Lie Algebra - Examples

Examples

• Any vector space V endowed with the identically zero Lie bracket becomes a Lie algebra. Such Lie algebras are called abelian, cf. below. Any one-dimensional Lie algebra over a field is abelian, by the antisymmetry of the Lie bracket.

• The three-dimensional Euclidean space R3 with the Lie bracket given by the cross product of vectors becomes a three-dimensional Lie algebra.

• The Heisenberg algebra H₃(R) is a three-dimensional Lie algebra with elements:

whose commutation relations are

It is explicitly exhibited as the space of 3×3 strictly upper-triangular matrices.

• The subspace of the general linear Lie algebra consisting of matrices of trace zero is a subalgebra, the special linear Lie algebra, denoted

• Any Lie group G defines an associated real Lie algebra . The definition in general is somewhat technical, but in the case of real matrix groups, it can be formulated via the exponential map, or the matrix exponent. The Lie algebra consists of those matrices X for which

for all real numbers t. The Lie bracket of is given by the commutator of matrices. As a concrete example, consider the special linear group SL(n,R), consisting of all n × n matrices with real entries and determinant 1. This is a matrix Lie group, and its Lie algebra consists of all n × n matrices with real entries and trace 0.

• The real vector space of all n × n skew-hermitian matrices is closed under the commutator and forms a real Lie algebra denoted . This is the Lie algebra of the unitary group U(n).

• An important class of infinite-dimensional real Lie algebras arises in differential topology. The space of smooth vector fields on a differentiable manifold M forms a Lie algebra, where the Lie bracket is defined to be the commutator of vector fields. One way of expressing the Lie bracket is through the formalism of Lie derivatives, which identifies a vector field X with a first order partial differential operator L_X acting on smooth functions by letting L_X(f) be the directional derivative of the function f in the direction of X. The Lie bracket of two vector fields is the vector field defined through its action on functions by the formula:

This Lie algebra is related to the pseudogroup of diffeomorphisms of M.

• The commutation relations between the x, y, and z components of the angular momentum operator in quantum mechanics form a representation of a complex three-dimensional Lie algebra, which is the complexification of the Lie algebra so(3) of the three-dimensional rotation group:

• A Kac–Moody algebra is an example of an infinite-dimensional Lie algebra.