Lie algebra, exponential

Lie algebra of type

A finite-dimensional real Lie algebra for any element of which the operator of adjoint representation (cf. Adjoint representation of a Lie group) does not have purely imaginary eigen values. The exponential mapping of the algebra into the corresponding simply-connected Lie group is a diffeomorphism, and is an exponential Lie group (cf. Lie group, exponential).

Every exponential Lie algebra is solvable (cf. Lie algebra, solvable). A nilpotent Lie algebra (cf. Lie algebra, nilpotent) over is an exponential Lie algebra. The class of exponential Lie algebras is intermediate between the classes of all solvable and all supersolvable Lie algebras (cf. Lie algebra, supersolvable); it is closed with respect to transition to subalgebras, quotient algebras and finite direct sums, but it is not closed with respect to extensions.

The simplest example of an exponential Lie algebra that is not a supersolvable Lie algebra is the three-dimensional Lie algebra with basis and multiplication specified by the formulas

where is a real matrix that has complex but not purely imaginary eigen values. The three-dimensional Lie algebra with basis and defining relations

is a solvable, but not an exponential Lie algebra.

A Lie algebra is exponential if and only if all roots of (cf. Root system) have the form , where and are real linear forms on and is proportional to (see ), or if has no quotient algebra containing a subalgebra isomorphic to .

For references see Lie group, exponential.