A is an arbitrary 2nx2n matrix.

• an analog construction can be done with arbitrary linear geometric maps, e.g. in the case so(n) the structure is the Euclidian metric g with matrix δ_ab the Kronecker delta, i.e. 1 for a=b and 0 otherwise.

Lie algebra associated to a Lie group

To a Lie group G over the real or complex numbers we can associate a Lie algebra in the following way. The vector space is the tangent space at the identity of G. This vector space is also canonically isomorphic to the left-invariant vector fields on G. The commutator of two left-invariant vector fields is again left-invariant and moreover k-linear and skew-symmetric. This endows the vector space with a bracket.

Remarks about infinite dimensional Lie algebras

In the above definition we did not restrict to finite dimensional vector spaces even though this is usually implied when talking about Lie algebra. In infinite dimensional Lie algebras there is the obvious question of continuity of the Lie bracket and one requires thus at least a topological vector space and often demands continuity of the bracket (and operations of the vector space).

The easiest version of infinite dimensional Lie algebras are Lie algebroids, and a particular example of that is the tangent bundle of a smooth manifold.