In the theory of Lie groups, the exponential map is a map from the Lie algebra g {\displaystyle {\mathfrak {g}}} of a Lie group G {\displaystyle G} to...

geometry) for a manifold with a Riemannian metric, exponential map (Lie theory) from a Lie algebra to a Lie group, More generally, in a manifold with an affine...

equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let X...

function's current value Exponential map (Riemannian geometry), in Riemannian geometry Exponential map (Lie theory), in Lie theory Exponential notation, also known...

integral Exponential integrator Exponential map (Lie theory) Exponential map (Riemannian geometry) Exponential map (discrete dynamical systems) Exponential notation...

theory of Lie groups, the exponential map is a map from the Lie algebra g of a Lie group G into G. In case G is a matrix Lie group, the exponential map...

transformation from the functor Lie to the identity functor on the category of Lie groups.) The exponential map from the Lie algebra to the Lie group is not always...

is a matrix Lie group, the expression e X {\displaystyle e^{X}} can be computed by the usual power series for the exponential. In any Lie group, there...

Cartan. The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence...

understood. For infinite-dimensional Lie algebras, Lie theory works less well. The exponential map need not be a local homeomorphism (for example, in...

In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold (or pseudo-Riemannian manifold) M to...

associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements...

group), then the Lie algebra g {\displaystyle {\mathfrak {g}}} consists of matrices and the exponential map is the matrix exponential exp ⁡ ( X ) = e X...

a Lie group with Lie algebra g {\displaystyle {\mathfrak {g}}} . Let exp : g → G {\displaystyle \exp :{\mathfrak {g}}\to G} be the exponential map. The...

mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices...

topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include the circle group T...

and complex Lie algebras. This difference can be traced to the properties of Frobenius automorphism and to the failure of the exponential map to establish...

In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension e is an enlargement of a given Lie algebra g by another...

metric Angle of parallelism Prime geodesic Geodesic flow Exponential map (Lie theory) Exponential map (Riemannian geometry) Injectivity radius Geodesic deviation...

to be commutative. There are different definitions used in group theory and ring theory. The commutator of two elements, g and h, of a group G, is the element...

connection Einstein manifold Euclidean geometry Exponential map Exponential map (Lie theory), Exponential map (Riemannian geometry) Finsler metric A generalization...

uses structure theory of semisimple Lie algebras. For classical Lie algebras there is a more explicit construction. Let g0 be a real Lie algebra of matrices...

In terms of Lie theory, the Rodrigues' formula provides an algorithm to compute the exponential map from the Lie algebra so(3) to its Lie group SO(3)...

subalgebras of their Lie algebras. The exponential map is onto in each case, since the polynomial function log ( eA eB ) lies in a given Lie subalgebra if A...

theory is specified by a choice of simple Lie group G known as the gauge group of the theory and also a number referred to as the level of the theory...

linearity. Since SO(3) is a matrix Lie group, its exponential map is defined using the standard matrix exponential series, { exp : s o ( 3 ) → SO ⁡ (...

a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. The list of simple Lie groups can...

symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold...

folds must lie on the creases. In the map folding problem, the paper is a map, divided by creases into rectangles, and the folds must again lie only along...

"complex Lie group" is defined as a complex analytic manifold that is also a group whose multiplication and inversion are each given by a holomorphic map. The...