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 (the identity matrix) in the lower left block, -Id in the upper right block, and 0 in the rest. An nxn matrix is an infinitesimal symmetry of J if

A^ToJ +JoA = 0.

• 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.

Classification

Lie algebras can be classified by the following properties:

Abelian, nilpotent, and solvable

Abelian means that the Lie bracket vanishes, i.e. [X,Y]=0. The only examples are the kⁿ.

Consider the definition of the lower central series of a Lie algebra g

g > [g,g] > [[g,g],g] > [[[g,g],g],g] > …

A Lie algebra is called nilpotent if the lower central series finally becomes 0. A series of length n means that arbitrary iterated commutators of length greater than n always vanish.

Next, consider the definition of the derived series of a Lie algebra g

D⁰g=g, Dⁿ⁺¹g=[Dⁿg,Dⁿg].

A Lie algebra is called solvable if its derived series finally becomes 0.

The complete classification of solvable Lie algebras is still an open problem and beyond the means of finitely many invariants. It is thus a problem that is not Turing computable.

Simple and Semisimple

The opposite of a solvable Lie algebra is a simple Lie algebra which means that it has no proper ideals. The classification of simple complex Lie algebras (see also the page about Lie algebra/root systems) states that these fall into four infinite series and five exceptional Lie algebras: A_n=sl_n+1(C), B_n=so(2n+1)⊗C, C_n=sp_n(C), D_n=so_2n⊗C, and the exeptions e₈, e₇, e₆, f₄, g₂.

A Lie algebra is semi-simple if every element can be written as the commutator of two elements, i.e. for the Lie algebra g, [g,g]=g. Simple Lie algebras are semi-simple. Moreover every semi-simple Lie algebra decomposes into the direct sum of its minimal ideals which are simple Lie algebras.

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 algebras. 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.